深入理解 Uniswap v3 智能合约 - Core（一）

adshao
发布于 2024-04-09 17:55
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此文详细介绍了 Uniswap v3 的智能合约，包括核心合约 UniswapV3Factory 和 UniswapV3Pool 的实现与功能，重点讲解了流动性管理、交易对创建、代币交换等操作的实现原理和代码示例，涵盖了预言机的使用及其扩展等技术细节，具有很强的实用性和参考价值。

## 深入理解 Uniswap v3 智能合约 （一）

### 概述

与Uniswap v2一样，Uniswap v3的合约也分为两类：

* [Uniswap-v3-core](#Uniswap-v3-core)
    - Uniswap v3的核心代码，实现了协议定义的所有功能，外部合约可直接与core合约交互
* [Uniswap-v3-periphery](https://learnblockchain.cn/article/13645)
    - 基于使用场景封装接口，如头寸（Position）管理、多路径代币交换等功能，Uniswap界面即与periphery合约交互

如果你希望以用户场景角度阅读本文，请直接从[Uniswap-v3-periphery](https://learnblockchain.cn/article/13645)开始，它包含了创建头寸、修改头寸流动性、交换代币等常用功能。

如果你希望从底层核心模块开始阅读，请从[Uniswap-v3-core](#Uniswap-v3-core)开始。

### Uniswap-v3-core

#### UniswapV3Factory.sol

工厂合约主要包含三个功能：

* [createPool](#createPool)：创建交易对池子
* [setOwner](#setOwner)：设置工厂合约Owner
* [enableFeeAmount](#enableFeeAmount)：添加手续费等级

##### createPool

创建一个Uniswap v3交易对池子，注意，由于Uniswap v3支持不同手续费等级，如0.05%、0.30%、1.00%等，因此一个交易对合约由`tokenA`、`tokenB`和`fee`（手续费）唯一确定。

> 计算交易对合约还需要：factory工厂合约地址、合约初始化代码的hash。

```solidity
/// @inheritdoc IUniswapV3Factory
function createPool(
    address tokenA,
    address tokenB,
    uint24 fee
) external override noDelegateCall returns (address pool) {
    require(tokenA != tokenB);
    (address token0, address token1) = tokenA < tokenB ? (tokenA, tokenB) : (tokenB, tokenA);
    require(token0 != address(0));
    int24 tickSpacing = feeAmountTickSpacing[fee];
    require(tickSpacing != 0);
    require(getPool[token0][token1][fee] == address(0));
    pool = deploy(address(this), token0, token1, fee, tickSpacing);
    getPool[token0][token1][fee] = pool;
    // populate mapping in the reverse direction, deliberate choice to avoid the cost of comparing addresses
    getPool[token1][token0][fee] = pool;
    emit PoolCreated(token0, token1, fee, tickSpacing, pool);
}
```

因为传入的`tokenA`和`tokenB`是无序的，首先对`tokenA`、`tokenB`排序，确保`tokenA < tokenB`。

通过手续费等级获取对应的`tickSpacing`：

```solidity
int24 tickSpacing = feeAmountTickSpacing[fee];
```

我们在《深入理解Uniswap v3白皮书》中介绍过`tickSpacing`的作用，每个手续费等级对应一个`tickSpacing`，只有被`tickSpacing`整除的`tick`才允许被初始化，`tickSpacing`越大，每个`tick`流动性越多，`tick`之间滑点越大，但会节省跨`tick`操作的gas。这里作为`Pool`的参数保存起来。

确认该交易对对应的手续费等级没有创建过：

```solidity
require(getPool[token0][token1][fee] == address(0));
```

创建（部署）交易对合约：

```solidity
pool = deploy(address(this), token0, token1, fee, tickSpacing);
```

deploy代码如下：

```solidity
/// @dev Deploys a pool with the given parameters by transiently setting the parameters storage slot and then
/// clearing it after deploying the pool.
/// @param factory The contract address of the Uniswap V3 factory
/// @param token0 The first token of the pool by address sort order
/// @param token1 The second token of the pool by address sort order
/// @param fee The fee collected upon every swap in the pool, denominated in hundredths of a bip
/// @param tickSpacing The spacing between usable ticks
function deploy(
    address factory,
    address token0,
    address token1,
    uint24 fee,
    int24 tickSpacing
) internal returns (address pool) {
    parameters = Parameters({factory: factory, token0: token0, token1: token1, fee: fee, tickSpacing: tickSpacing});
    pool = address(new UniswapV3Pool{salt: keccak256(abi.encode(token0, token1, fee))}());
    delete parameters;
}
```

我们在Uniswap v2中提到，为了确保交易对合约地址的可计算性和唯一性，Uniswap v2使用`CREATE2`操作码创建交易对合约；从Solidity 0.6.2版本开始（[Github PR](https://github.com/ethereum/solidity/pull/8177)），支持在`new`方法中传递`salt`参数实现`CREATE2`功能；`salt`参数确保了合约地址的唯一性和可计算性。从代码可知，Uniswap v3交易对合约使用token0、token1、fee唯一确定一个交易对合约，比如，根据ETH-USDC 0.05%手续费（以及工厂合约地址、初始化代码hash）等信息，可计算交易对合约地址。

最后，保存交易对合约地址到`getPool`变量中：

```solidity
getPool[token0][token1][fee] = pool;
// populate mapping in the reverse direction, deliberate choice to avoid the cost of comparing addresses
getPool[token1][token0][fee] = pool;
```

##### setOwner

设置工厂合约owner，owner具有以下权限：

* [setOwner](#setOwner)：修改owner
* [enableFeeAmount](#enableFeeAmount)：添加手续费等级
* [setFeeProtocol](#setFeeProtocol)：修改某个交易对的协议手续费比例
* [collectProtocol](#collectProtocol)：收集某个交易对的协议手续费

首先判断请求由当前owner发起，确认后修改owner：

```solidity
/// @inheritdoc IUniswapV3Factory
function setOwner(address _owner) external override {
    require(msg.sender == owner);
    emit OwnerChanged(owner, _owner);
    owner = _owner;
}
```

##### enableFeeAmount

Uniswap v3默认支持三种手续费等级：0.05%、0.30%和1.00%，对应的fee值分别为500、3000和10000；fee的基本单位是百分之一基点，即0.01 bp = $10^{-6}$。

手续费百分比计算公式为：

$$
f_{ratio} = \frac{fee}{1,000,000}
$$

```solidity
/// @inheritdoc IUniswapV3Factory
function enableFeeAmount(uint24 fee, int24 tickSpacing) public override {
    require(msg.sender == owner);
    require(fee < 1000000);
    // tick spacing is capped at 16384 to prevent the situation where tickSpacing is so large that
    // TickBitmap#nextInitializedTickWithinOneWord overflows int24 container from a valid tick
    // 16384 ticks represents a >5x price change with ticks of 1 bips
    require(tickSpacing > 0 && tickSpacing < 16384);
    require(feeAmountTickSpacing[fee] == 0);

feeAmountTickSpacing[fee] = tickSpacing;
    emit FeeAmountEnabled(fee, tickSpacing);
}
```

#### UniswapV3Pool.sol

这是Uniswap v3的主要代码，定义了交易对池子的功能：

* [initialize](#initialize)：初始化交易对
* [mint](#mint)：添加流动性
* [burn](#burn)：移除流动性
* [swap](#swap)：交换代币
* [flash](#flash)：闪电贷
* [collect](#collect)：取回代币
* [increaseObservationCardinalityNext](#increaseObservationCardinalityNext)：扩展预言机空间
* [observe](#observe)：获取预言机数据

此外，factory（工厂合约）owner还可以调用以下两个方法：

* [setFeeProtocol](#setFeeProtocol)：修改某个交易对的协议手续费比例
* [collectProtocol](#collectProtocol)：收集某个交易对的协议手续费

![](https://img.learnblockchain.cn/2025/04/04/uniswapV3Pool.png)

##### initialize

创建完交易对后，需要调用`initialize`方法初始化合约，才能正常使用交易对功能。

该方法初始化`slot0`变量：

```solidity
/// @inheritdoc IUniswapV3PoolActions
/// @dev not locked because it initializes unlocked
function initialize(uint160 sqrtPriceX96) external override {
    require(slot0.sqrtPriceX96 == 0, 'AI');

int24 tick = TickMath.getTickAtSqrtRatio(sqrtPriceX96);

(uint16 cardinality, uint16 cardinalityNext) = observations.initialize(_blockTimestamp());

slot0 = Slot0({
        sqrtPriceX96: sqrtPriceX96,
        tick: tick,
        observationIndex: 0,
        observationCardinality: cardinality,
        observationCardinalityNext: cardinalityNext,
        feeProtocol: 0,
        unlocked: true
    });

emit Initialize(sqrtPriceX96, tick);
}
```

`slot0`定义如下：

* `sqrtPriceX96`：交易对当前的开根号价格 $\sqrt{P}$
* `tick`：当前 $\sqrt{P}$ 对应的tick，使用[getTickAtSqrtRatio](#getTickAtSqrtRatio)计算得出
* `observationIndex`：最近更新的（预言机）观测点数组序号
* `observationCardinality`：（预言机）观测点数组容量，最大65536，初始时为1
* `observationCardinalityNext`：下一个（预言机）观测点数组容量，如果手动扩容容量，会更新这个值，初始时为1
* `feeProtocol`：协议手续费比例，可以分别为`token0`和`token1`设置交易手续费中分给协议的比例
* `unlocked`：当前交易对合约是否非锁定状态

##### mint

该方法实现添加流动性功能。实际上，首次添加流动性和后续增加流动性，都会使用该方法。

`mint`方法参数如下：

* `recipient`：头寸接收者（owner）
* `tickLower`：流动性区间低点
* `tickUpper`：流动性区间高点
* `amount`：流动性数量
* `data`：回调参数

```solidity
/// @inheritdoc IUniswapV3PoolActions
/// @dev noDelegateCall is applied indirectly via _modifyPosition
function mint(
    address recipient,
    int24 tickLower,
    int24 tickUpper,
    uint128 amount,
    bytes calldata data
) external override lock returns (uint256 amount0, uint256 amount1) {
    require(amount > 0);
    (, int256 amount0Int, int256 amount1Int) =
        _modifyPosition(
            ModifyPositionParams({
                owner: recipient,
                tickLower: tickLower,
                tickUpper: tickUpper,
                liquidityDelta: int256(amount).toInt128()
            })
        );

amount0 = uint256(amount0Int);
    amount1 = uint256(amount1Int);

uint256 balance0Before;
    uint256 balance1Before;
    if (amount0 > 0) balance0Before = balance0();
    if (amount1 > 0) balance1Before = balance1();
    IUniswapV3MintCallback(msg.sender).uniswapV3MintCallback(amount0, amount1, data);
    if (amount0 > 0) require(balance0Before.add(amount0) <= balance0(), 'M0');
    if (amount1 > 0) require(balance1Before.add(amount1) <= balance1(), 'M1');

emit Mint(msg.sender, recipient, tickLower, tickUpper, amount, amount0, amount1);
}
```

`mint`方法主要的逻辑都在[_modifyPosition](#_modifyPosition)中，其返回的`amount0Int`和`amount1Int`表示:如果添加`amount`数量的流动性，则需要分别向交易对合约转入的`token0`和`token1`的代币数量。

调用方需在`uniswapV3MintCallback`完成代币的转入操作；调用`mint`方法的合约需要实现`IUniswapV3MintCallback`接口，Uniswap v3在[periphery](https://hackmd.io/cTPg4x2TR4WthYEF8anLug)合约的`NonfungiblePositionManager.sol`实现该接口。

> 因为`mint`调用方需要实现接口方法，因此个人ETH账户（EOA）无法调用该方法。

```solidity
IUniswapV3MintCallback(msg.sender).uniswapV3MintCallback(amount0, amount1, data);
```

###### _modifyPosition

继续来看`_modifyPosition`：

```solidity
/// @dev Effect some changes to a position
/// @param params the position details and the change to the position's liquidity to effect
/// @return position a storage pointer referencing the position with the given owner and tick range
/// @return amount0 the amount of token0 owed to the pool, negative if the pool should pay the recipient
/// @return amount1 the amount of token1 owed to the pool, negative if the pool should pay the recipient
function _modifyPosition(ModifyPositionParams memory params)
    private
    noDelegateCall
    returns (
        Position.Info storage position,
        int256 amount0,
        int256 amount1
    )
{
    checkTicks(params.tickLower, params.tickUpper);

Slot0 memory _slot0 = slot0; // SLOAD for gas optimization

position = _updatePosition(
        params.owner,
        params.tickLower,
        params.tickUpper,
        params.liquidityDelta,
        _slot0.tick
    );
```

先通过[_updatePosition](#_updatePosition)更新头寸信息，我们在下一节会具体介绍。

```solidity
    if (params.liquidityDelta != 0) {
        if (_slot0.tick < params.tickLower) {
            // current tick is below the passed range; liquidity can only become in range by crossing from left to
            // right, when we'll need _more_ token0 (it's becoming more valuable) so user must provide it
            amount0 = SqrtPriceMath.getAmount0Delta(
                TickMath.getSqrtRatioAtTick(params.tickLower),
                TickMath.getSqrtRatioAtTick(params.tickUpper),
                params.liquidityDelta
            );
        } else if (_slot0.tick < params.tickUpper) {
            // current tick is inside the passed range
            uint128 liquidityBefore = liquidity; // SLOAD for gas optimization

// write an oracle entry
            (slot0.observationIndex, slot0.observationCardinality) = observations.write(
                _slot0.observationIndex,
                _blockTimestamp(),
                _slot0.tick,
                liquidityBefore,
                _slot0.observationCardinality,
                _slot0.observationCardinalityNext
            );

amount0 = SqrtPriceMath.getAmount0Delta(
                _slot0.sqrtPriceX96,
                TickMath.getSqrtRatioAtTick(params.tickUpper),
                params.liquidityDelta
            );
            amount1 = SqrtPriceMath.getAmount1Delta(
                TickMath.getSqrtRatioAtTick(params.tickLower),
                _slot0.sqrtPriceX96,
                params.liquidityDelta
            );

liquidity = LiquidityMath.addDelta(liquidityBefore, params.liquidityDelta);
        } else {
            // current tick is above the passed range; liquidity can only become in range by crossing from right to
            // left, when we'll need _more_ token1 (it's becoming more valuable) so user must provide it
            amount1 = SqrtPriceMath.getAmount1Delta(
                TickMath.getSqrtRatioAtTick(params.tickLower),
                TickMath.getSqrtRatioAtTick(params.tickUpper),
                params.liquidityDelta
            );
        }
    }
}
```

代码的下半部分则主要通过[getAmount0Delta](#getAmount0Delta)和[getAmount1Delta](#getAmount1Delta)计算该流动性需要分别提供的`token0`和`token1`的数量，即`amount0`和`amount1`。

具体地，当你提供流动性的区间大于当前`tick` $i_c$ 时，因为`tick`大小与 $\sqrt{P}$（即 $\sqrt{\frac{y}{x}}$ ）成正比，意味着在大于 $i_c$ 的区间， $x$ 的价值更高（需要更少的 $x$ ），因此添加流动性时需在该部分提供 $x$ 代币，即`amount0`数量的`token0`；反之，则提供 $y$ 代币，即`amount1`的`token1`。

如下所示：

$$
\begin{cases}i_c, ..., \overbrace{i_l, ..., i_u}^{amount0} & \text{$i_c < i_l$}\\
\overbrace{i_l, ...}^{amount1}, i_c, \overbrace{..., i_u}^{amount0} & \text{$i_l \leq i_c < i_u$}\\
\overbrace{i_l, ..., i_u}^{amount1}, ..., i_c & \text{$i_u \leq i_c$}\end{cases}
$$

其中， $i_l$ ,  $i_u$ 为提供流动性价格区间的边界， $i_c$ 为当前价格对应的`tick`。

如果当前价格在区间中，即 $i_l \leq i_c < i_u$ 时，`_modifyPosition`会记录一次（预言机）观测点数据，因为此时区间的流动性发生了变化，需要记录每流动性的持续时间`secondsPerLiquidityCumulativeX128`：

```solidity
// write an oracle entry
(slot0.observationIndex, slot0.observationCardinality) = observations.write(
    _slot0.observationIndex,
    _blockTimestamp(),
    _slot0.tick,
    liquidityBefore,
    _slot0.observationCardinality,
    _slot0.observationCardinalityNext
);
```

在计算`amount0`和`amount1`后，更新当前交易对的全局活跃流动性`liquidity`：

```solidity
liquidity = LiquidityMath.addDelta(liquidityBefore, params.liquidityDelta);
```

这个全局流动性会在[swap](#swap)时用到。

###### _updatePosition

`_modifyPosition`中的`_updatePosition`代码如下：

```solidity
/// @dev Gets and updates a position with the given liquidity delta
/// @param owner the owner of the position
/// @param tickLower the lower tick of the position's tick range
/// @param tickUpper the upper tick of the position's tick range
/// @param tick the current tick, passed to avoid sloads
function _updatePosition(
    address owner,
    int24 tickLower,
    int24 tickUpper,
    int128 liquidityDelta,
    int24 tick
) private returns (Position.Info storage position) {
    position = positions.get(owner, tickLower, tickUpper);

uint256 _feeGrowthGlobal0X128 = feeGrowthGlobal0X128; // SLOAD for gas optimization
    uint256 _feeGrowthGlobal1X128 = feeGrowthGlobal1X128; // SLOAD for gas optimization

// if we need to update the ticks, do it
    bool flippedLower;
    bool flippedUpper;
    if (liquidityDelta != 0) {
        uint32 time = _blockTimestamp();
        (int56 tickCumulative, uint160 secondsPerLiquidityCumulativeX128) =
            observations.observeSingle(
                time,
                0,
                slot0.tick,
                slot0.observationIndex,
                liquidity,
                slot0.observationCardinality
            );
```

`observations.observeSingle`计算从最后一次观测点到现在的累积tick`tickCumulative`和累积每份流动性的持续时间`secondsPerLiquidityCumulativeX128`。

```solidity
        flippedLower = ticks.update(
            tickLower,
            tick,
            liquidityDelta,
            _feeGrowthGlobal0X128,
            _feeGrowthGlobal1X128,
            secondsPerLiquidityCumulativeX128,
            tickCumulative,
            time,
            false,
            maxLiquidityPerTick
        );
        flippedUpper = ticks.update(
            tickUpper,
            tick,
            liquidityDelta,
            _feeGrowthGlobal0X128,
            _feeGrowthGlobal1X128,
            secondsPerLiquidityCumulativeX128,
            tickCumulative,
            time,
            true,
            maxLiquidityPerTick
        );
        if (flippedLower) {
            tickBitmap.flipTick(tickLower, tickSpacing);
        }
        if (flippedUpper) {
            tickBitmap.flipTick(tickUpper, tickSpacing);
        }
    }
```

接着使用`ticks.update`分别更新`tickLower`（价格区间低点）和`tickUpper`（价格区间高点）的状态，具体请参考[Tick.update](#update)。

如果对应`tick`的流动性从从0到有，或从有到0，则表示该`tick`需要被翻转。如果该`tick`未被标记为初始化，则标记为初始化；否则，将其取消初始化；这里用到`tickBitmap.flipTick`方法，请参考[TickBitmap.flipTick](#flipTick)。

```solidity
    (uint256 feeGrowthInside0X128, uint256 feeGrowthInside1X128) =
        ticks.getFeeGrowthInside(tickLower, tickUpper, tick, _feeGrowthGlobal0X128, _feeGrowthGlobal1X128);
```

接着，计算该价格区间的累积每流动性手续费。

```solidity
    position.update(liquidityDelta, feeGrowthInside0X128, feeGrowthInside1X128);
```

更新头寸（Position）信息，这里主要更新了头寸的应收手续费`tokensOwed0`和`tokensOwed1`，以及头寸流动性`liquidity`，请参考[Position.update](#update1)。

```solidity
    // clear any tick data that is no longer needed
    if (liquidityDelta < 0) {
        if (flippedLower) {
            ticks.clear(tickLower);
        }
        if (flippedUpper) {
            ticks.clear(tickUpper);
        }
    }
}
```

如果是移除流动性，并且`tick`被翻转，则调用[clear](#Tick.clear)清空`tick`状态。

最后，回到`mint`方法，调用者需要确保在`uniswapV3MintCallback`方法中，将这里计算出的`amount0`和`amount1`数量的`token0`和`token1`代币转入交易对合约。

总结`mint`方法的主要工作如下：

1. 更新价格区间端点（lower, upper）的信息：`ticks.update`
2. 如果Tick状态翻转，则更新位图的标识位，设置为“已初始化”或“未初始化”：`tickBitmap.flipTick`
3. 更新头寸（Position）信息：`positions.update`
4. 如果当前tick位于价格区间中，则：
    - 写入一次预言机观测点：`observations.write`
    - 更新全局活跃流动性：`liquidity`
5. 调用方向交易对合约转账：`uniswapV3MintCallback`

##### burn

销毁流动性（`burn`）的逻辑与添加流动性（`mint`）几乎完全一样，唯一的区别是`liquidityDelta`是负的。

```solidity
/// @inheritdoc IUniswapV3PoolActions
/// @dev noDelegateCall is applied indirectly via _modifyPosition
function burn(
    int24 tickLower,
    int24 tickUpper,
    uint128 amount
) external override lock returns (uint256 amount0, uint256 amount1) {
    (Position.Info storage position, int256 amount0Int, int256 amount1Int) =
        _modifyPosition(
            ModifyPositionParams({
                owner: msg.sender,
                tickLower: tickLower,
                tickUpper: tickUpper,
                liquidityDelta: -int256(amount).toInt128()
            })
        );

amount0 = uint256(-amount0Int);
    amount1 = uint256(-amount1Int);

if (amount0 > 0 || amount1 > 0) {
        (position.tokensOwed0, position.tokensOwed1) = (
            position.tokensOwed0 + uint128(amount0),
            position.tokensOwed1 + uint128(amount1)
        );
    }

emit Burn(msg.sender, tickLower, tickUpper, amount, amount0, amount1);
}
```

这里与`mint`使用同一个[_modifyPosition](#_modifyPosition)方法。

需注意，当销毁（部分）流动性后，代币并没有转回到调用方，而是以未领取代币的形式记在头寸（Position）上。

##### swap

`swap`方法是Uniswap v3代码的核心，该方法实现两个代币的交换，从`token0`交换到`token1`，或者相反。

相比Uniswap v2的同质化流动性，我们重点关注在`swap`过程中，价格如何变化，以及如何影响流动性的。

先来看`swap`方法的几个参数：

* `recipient`：交易后的代币接收者
* `zeroForOne`：如果从`token0`交换`token1`则为`true`，从`token1`交换`token0`则为`false`
* `amountSpecified`: 指定的代币数量，如果为正，表示希望输入的代币数量；如果为负，则表示希望输出的代币数量
* `sqrtPriceLimitX96`：能够承受的价格上限（或下限），格式为`Q64.96`；如果从`token0`到`token1`，则表示`swap`过程中的价格下限；如果从`token1`到`token0`，则表示价格上限；如果价格超过该值，则`swap`失败
* `data`：回调参数

```solidity
/// @inheritdoc IUniswapV3PoolActions
function swap(
    address recipient,
    bool zeroForOne,
    int256 amountSpecified,
    uint160 sqrtPriceLimitX96,
    bytes calldata data
) external override noDelegateCall returns (int256 amount0, int256 amount1) {
    require(amountSpecified != 0, 'AS');

Slot0 memory slot0Start = slot0;

require(slot0Start.unlocked, 'LOK');
    require(
        zeroForOne
            ? sqrtPriceLimitX96 < slot0Start.sqrtPriceX96 && sqrtPriceLimitX96 > TickMath.MIN_SQRT_RATIO
            : sqrtPriceLimitX96 > slot0Start.sqrtPriceX96 && sqrtPriceLimitX96 < TickMath.MAX_SQRT_RATIO,
        'SPL'
    );

slot0.unlocked = false;

SwapCache memory cache =
        SwapCache({
            liquidityStart: liquidity,
            blockTimestamp: _blockTimestamp(),
            feeProtocol: zeroForOne ? (slot0Start.feeProtocol % 16) : (slot0Start.feeProtocol >> 4),
            secondsPerLiquidityCumulativeX128: 0,
            tickCumulative: 0,
            computedLatestObservation: false
        });

bool exactInput = amountSpecified > 0;

SwapState memory state =
        SwapState({
            amountSpecifiedRemaining: amountSpecified,
            amountCalculated: 0,
            sqrtPriceX96: slot0Start.sqrtPriceX96,
            tick: slot0Start.tick,
            feeGrowthGlobalX128: zeroForOne ? feeGrowthGlobal0X128 : feeGrowthGlobal1X128,
            protocolFee: 0,
            liquidity: cache.liquidityStart
        });
```

上面代码主要是初始化状态相关的。

因为 $\sqrt{P} = \sqrt{\frac{y}{x}}$ ，当`zeroForOne = true`，即从`token0`到`token1`时，swap过程中 Pool 的 $x$ 变多， $y$ 变少，因此 $\sqrt{P}$ 逐渐减小，所以指定的价格极限`sqrtPriceLimitX96`需要小于当前市场价格`sqrtPriceX96`。

另外，需要注意几个关键数据：

* 初始交易价格`state.sqrtPriceX96`为：`slot0.sqrtPriceX96`
    - 这里并没有使用`slot0.tick`计算初始价格，因为计算出来的值与`slot0.sqrtPriceX96`可能不一致，我们在后面代码会看到，`slot0.tick`不能作为当前价格
* 初始可用流动性`state.liquidity`为：`liquidity`，也就是我们在[mint](#mint)或[burn](#burn)时更新的全局可用流动性

根据`zeroForOne`和`exactInput`，可以有四种`swap`组合：

|zeroForOne|exactInput|swap|
|---|---|---|
|true|true|输入固定数量`token0`，输出最大数量`token1`|
|true|false|输入最小数量`token0`，输出固定数量`token1`|
|false|true|输入固定数量`token1`，输出最大数量`token0`|
|false|false|输入最小数量`token1`，输出固定数量`token0`|

一个完整的`swap`可以由多个`step`组成，代码如下：

```solidity
// continue swapping as long as we haven't used the entire input/output and haven't reached the price limit
while (state.amountSpecifiedRemaining != 0 && state.sqrtPriceX96 != sqrtPriceLimitX96) {
    StepComputations memory step;

step.sqrtPriceStartX96 = state.sqrtPriceX96;

(step.tickNext, step.initialized) = tickBitmap.nextInitializedTickWithinOneWord(
        state.tick,
        tickSpacing,
        zeroForOne
    );

// ensure that we do not overshoot the min/max tick, as the tick bitmap is not aware of these bounds
    if (step.tickNext < TickMath.MIN_TICK) {
        step.tickNext = TickMath.MIN_TICK;
    } else if (step.tickNext > TickMath.MAX_TICK) {
        step.tickNext = TickMath.MAX_TICK;
    }

// get the price for the next tick
    step.sqrtPriceNextX96 = TickMath.getSqrtRatioAtTick(step.tickNext);

// compute values to swap to the target tick, price limit, or point where input/output amount is exhausted
    (state.sqrtPriceX96, step.amountIn, step.amountOut, step.feeAmount) = SwapMath.computeSwapStep(
        state.sqrtPriceX96,
        (zeroForOne ? step.sqrtPriceNextX96 < sqrtPriceLimitX96 : step.sqrtPriceNextX96 > sqrtPriceLimitX96)
            ? sqrtPriceLimitX96
            : step.sqrtPriceNextX96,
        state.liquidity,
        state.amountSpecifiedRemaining,
        fee
    );

if (exactInput) {
        state.amountSpecifiedRemaining -= (step.amountIn + step.feeAmount).toInt256();
        state.amountCalculated = state.amountCalculated.sub(step.amountOut.toInt256());
    } else {
        state.amountSpecifiedRemaining += step.amountOut.toInt256();
        state.amountCalculated = state.amountCalculated.add((step.amountIn + step.feeAmount).toInt256());
    }

// if the protocol fee is on, calculate how much is owed, decrement feeAmount, and increment protocolFee
    if (cache.feeProtocol > 0) {
        uint256 delta = step.feeAmount / cache.feeProtocol;
        step.feeAmount -= delta;
        state.protocolFee += uint128(delta);
    }

// update global fee tracker
    if (state.liquidity > 0)
        state.feeGrowthGlobalX128 += FullMath.mulDiv(step.feeAmount, FixedPoint128.Q128, state.liquidity);

// shift tick if we reached the next price
    if (state.sqrtPriceX96 == step.sqrtPriceNextX96) {
        // if the tick is initialized, run the tick transition
        if (step.initialized) {
            // check for the placeholder value, which we replace with the actual value the first time the swap
            // crosses an initialized tick
            if (!cache.computedLatestObservation) {
                (cache.tickCumulative, cache.secondsPerLiquidityCumulativeX128) = observations.observeSingle(
                    cache.blockTimestamp,
                    0,
                    slot0Start.tick,
                    slot0Start.observationIndex,
                    cache.liquidityStart,
                    slot0Start.observationCardinality
                );
                cache.computedLatestObservation = true;
            }
            int128 liquidityNet =
                ticks.cross(
                    step.tickNext,
                    (zeroForOne ? state.feeGrowthGlobalX128 : feeGrowthGlobal0X128),
                    (zeroForOne ? feeGrowthGlobal1X128 : state.feeGrowthGlobalX128),
                    cache.secondsPerLiquidityCumulativeX128,
                    cache.tickCumulative,
                    cache.blockTimestamp
                );
            // if we're moving leftward, we interpret liquidityNet as the opposite sign
            // safe because liquidityNet cannot be type(int128).min
            if (zeroForOne) liquidityNet = -liquidityNet;

state.liquidity = LiquidityMath.addDelta(state.liquidity, liquidityNet);
        }

state.tick = zeroForOne ? step.tickNext - 1 : step.tickNext;
    } else if (state.sqrtPriceX96 != step.sqrtPriceStartX96) {
        // recompute unless we're on a lower tick boundary (i.e. already transitioned ticks), and haven't moved
        state.tick = TickMath.getTickAtSqrtRatio(state.sqrtPriceX96);
    }
}
```

整理成伪代码（pseudo code）如下：

```python
loop if 剩余代币 != 0 and 当前价格 != 最小（或最大）价格:
    // step
    初始价格 := 上一个step的价格
    下一个tick := 根据当前tick，寻找最近的已初始化的tick，或者本组最后一个未初始化的tick
    目标价格 := 根据下一个tick计算的价格
    交换后的价格, 消耗的输入代币数量, 得到的输出代币数量, 交易手续费 := 完成一步交换(初始价格, 目标价格, 可用流动性, 剩余代币)

更新 剩余代币
    更新 协议手续费

if 交换后价格 == 目标价格:
        if tick已初始化：
            价格穿越该tick，更新tick相关字段
            更新 可用流动性

当前tick := 下一个tick - 1
    else if 交换后价格 != 初始价格:
        当前tick := 根据交换后价格计算tick
```

首先需要根据当前`tick`寻找下一个`tick`，即`tickNext`，具体逻辑可参考：[tickBitmap.nextInitializedTickWithinOneWord](#nextInitializedTickWithinOneWord)。

计算当前step的目标价格：

```solidity
// get the price for the next tick
step.sqrtPriceNextX96 = TickMath.getSqrtRatioAtTick(step.tickNext);
```

计算本次（step）交换的输入输出（即执行一次交换）：

```solidity
// compute values to swap to the target tick, price limit, or point where input/output amount is exhausted
(state.sqrtPriceX96, step.amountIn, step.amountOut, step.feeAmount) = SwapMath.computeSwapStep(
    state.sqrtPriceX96,
    (zeroForOne ? step.sqrtPriceNextX96 < sqrtPriceLimitX96 : step.sqrtPriceNextX96 > sqrtPriceLimitX96)
        ? sqrtPriceLimitX96
        : step.sqrtPriceNextX96,
    state.liquidity,
    state.amountSpecifiedRemaining,
    fee
);
```

`SwapMath.computeSwapStep`将根据当前价格、目标价格、可用流动性、可用输入代币等数据，计算本次交换能最多成交的输入代币数量（`amountIn`），输出代币数量（`amountOut`），手续费（`feeAmount`）和成交后价格（`sqrtRatioNextX96`）。请参考[computeSwapStep](#computeSwapStep)。

保存本次交易的`amountIn`和`amountOut`：

```solidity
if (exactInput) {
    state.amountSpecifiedRemaining -= (step.amountIn + step.feeAmount).toInt256();
    state.amountCalculated = state.amountCalculated.sub(step.amountOut.toInt256());
} else {
    state.amountSpecifiedRemaining += step.amountOut.toInt256();
    state.amountCalculated = state.amountCalculated.add((step.amountIn + step.feeAmount).toInt256());
}
```

* 如果是指定输入代币数量（`token0`或`token1`）
    - `amountSpecifiedRemaining`表示（扣除手续费后）剩余可用输入代币数量
    - `amountCalculated`表示已输出代币数量（注意，这里是负值）
* 如果是指定输出代币数量（`token0`或`token1`）
    - `amountSpecifiedRemaining`表示剩余需要输出的代币数量（初始为负值，因此每次交换后需要`+= step.amountOut`），直到为0
    - `amountCalculated`表示（加入手续费后）已使用的输入代币数量

计算协议手续费：

```solidity
// if the protocol fee is on, calculate how much is owed, decrement feeAmount, and increment protocolFee
if (cache.feeProtocol > 0) {
    uint256 delta = step.feeAmount / cache.feeProtocol;
    step.feeAmount -= delta;
    state.protocolFee += uint128(delta);
}
```

如果开启了协议手续费，则从交易手续费中拆出协议手续费。注意，协议手续费的值`feeProtocol`表示交易手续费的 $\frac{1}{n}$。

```solidity
// update global fee tracker
if (state.liquidity > 0)
    state.feeGrowthGlobalX128 += FullMath.mulDiv(step.feeAmount, FixedPoint128.Q128, state.liquidity);
```

计算（每流动性）全局累积手续费。

```solidity
// shift tick if we reached the next price
if (state.sqrtPriceX96 == step.sqrtPriceNextX96) {
    // if the tick is initialized, run the tick transition
    if (step.initialized) {
        // check for the placeholder value, which we replace with the actual value the first time the swap
        // crosses an initialized tick
        if (!cache.computedLatestObservation) {
            (cache.tickCumulative, cache.secondsPerLiquidityCumulativeX128) = observations.observeSingle(
                cache.blockTimestamp,
                0,
                slot0Start.tick,
                slot0Start.observationIndex,
                cache.liquidityStart,
                slot0Start.observationCardinality
            );
            cache.computedLatestObservation = true;
        }
        int128 liquidityNet =
            ticks.cross(
                step.tickNext,
                (zeroForOne ? state.feeGrowthGlobalX128 : feeGrowthGlobal0X128),
                (zeroForOne ? feeGrowthGlobal1X128 : state.feeGrowthGlobalX128),
                cache.secondsPerLiquidityCumulativeX128,
                cache.tickCumulative,
                cache.blockTimestamp
            );
        // if we're moving leftward, we interpret liquidityNet as the opposite sign
        // safe because liquidityNet cannot be type(int128).min
        if (zeroForOne) liquidityNet = -liquidityNet;

state.liquidity = LiquidityMath.addDelta(state.liquidity, liquidityNet);
    }

state.tick = zeroForOne ? step.tickNext - 1 : step.tickNext;
} else if (state.sqrtPriceX96 != step.sqrtPriceStartX96) {
    // recompute unless we're on a lower tick boundary (i.e. already transitioned ticks), and haven't moved
    state.tick = TickMath.getTickAtSqrtRatio(state.sqrtPriceX96);
}
```

* 如果本次交换后的价格达到目标价格（即根据寻找的下一个`tick`计算的价格）：
    * 如果该`tick`已经初始化，则：
        - 通过`ticks.cross`方法穿越该`tick`，反向设置相关`Outside`变量的数据
        - 使用`tick`净流动性`liquidityNet`更新可用流动性`state.liquidity`
            - 由于在[mint](#mint)初始化`tick`时，`tickLower`的`liquidityNet`是正的，即`liquidityDelta`；`tickUpper`的`liquidityNet`是负的，即`-liquidityDelta`；因此这里需要根据`zeroForOne`的值来调整`liquidityNet`的正负
            - 当`zeroForOne = true`时，随着交易的进行， Pool 中 $x$ 变多， $y$ 变少，价格 $\sqrt{P}$ 逐渐变小，`tick`朝 lower 方向移动，如果穿越了`tickLower`，意味着离开区间，因此需要减少流动性；反之，如果穿越了`tickUpper`，意味着进入区间，因此需要增加流动性；即都使用 `-liquidityNet`
            - 当`zeroForOne = false`时，随着交易的进行， Pool 中 $y$ 变多， $x$ 变少，价格 $\sqrt{P}$ 逐渐变大，`tick`朝 upper 方向移动，如果穿越了`tickUpper`，意味着离开区间，因此需要减少流动性；反之，如果穿越了`tickLower`，意味着进入区间，因此需要增加流动性；即都使用 `liquidityNet`
    * 移动当前`tick`到下一个`tick`
* 如果交换后的价格没有达到本次目标价格，但是又不等于初始价格，即表示此时交易结束：
    * 使用交换后的价格计算最新的`tick`值

重复上述step，直到交换完全结束。

完成交换后，更新全局状态：

```solidity
// update tick and write an oracle entry if the tick change
if (state.tick != slot0Start.tick) {
    (uint16 observationIndex, uint16 observationCardinality) =
        observations.write(
            slot0Start.observationIndex,
            cache.blockTimestamp,
            slot0Start.tick,
            cache.liquidityStart,
            slot0Start.observationCardinality,
            slot0Start.observationCardinalityNext
        );
    (slot0.sqrtPriceX96, slot0.tick, slot0.observationIndex, slot0.observationCardinality) = (
        state.sqrtPriceX96,
        state.tick,
        observationIndex,
        observationCardinality
    );
} else {
    // otherwise just update the price
    slot0.sqrtPriceX96 = state.sqrtPriceX96;
}
```

* 如果交换后的`tick`与交换前的`tick`不同：
    * 记录一次（预言机）观测点数据，因为`tickCumulative`发生了改变
    * 更新`slot0.sqrtPriceX96`, `slot0.tick`等值，注意此时`sqrtPriceX96`与`tick`并不一定对应，`sqrtPriceX96`才能准确反映当前价格
* 如果交换前后`tick`值相同，则只需要修改价格：
    * 更新`slot0.sqrtPriceX96`

同样，如果全局流动性发生改变，则更新`liquidity`：

```solidity
// update liquidity if it changed
if (cache.liquidityStart != state.liquidity) liquidity = state.liquidity;
```

更新累积手续费和协议手续费：

```solidity
// update fee growth global and, if necessary, protocol fees
// overflow is acceptable, protocol has to withdraw before it hits type(uint128).max fees
if (zeroForOne) {
    feeGrowthGlobal0X128 = state.feeGrowthGlobalX128;
    if (state.protocolFee > 0) protocolFees.token0 += state.protocolFee;
} else {
    feeGrowthGlobal1X128 = state.feeGrowthGlobalX128;
    if (state.protocolFee > 0) protocolFees.token1 += state.protocolFee;
}
```

注意，如果是从`token0`交换`token1`，则只能收取`token0`作为手续费；反之，只能收取`token1`作为手续费。

```solidity
(amount0, amount1) = zeroForOne == exactInput
    ? (amountSpecified - state.amountSpecifiedRemaining, state.amountCalculated)
    : (state.amountCalculated, amountSpecified - state.amountSpecifiedRemaining);
```

计算本次交换需要的具体`amount0`和`amount1`。

```solidity
// do the transfers and collect payment
if (zeroForOne) {
    if (amount1 < 0) TransferHelper.safeTransfer(token1, recipient, uint256(-amount1));

uint256 balance0Before = balance0();
    IUniswapV3SwapCallback(msg.sender).uniswapV3SwapCallback(amount0, amount1, data);
    require(balance0Before.add(uint256(amount0)) <= balance0(), 'IIA');
} else {
    if (amount0 < 0) TransferHelper.safeTransfer(token0, recipient, uint256(-amount0));

uint256 balance1Before = balance1();
    IUniswapV3SwapCallback(msg.sender).uniswapV3SwapCallback(amount0, amount1, data);
    require(balance1Before.add(uint256(amount1)) <= balance1(), 'IIA');
}

emit Swap(msg.sender, recipient, amount0, amount1, state.sqrtPriceX96, state.liquidity, state.tick);
slot0.unlocked = true;
```

合约将输出代币转账给`recipient`，同时，调用方需要在`uniswapV3SwapCallback`方法将输入代币转给交易对合约：

至此，整个`swap`流程就结束了。

##### flash

本方法实现Uniswap v3闪电贷功能。

方法参数：

* `recipient`：闪电贷接收者
* `amount0`：借出`token0`的数量
* `amount1`：借出`token1`的数量
* `data`：回调方法参数

```solidity
/// @inheritdoc IUniswapV3PoolActions
function flash(
    address recipient,
    uint256 amount0,
    uint256 amount1,
    bytes calldata data
) external override lock noDelegateCall {
    uint128 _liquidity = liquidity;
    require(_liquidity > 0, 'L');

uint256 fee0 = FullMath.mulDivRoundingUp(amount0, fee, 1e6);
    uint256 fee1 = FullMath.mulDivRoundingUp(amount1, fee, 1e6);
```

闪电贷手续费与`swap`手续费相同，都是 $\frac{fee}{10^6}$。

```solidity
    uint256 balance0Before = balance0();
    uint256 balance1Before = balance1();

if (amount0 > 0) TransferHelper.safeTransfer(token0, recipient, amount0);
    if (amount1 > 0) TransferHelper.safeTransfer(token1, recipient, amount1);

IUniswapV3FlashCallback(msg.sender).uniswapV3FlashCallback(fee0, fee1, data);

uint256 balance0After = balance0();
    uint256 balance1After = balance1();

require(balance0Before.add(fee0) <= balance0After, 'F0');
    require(balance1Before.add(fee1) <= balance1After, 'F1');
```

向收款人转入贷出的代币数量，`flash`方法的调用方需实现`IUniswapV3FlashCallback.uniswapV3FlashCallback`接口方法，并在该方法中归还代币，包含手续费。

```solidity
    // sub is safe because we know balanceAfter is gt balanceBefore by at least fee
    uint256 paid0 = balance0After - balance0Before;
    uint256 paid1 = balance1After - balance1Before;

if (paid0 > 0) {
        uint8 feeProtocol0 = slot0.feeProtocol % 16;
        uint256 fees0 = feeProtocol0 == 0 ? 0 : paid0 / feeProtocol0;
        if (uint128(fees0) > 0) protocolFees.token0 += uint128(fees0);
        feeGrowthGlobal0X128 += FullMath.mulDiv(paid0 - fees0, FixedPoint128.Q128, _liquidity);
    }
    if (paid1 > 0) {
        uint8 feeProtocol1 = slot0.feeProtocol >> 4;
        uint256 fees1 = feeProtocol1 == 0 ? 0 : paid1 / feeProtocol1;
        if (uint128(fees1) > 0) protocolFees.token1 += uint128(fees1);
        feeGrowthGlobal1X128 += FullMath.mulDiv(paid1 - fees1, FixedPoint128.Q128, _liquidity);
    }

emit Flash(msg.sender, recipient, amount0, amount1, paid0, paid1);
}
```

根据收取的手续费，计算协议手续费（注意，`token0`和`token1`的协议手续费是单独设置的，参考：[setFeeProtocol](#setFeeProtocol)），最后更新`protocolFees`和`feeGrowthGlobal1X128`。

##### collect

该方法实现取回代币功能，包括销毁流动性记录的代币和手续费代币。

参数如下：

* `recipient`：代币接收者
* `tickLower`：头寸低点
* `tickUpper`：头寸高点
* `amount0Requested`：请求取回的`token0`数量
* `amount1Requested`：请求取回的`token1`数量

```solidity
/// @inheritdoc IUniswapV3PoolActions
function collect(
    address recipient,
    int24 tickLower,
    int24 tickUpper,
    uint128 amount0Requested,
    uint128 amount1Requested
) external override lock returns (uint128 amount0, uint128 amount1) {
    // we don't need to checkTicks here, because invalid positions will never have non-zero tokensOwed{0,1}
    Position.Info storage position = positions.get(msg.sender, tickLower, tickUpper);

amount0 = amount0Requested > position.tokensOwed0 ? position.tokensOwed0 : amount0Requested;
    amount1 = amount1Requested > position.tokensOwed1 ? position.tokensOwed1 : amount1Requested;

if (amount0 > 0) {
        position.tokensOwed0 -= amount0;
        TransferHelper.safeTransfer(token0, recipient, amount0);
    }
    if (amount1 > 0) {
        position.tokensOwed1 -= amount1;
        TransferHelper.safeTransfer(token1, recipient, amount1);
    }

emit Collect(msg.sender, recipient, tickLower, tickUpper, amount0, amount1);
}
```

上述代码比较简单，这里不再展开。需要注意，如果希望取回所有代币，则需要指定比`tokensOwned`更大的数，比如可以使用`type(uint128).max`。

##### increaseObservationCardinalityNext

扩容预言机观测点的可写入空间，该方法调用`Oracle.sol`的[grow](#grow)方法实现扩容。

```solidity
/// @inheritdoc IUniswapV3PoolActions
function increaseObservationCardinalityNext(uint16 observationCardinalityNext)
    external
    override
    lock
    noDelegateCall
{
    uint16 observationCardinalityNextOld = slot0.observationCardinalityNext; // for the event
    uint16 observationCardinalityNextNew =
        observations.grow(observationCardinalityNextOld, observationCardinalityNext);
    slot0.observationCardinalityNext = observationCardinalityNextNew;
    if (observationCardinalityNextOld != observationCardinalityNextNew)
        emit IncreaseObservationCardinalityNext(observationCardinalityNextOld, observationCardinalityNextNew);
}
```

##### observe

批量获取指定时间的观测点数据，该方法调用`Oracle.sol`的[observe](#observe2)实现。

```solidity
/// @inheritdoc IUniswapV3PoolDerivedState
function observe(uint32[] calldata secondsAgos)
    external
    view
    override
    noDelegateCall
    returns (int56[] memory tickCumulatives, uint160[] memory secondsPerLiquidityCumulativeX128s)
{
    return
        observations.observe(
            _blockTimestamp(),
            secondsAgos,
            slot0.tick,
            slot0.observationIndex,
            liquidity,
            slot0.observationCardinality
        );
}
```

##### setFeeProtocol

设置协议手续费的比例，该方法仅允许工厂合约的owner执行。

注意，需要分别设置`token0`和`token1`的协议手续费比例，该比例是交易手续费的占比，合法值为0（不开启协议手续费）或者 $4 \leq n \leq 10$，也就是可以设置协议手续费为交易手续费的 $\frac{1}{n}$。

```solidity
/// @inheritdoc IUniswapV3PoolOwnerActions
function setFeeProtocol(uint8 feeProtocol0, uint8 feeProtocol1) external override lock onlyFactoryOwner {
    require(
        (feeProtocol0 == 0 || (feeProtocol0 >= 4 && feeProtocol0 <= 10)) &&
            (feeProtocol1 == 0 || (feeProtocol1 >= 4 && feeProtocol1 <= 10))
    );
    uint8 feeProtocolOld = slot0.feeProtocol;
    slot0.feeProtocol = feeProtocol0 + (feeProtocol1 << 4);
    emit SetFeeProtocol(feeProtocolOld % 16, feeProtocolOld >> 4, feeProtocol0, feeProtocol1);
}
```

`slot0.feeProtocol`类型为`uint8`，保存两种代币的协议手续费比例，高4位为`token1`，低4位为`token0`：

$$
slot0.feeProtocol = \overbrace{0000}^{fee1}\overbrace{0000}^{fee0}
$$

因此，`feeProtocolOld % 16`表示`token0`的协议手续费比例`fee0`，`feeProtocolOld >> 4`表示`token1`的协议手续费比例`fee1`。

##### collectProtocol

取回协议手续费，该方法仅允许工厂合约的owner执行。

协议手续费有两个来源：

* `swap`产生的交易手续费
* `flash`产生的闪电贷手续费

```solidity
/// @inheritdoc IUniswapV3PoolOwnerActions
function collectProtocol(
    address recipient,
    uint128 amount0Requested,
    uint128 amount1Requested
) external override lock onlyFactoryOwner returns (uint128 amount0, uint128 amount1) {
    amount0 = amount0Requested > protocolFees.token0 ? protocolFees.token0 : amount0Requested;
    amount1 = amount1Requested > protocolFees.token1 ? protocolFees.token1 : amount1Requested;

if (amount0 > 0) {
        if (amount0 == protocolFees.token0) amount0--; // ensure that the slot is not cleared, for gas savings
        protocolFees.token0 -= amount0;
        TransferHelper.safeTransfer(token0, recipient, amount0);
    }
    if (amount1 > 0) {
        if (amount1 == protocolFees.token1) amount1--; // ensure that the slot is not cleared, for gas savings
        protocolFees.token1 -= amount1;
        TransferHelper.safeTransfer(token1, recipient, amount1);
    }

emit CollectProtocol(msg.sender, recipient, amount0, amount1);
}
```

上述代码比较简单，此处不再展开。

#### Tick.sol

Tick.sol管理Tick内部状态。

##### tickSpacingToMaxLiquidityPerTick

根据`tickSpacing`计算每个`tick`最大流动性，只有能够被`tickSpacing`整除的`tick`才能够存放流动性：

```solidity
/// @notice Derives max liquidity per tick from given tick spacing
/// @dev Executed within the pool constructor
/// @param tickSpacing The amount of required tick separation, realized in multiples of `tickSpacing`
///     e.g., a tickSpacing of 3 requires ticks to be initialized every 3rd tick i.e., ..., -6, -3, 0, 3, 6, ...
/// @return The max liquidity per tick
function tickSpacingToMaxLiquidityPerTick(int24 tickSpacing) internal pure returns (uint128) {
    int24 minTick = (TickMath.MIN_TICK / tickSpacing) * tickSpacing;
    int24 maxTick = (TickMath.MAX_TICK / tickSpacing) * tickSpacing;
    uint24 numTicks = uint24((maxTick - minTick) / tickSpacing) + 1;
    return type(uint128).max / numTicks;
}
```

##### getFeeGrowthInside

计算两个`tick`区间内部的每流动性累积手续费，该方法实现白皮书公式6.17-6.19:

$$
f_a(i) = \begin{cases} f_g - f_o(i) & \text{$i_c \geq i$}\\
f_o(i) & \text{$i_c < i$} \end{cases} \quad \text{(6.17)}
$$

$$
f_b(i) = \begin{cases} f_o(i) & \text{$i_c \geq i$}\\
f_g - f_o(i) & \text{$i_c < i$}\end{cases} \quad \text{(6.18)}
$$

$$
f_r = f_g - f_b(i_l) - f_a(i_u) \quad \text{(6.19)}
$$

代码如下：

```solidity
/// @notice Retrieves fee growth data
/// @param self The mapping containing all tick information for initialized ticks
/// @param tickLower The lower tick boundary of the position
/// @param tickUpper The upper tick boundary of the position
/// @param tickCurrent The current tick
/// @param feeGrowthGlobal0X128 The all-time global fee growth, per unit of liquidity, in token0
/// @param feeGrowthGlobal1X128 The all-time global fee growth, per unit of liquidity, in token1
/// @return feeGrowthInside0X128 The all-time fee growth in token0, per unit of liquidity, inside the position's tick boundaries
/// @return feeGrowthInside1X128 The all-time fee growth in token1, per unit of liquidity, inside the position's tick boundaries
function getFeeGrowthInside(
    mapping(int24 => Tick.Info) storage self,
    int24 tickLower,
    int24 tickUpper,
    int24 tickCurrent,
    uint256 feeGrowthGlobal0X128,
    uint256 feeGrowthGlobal1X128
) internal view returns (uint256 feeGrowthInside0X128, uint256 feeGrowthInside1X128) {
    Info storage lower = self[tickLower];
    Info storage upper = self[tickUpper];

// calculate fee growth below
    uint256 feeGrowthBelow0X128;
    uint256 feeGrowthBelow1X128;
    if (tickCurrent >= tickLower) {
        feeGrowthBelow0X128 = lower.feeGrowthOutside0X128;
        feeGrowthBelow1X128 = lower.feeGrowthOutside1X128;
    } else {
        feeGrowthBelow0X128 = feeGrowthGlobal0X128 - lower.feeGrowthOutside0X128;
        feeGrowthBelow1X128 = feeGrowthGlobal1X128 - lower.feeGrowthOutside1X128;
    }

// calculate fee growth above
    uint256 feeGrowthAbove0X128;
    uint256 feeGrowthAbove1X128;
    if (tickCurrent < tickUpper) {
        feeGrowthAbove0X128 = upper.feeGrowthOutside0X128;
        feeGrowthAbove1X128 = upper.feeGrowthOutside1X128;
    } else {
        feeGrowthAbove0X128 = feeGrowthGlobal0X128 - upper.feeGrowthOutside0X128;
        feeGrowthAbove1X128 = feeGrowthGlobal1X128 - upper.feeGrowthOutside1X128;
    }

feeGrowthInside0X128 = feeGrowthGlobal0X128 - feeGrowthBelow0X128 - feeGrowthAbove0X128;
    feeGrowthInside1X128 = feeGrowthGlobal1X128 - feeGrowthBelow1X128 - feeGrowthAbove1X128;
}
```

首先根据当前`tickCurrent`，分别计算`tickLower`和`tickUpper`的 $f_a$ ,  $f_b$ ，最后计算出区间内手续费 $f_r$：

$$
\underbrace{\overbrace{..., i_l - 1}^{f_b(i_l)}, \overbrace{i_l, i_l + 1, ..., i_u - 1, i_u}^{f_r}, \overbrace{i_u + 1, ...}^{f_a(i_u)}}_{f_g}
$$

为什么需要计算区间内每流动性累计手续费呢？因为每个头寸（Position）会在`mint`/`burn`时根据该值计算自己的应收手续费：

$$
liquidityDelta \cdot (feeGrowthInside - feeGrowthInsideLast)
$$

##### update

更新`tick`状态，并返回该`tick`是否翻转`flipped`：

```solidity
/// @notice Updates a tick and returns true if the tick was flipped from initialized to uninitialized, or vice versa
/// @param self The mapping containing all tick information for initialized ticks
/// @param tick The tick that will be updated
/// @param tickCurrent The current tick
/// @param liquidityDelta A new amount of liquidity to be added (subtracted) when tick is crossed from left to right (right to left)
/// @param feeGrowthGlobal0X128 The all-time global fee growth, per unit of liquidity, in token0
/// @param feeGrowthGlobal1X128 The all-time global fee growth, per unit of liquidity, in token1
/// @param secondsPerLiquidityCumulativeX128 The all-time seconds per max(1, liquidity) of the pool
/// @param tickCumulative The tick * time elapsed since the pool was first initialized
/// @param time The current block timestamp cast to a uint32
/// @param upper true for updating a position's upper tick, or false for updating a position's lower tick
/// @param maxLiquidity The maximum liquidity allocation for a single tick
/// @return flipped Whether the tick was flipped from initialized to uninitialized, or vice versa
function update(
    mapping(int24 => Tick.Info) storage self,
    int24 tick,
    int24 tickCurrent,
    int128 liquidityDelta,
    uint256 feeGrowthGlobal0X128,
    uint256 feeGrowthGlobal1X128,
    uint160 secondsPerLiquidityCumulativeX128,
    int56 tickCumulative,
    uint32 time,
    bool upper,
    uint128 maxLiquidity
) internal returns (bool flipped) {
    Tick.Info storage info = self[tick];

uint128 liquidityGrossBefore = info.liquidityGross;
    uint128 liquidityGrossAfter = LiquidityMath.addDelta(liquidityGrossBefore, liquidityDelta);

require(liquidityGrossAfter <= maxLiquidity, 'LO');

flipped = (liquidityGrossAfter == 0) != (liquidityGrossBefore == 0);
```

如果`tick`从无流动性到有流动性，或者从有流动性变成无流动性，则表示`tick`需要翻转`flipped`。

```solidity
    if (liquidityGrossBefore == 0) {
        // by convention, we assume that all growth before a tick was initialized happened _below_ the tick
        if (tick <= tickCurrent) {
            info.feeGrowthOutside0X128 = feeGrowthGlobal0X128;
            info.feeGrowthOutside1X128 = feeGrowthGlobal1X128;
            info.secondsPerLiquidityOutsideX128 = secondsPerLiquidityCumulativeX128;
            info.tickCumulativeOutside = tickCumulative;
            info.secondsOutside = time;
        }
        info.initialized = true;
    }
```

如果`tick`之前没有流动性，则进行初始化；对于小于当前`tickCurrent`的`tick`，设置`Outside`等变量。

```solidity
    info.liquidityGross = liquidityGrossAfter;
```

`liquidityGross`表示总流动性，用于判断`tick`是否需要初始化：

* 如果`mint`，则增加流动性；如果`burn`，则减少流动性
* 该变量与`tick`在不同头寸中是否作为边界低点或高点无关，只与`mint`或`burn`操作有关
* 如果一个`tick`同时被用作`tickLower`和`tickUpper`，则其`liquidityNet`可能是0，但`liquidityGross`仍然会大于0，因此不需要再次初始化

```solidity
    // when the lower (upper) tick is crossed left to right (right to left), liquidity must be added (removed)
    info.liquidityNet = upper
        ? int256(info.liquidityNet).sub(liquidityDelta).toInt128()
        : int256(info.liquidityNet).add(liquidityDelta).toInt128();
}
```

`liquidityNet`表示净流动性，当`swap`穿越`tick`时，用于更新全局可用流动性`liquidity`：

* 如果作为`tickLower`，即边界低点（左边界点），则增加`liquidityDelta`（`mint`时为正，`burn`时为负）
* 如果作为`tickUpper`，即边界高点（右边界点），则减少`liquidityDelta`（`mint`时为正，`burn`时为负）

##### clear

当`tick`翻转后，如果没有流动性关联该`tick`，即`liquidityGross = 0`，则清空`tick`状态：

```solidity
/// @notice Clears tick data
/// @param self The mapping containing all initialized tick information for initialized ticks
/// @param tick The tick that will be cleared
function clear(mapping(int24 => Tick.Info) storage self, int24 tick) internal {
    delete self[tick];
}
```

##### cross

当`tick`被穿越时，需要翻转`Outside`等变量的方向，如白皮书公式6.20:

$$
f_o(i) := f_g - f_o(i) \quad \text{(6.20)}
$$

这些变量在[getFeeGrowthInside](#getFeeGrowthInside)等方法被用到。

```solidity
/// @notice Transitions to next tick as needed by price movement
/// @param self The mapping containing all tick information for initialized ticks
/// @param tick The destination tick of the transition
/// @param feeGrowthGlobal0X128 The all-time global fee growth, per unit of liquidity, in token0
/// @param feeGrowthGlobal1X128 The all-time global fee growth, per unit of liquidity, in token1
/// @param secondsPerLiquidityCumulativeX128 The current seconds per liquidity
/// @param tickCumulative The tick * time elapsed since the pool was first initialized
/// @param time The current block.timestamp
/// @return liquidityNet The amount of liquidity added (subtracted) when tick is crossed from left to right (right to left)
function cross(
    mapping(int24 => Tick.Info) storage self,
    int24 tick,
    uint256 feeGrowthGlobal0X128,
    uint256 feeGrowthGlobal1X128,
    uint160 secondsPerLiquidityCumulativeX128,
    int56 tickCumulative,
    uint32 time
) internal returns (int128 liquidityNet) {
    Tick.Info storage info = self[tick];
    info.feeGrowthOutside0X128 = feeGrowthGlobal0X128 - info.feeGrowthOutside0X128;
    info.feeGrowthOutside1X128 = feeGrowthGlobal1X128 - info.feeGrowthOutside1X128;
    info.secondsPerLiquidityOutsideX128 = secondsPerLiquidityCumulativeX128 - info.secondsPerLiquidityOutsideX128;
    info.tickCumulativeOutside = tickCumulative - info.tickCumulativeOutside;
    info.secondsOutside = time - info.secondsOutside;
    liquidityNet = info.liquidityNet;
}
```

#### TickMath.sol

TickMath主要包含两个方法：

* [getSqrtRatioAtTick](#getsqrtratioattick)：根据tick计算开根号价格 $\sqrt{P}$
* [getTickAtSqrtRatio](#gettickatsqrtratio)：根据开根号价格 $\sqrt{P}$ 计算tick

##### getSqrtRatioAtTick

该方法对应白皮书公式6.2:

$$
\sqrt{p}(i) = \sqrt{1.0001}^i = 1.0001^{\frac{i}{2}}
$$

其中， $i$ 即为`tick`。

因为Uniswap v3支持的价格（ $\frac{token1}{token0}$ ）区间为 $[2^{-128}, 2^{128}]$ ，根据白皮书公式6.1:

$$
p(i) = 1.0001^i
$$

因此，对应的最大tick（MAX_TICK）为：

$$
i = \lfloor log_{1.0001}{p(i)} \rfloor = \lfloor log_{1.0001}{2^{128}} \rfloor = \lfloor 887272.7517970635 \rfloor = 887272
$$

最小tick（MIN_TICK）为：

$$
i = \lceil log_{1.0001}{2^{-128}} \rceil = \lceil -887272.7517970635 \rceil = -887272
$$

假设 $i$ $\geq 0$，对于一个给定的tick $i$，它总可以表示为二进制，因此以下式子总是成立：

$$
\begin{cases} i = \sum_{n=0}^{19}{(x_n \cdot 2^n)} = x_0 \cdot 1 + x_1 \cdot 2 + x_2 \cdot 4 + ... + x_{19}\cdot 524288 \\
\forall x_n \in \{0, 1\} \end{cases} \quad \text{(1.1)}
$$

其中， $x_n$ 为 $i$ 的二进制位。如 $i=6$ ，其对应的二进制为：`000000000000000000000110`，则 $x_1 = 1, x_2 = 1$ ，其余 $x_n$ 均为0。

同样可以推出 $i < 0$ 也可以用类似的公式表示。

我们先看 $i < 0$ 的情况：

如果 $i < 0$，则：

$$
\sqrt{p}(i) = 1.0001^{\frac{i}{2}} = 1.0001^{-\frac{|i|}{2}} = \frac{1}{1.0001^{\frac{|i|}{2}}} = \frac{1}{1.0001^{\frac{1}{2}(\sum_{n=0}^{19}{(x_n \cdot 2^n)})}} \\
= \frac{1}{1.0001^{\frac{1}{2} \cdot x_0}} \cdot \frac{1}{1.0001^{\frac{2}{2} \cdot x_1}} \cdot \frac{1}{1.0001^{\frac{4}{2} \cdot x_2}} \cdot ... \cdot \frac{1}{1.0001^{\frac{524288}{2} \cdot x_{19}}}
$$

根据二进制位 $x_n$ 的值，可以总结如下：

$$
\frac{1}{1.0001^{\frac{x_n \cdot 2^n}{2}}} \begin{cases} = 1 & \text{$x_n = 0, n \geq 0, i < 0$}\\
< 1 & \text{$x_n = 1, n \geq 0, i < 0$}
\end{cases}
$$

为了最小化精度误差，在计算过程中，使用`Q128.128`（128位定点数）表示中间价格，对于每一个价格 $p$ ，均需要左移128位。由于 $i < 0, x_n = 1$ 时， $\frac{1}{1.0001^{\frac{x_n \cdot 2^n}{2}}} < 1$ ，因此在连续乘积过程中不会有溢出问题。

可以总结计算 $\sqrt{p}(i)$ 的方法：

* 初始值为1，从第0位开始，从低位到高位（从右往左）循环遍历 $i$ 的二进制比特位
* 如果该位不为0，则乘以对应的 $\frac{2^{128}}{1.0001^{\frac{2^n}{2}}}$ ，其中 $2^{128}$ 表示左移128位
* 如果该位为0，则乘以1，可以省略

```solidity
/// @notice Calculates sqrt(1.0001^tick) * 2^96
/// @dev Throws if |tick| > max tick
/// @param tick The input tick for the above formula
/// @return sqrtPriceX96 A Fixed point Q64.96 number representing the sqrt of the ratio of the two assets (token1/token0)
/// at the given tick
function getSqrtRatioAtTick(int24 tick) internal pure returns (uint160 sqrtPriceX96) {
    uint256 absTick = tick < 0 ? uint256(-int256(tick)) : uint256(int256(tick));
    require(absTick <= uint256(MAX_TICK), 'T');

// 如果第0位非0，则ratio = 0xfffcb933bd6fad37aa2d162d1a594001 ，即：2^128 / 1.0001^0.5
    uint256 ratio = absTick & 0x1 != 0 ? 0xfffcb933bd6fad37aa2d162d1a594001 : 0x100000000000000000000000000000000;
    // 如果第1位非0，则乘以 0xfff97272373d413259a46990580e213a ，即：2^128 / 1.0001^1，因为两个乘数均为Q128.128，最终结果多乘了2^128，因此需要右移128
    if (absTick & 0x2 != 0) ratio = (ratio * 0xfff97272373d413259a46990580e213a) >> 128;
    // 如果第2位非0，则乘以 0xfff2e50f5f656932ef12357cf3c7fdcc ，即：2^128 / 1.0001^2，
    if (absTick & 0x4 != 0) ratio = (ratio * 0xfff2e50f5f656932ef12357cf3c7fdcc) >> 128;
    // 以此类推
    if (absTick & 0x8 != 0) ratio = (ratio * 0xffe5caca7e10e4e61c3624eaa0941cd0) >> 128;
    if (absTick & 0x10 != 0) ratio = (ratio * 0xffcb9843d60f6159c9db58835c926644) >> 128;
    if (absTick & 0x20 != 0) ratio = (ratio * 0xff973b41fa98c081472e6896dfb254c0) >> 128;
    if (absTick & 0x40 != 0) ratio = (ratio * 0xff2ea16466c96a3843ec78b326b52861) >> 128;
    if (absTick & 0x80 != 0) ratio = (ratio * 0xfe5dee046a99a2a811c461f1969c3053) >> 128;
    if (absTick & 0x100 != 0) ratio = (ratio * 0xfcbe86c7900a88aedcffc83b479aa3a4) >> 128;
    if (absTick & 0x200 != 0) ratio = (ratio * 0xf987a7253ac413176f2b074cf7815e54) >> 128;
    if (absTick & 0x400 != 0) ratio = (ratio * 0xf3392b0822b70005940c7a398e4b70f3) >> 128;
    if (absTick & 0x800 != 0) ratio = (ratio * 0xe7159475a2c29b7443b29c7fa6e889d9) >> 128;
    if (absTick & 0x1000 != 0) ratio = (ratio * 0xd097f3bdfd2022b8845ad8f792aa5825) >> 128;
    if (absTick & 0x2000 != 0) ratio = (ratio * 0xa9f746462d870fdf8a65dc1f90e061e5) >> 128;
    if (absTick & 0x4000 != 0) ratio = (ratio * 0x70d869a156d2a1b890bb3df62baf32f7) >> 128;
    if (absTick & 0x8000 != 0) ratio = (ratio * 0x31be135f97d08fd981231505542fcfa6) >> 128;
    if (absTick & 0x10000 != 0) ratio = (ratio * 0x9aa508b5b7a84e1c677de54f3e99bc9) >> 128;
    if (absTick & 0x20000 != 0) ratio = (ratio * 0x5d6af8dedb81196699c329225ee604) >> 128;
    if (absTick & 0x40000 != 0) ratio = (ratio * 0x2216e584f5fa1ea926041bedfe98) >> 128;
    // 如果第19位非0，因为（2^19 = 0x80000=524288），则乘以 0x2216e584f5fa1ea926041bedfe98，即：2^128 / 1.0001^(524288/2)
    // tick的最大值为887272，因此其二进制最多只需要20位表示，从0开始计数，最后一位为第19位。
    if (absTick & 0x80000 != 0) ratio = (ratio * 0x48a170391f7dc42444e8fa2) >> 128;

if (tick > 0) ratio = type(uint256).max / ratio;

// this divides by 1<<32 rounding up to go from a Q128.128 to a Q128.96.
    // we then downcast because we know the result always fits within 160 bits due to our tick input constraint
    // we round up in the division so getTickAtSqrtRatio of the output price is always consistent
    sqrtPriceX96 = uint160((ratio >> 32) + (ratio % (1 << 32) == 0 ? 0 : 1));
}
```

假设 $i > 0$ 时：

$$
\sqrt{p_{Q128128}(i)} = 2^{128} \cdot \sqrt{p(i)} = 2^{128} \cdot 1.0001^{\frac{i}{2}} \\
= \frac{2^{128}}{1.0001^{-\frac{i}{2}}} = \frac{2^{256}}{2^{128} \cdot \sqrt{p(-i)}} = \frac{2^{256}}{\sqrt{p_{Q128128}(-i)}}
$$

因此，只需要算出 $i < 0$ 时的 ratio 值，使用 $2^{256}$ 除以ratio即可得出 $i > 0$ 时，使用`Q128.128`表示的ratio值：
```solidity
if (tick > 0) ratio = type(uint256).max / ratio;
```

代码最后一行将ratio右移32位，转化为`Q128.96`格式的定点数：

```solidity
sqrtPriceX96 = uint160((ratio >> 32) + (ratio % (1 << 32) == 0 ? 0 : 1));
```

这里算的是开根号价格 $\sqrt{p}$ ，由于价格 $p$ 最大为 $2^{128}$ ，因此 $\sqrt{p}$ 最大为 $2^{64}$ ，也就是整数部分最大只需要64位表示，因此最终的sqrtPriceX96一定可以用160位（64+96，即`Q64.96`格式的定点数）表示。

##### getTickAtSqrtRatio

该方法对应白皮书中的公式6.8：

$$
i_c = \lfloor \log_{\sqrt{1.0001}} \sqrt{P} \rfloor
$$

本方法涉及在Solidity中计算对数，根据对数公式，可以推出：

$$
\log_{\sqrt{1.0001}} \sqrt{P} = \frac{log_2{\sqrt{P}}}{log_2{\sqrt{1.0001}}} = log_2{\sqrt{P}}  \cdot log_{\sqrt{1.0001}}{2}
$$

由于 $log_{\sqrt{1.0001}}{2}$ 是一个常数，因此我们只需要计算 $log_2{\sqrt{P}}$ 即可。

将输入的参数为 $\sqrt{P}$ 看作 $x$ ，问题转化为求 $log_2{x}$。

把结果分为整数部分 $n$ 和小数部分 $m$ ，则：

$$
n \leq log_2{x} = n + m < n + 1
$$

###### 整数部分

对于 $n$ ，因为：

$$
2^n \leq x < 2^{n+1}
$$

可以通过二分查找找到 $n$ 值：

* 对于256位的数， $0 \leq n < 256$ ，可以用8位比特表示 $n$
* 从二进制表示的第8位（k=7）到第1位（k=0）（从最高位到最低位），依次比较 $x$ 是否大于 $2^{2^{k}} - 1$ ，如果大于则标记该位为1，并右移 $2^k$ 位；否则标记0
* 最终标记后的8位二进制即为 $n$ 值

使用Python代码描述如下：

```python
def find_msb(x):
    msb = 0
    for k in reversed(range(8)): // k = 7, 6, 5. 4, 3, 2, 1, 0
        if x > 2 ** (2 ** k) - 1:
            msb += 2 ** k // 标记该位为1，即加上 2 ** k
            x /= 2 ** (2 ** k) // 右移 2 ** k 位
    return msb
```

Uniswap v3中的Solidity代码如下（请参考代码中注释）：

```solidity
/// @notice Calculates the greatest tick value such that getRatioAtTick(tick) <= ratio
/// @dev Throws in case sqrtPriceX96 < MIN_SQRT_RATIO, as MIN_SQRT_RATIO is the lowest value getRatioAtTick may
/// ever return.
/// @param sqrtPriceX96 The sqrt ratio for which to compute the tick as a Q64.96
/// @return tick The greatest tick for which the ratio is less than or equal to the input ratio
function getTickAtSqrtRatio(uint160 sqrtPriceX96) internal pure returns (int24 tick) {
    // second inequality must be < because the price can never reach the price at the max tick
    require(sqrtPriceX96 >= MIN_SQRT_RATIO && sqrtPriceX96 < MAX_SQRT_RATIO, 'R');
    uint256 ratio = uint256(sqrtPriceX96) << 32; // 左移32位，转化为Q128.128格式

uint256 r = ratio;
    uint256 msb = 0;

assembly {
        // 如果大于2 ** (2 ** 7) - 1，则保存临时变量：2 ** 7
        let f := shl(7, gt(r, 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF))
        // msb += 2 ** 7
        msb := or(msb, f)
        // r /= (2 ** (2 ** 7))，即右移 2 ** 7
        r := shr(f, r)
    }
    assembly {
        let f := shl(6, gt(r, 0xFFFFFFFFFFFFFFFF))
        msb := or(msb, f)
        r := shr(f, r)
    }
    assembly {
        let f := shl(5, gt(r, 0xFFFFFFFF))
        msb := or(msb, f)
        r := shr(f, r)
    }
    assembly {
        let f := shl(4, gt(r, 0xFFFF))
        msb := or(msb, f)
        r := shr(f, r)
    }
    assembly {
        let f := shl(3, gt(r, 0xFF))
        msb := or(msb, f)
        r := shr(f, r)
    }
    assembly {
        let f := shl(2, gt(r, 0xF))
        msb := or(msb, f)
        r := shr(f, r)
    }
    assembly {
        let f := shl(1, gt(r, 0x3))
        msb := or(msb, f)
        r := shr(f, r)
    }
    assembly {
        let f := gt(r, 0x1)
        msb := or(msb, f)
    }
```

###### 小数部分

对于小数部分 $m$：

$$
0 \leq m = log_2{x} - n = log_2{\frac{x}{2^n}} < 1 \quad \text{(1.2)}
$$

其中， $n$ 为上文算出的msb，即整数部分。

我们先将 $\frac{x}{2^n}$ 看做一个整体 $r$，则：

$$
0 \leq log_2{r} < 1
$$

$$
1 \leq r = \frac{x}{2^n} < 2
$$

这里我们希望求出 $log_2{r}$ ，如果能够将 $log_2{r}$ 表示成一个不断收敛的数列，当小数位足够多时，就可以近似求出 $log_2{r}$ 的值。

根据对数公式，我们可以推导以下两个等式：

$$
log_2{r} = \frac{2 \cdot log_2{r}}{2} = \frac{log_2{r^2}}{2} \quad \text{(1.3)}
$$

$$
log_2{r} = log_2{2 \cdot \frac{r}{2}} = 1 + log_2{\frac{r}{2}} \quad \text{(1.4)}
$$

我们循环套用上述两个公式，可以整理以下方法：

1. 因为初始时 $log_2{r} < 1$，因此先应用公式1.3，将问题转化为求 $log_2{r^2}$ ，注意此时基数为 $\frac{1}{2}$ ；
    - 事实上，每一次进入步骤1，新的基数都是上一次基数的 $\frac{1}{2}$ ，比如第二次进入步骤1的基数为 $\frac{1}{4}$ ，以此类推。
2. 如果 $r^2$ >= 2 ，则应用公式1.4，分离出1，并将问题转化为求 $log_2{\frac{r^2}{2}}$ ；
    - 因为公式1.4是在公式1.3之后判断，因此这里的1需要乘以上一次步骤1的基数，如果是第一次则记录 $\frac{1}{2}$ ，第二次则记录 $\frac{1}{4}$ ，以此类推；
    - 因为 $1 \leq r < 2$ ，且 $2 \leq r^2 < 4$ ，因此 $1 \leq \frac{r^2}{2} < 2$ ，将 $\frac{r^2}{2}$ 看做一个整体 $r$ ，又回到步骤1求解 $log_2{r}$ ，并且 $1 \leq r < 2$ 。
3. 如果 $r^2 < 2$ ，则回到步骤1继续。

可以将上述步骤总结为以下公式：

$$
log_2{r} = m_1 \cdot \frac{1}{2} + m_2 \cdot \frac{1}{4} + ... + m_n \cdot \frac{1}{2^n} = \sum^{\infty}_{i=1}(m_i \cdot \frac{1}{2^i}) \quad \text{(1.5)}
$$

其中， $\forall m_i \in \{0, 1\}$。

这其实就是小数的二进制表示法，小数的二进制第一位表示为 $2^{-1}$ ，第二位为 $2^{-2}$ ，以此类推。而在我们上述计算 $log_2{r}$ 的步骤中，如果进入步骤2，则相当于标记该位为1；如果进入步骤3，则相当于标记该位为0。

重复以上步骤的过程，即为确认小数部分二进制位从高位到低位（从左到右）每一位的值，每一个循环确认一位。循环次数越多，计算得出的 $log_2{r}$ 精度越高。

我们继续看 Uniswap v3 中计算小数部分的代码：

```solidity
        if (msb >= 128) r = ratio >> (msb - 127);
        else r = ratio << (127 - msb);
```

这里msb即为整数部分 $n$ 。因为ratio是`Q128.128`，如果`msb >= 128`则表示`ratio >= 1`，因此需要右移整数位数得到小数部分`ratio >> msb`；`-127`表示左移127位，使用`Q129.127`表示小数部分；同样，如果`msb < 128`，则表示`ratio < 1`，其本身就只有小数部分，因此通过左移`127 - msb`位，将小数部分凑齐127位，也用`Q129.127`表示小数部分。

实际上，`ratio >> msb`即为公式1.2中的 $\frac{x}{2^n}$ ，也就是步骤1中的 $r$ ，在后续迭代算法（步骤1-3）中需要用到。

```solidity
    int256 log_2 = (int256(msb) - 128) << 64;
```
因为msb是基于`Q128.128`的ratio计算的，`int256(msb) - 128`表示 $n$ 的真正值。`<< 64`使用`Q192.64`表示 $n$ 。
这一行代码实际上是使用`Q192.64`保存整数部分的值。

下面代码循环计算二进制表示的小数部分的前14位小数：

```solidity
    assembly {
        // 根据步骤1，计算r^2，右移127位是因为两个r都是Q129.127
        r := shr(127, mul(r, r))
        // 因为1 <= r^2 < 4，仅需2位表示r^2的整数，
        // 因此从右往左数第129和128位表示r^2的整数部分，
        // 右移128位，仅剩129位，
        // 该值为1，则表示r >= 2；该值为0，则表示r < 2
        let f := shr(128, r)
        // 如果f == 1，则log_2 += Q192.64的1/2
        log_2 := or(log_2, shl(63, f))
        // 根据步骤2（即公式1.4），如果r >= 2（即f == 1），则r /= 2；否则不操作，即步骤3
        r := shr(f, r)
    }
    // 重复进行上述过程
    assembly {
        r := shr(127, mul(r, r))
        let f := shr(128, r)
        log_2 := or(log_2, shl(62, f))
        r := shr(f, r)
    }
    assembly {
        r := shr(127, mul(r, r))
        let f := shr(128, r)
        log_2 := or(log_2, shl(61, f))
        r := shr(f, r)
    }
    assembly {
        r := shr(127, mul(r, r))
        let f := shr(128, r)
        log_2 := or(log_2, shl(60, f))
        r := shr(f, r)
    }
    assembly {
        r := shr(127, mul(r, r))
        let f := shr(128, r)
        log_2 := or(log_2, shl(59, f))
        r := shr(f, r)
    }
    assembly {
        r := shr(127, mul(r, r))
        let f := shr(128, r)
        log_2 := or(log_2, shl(58, f))
        r := shr(f, r)
    }
    assembly {
        r := shr(127, mul(r, r))
        let f := shr(128, r)
        log_2 := or(log_2, shl(57, f))
        r := shr(f, r)
    }
    assembly {
        r := shr(127, mul(r, r))
        let f := shr(128, r)
        log_2 := or(log_2, shl(56, f))
        r := shr(f, r)
    }
    assembly {
        r := shr(127, mul(r, r))
        let f := shr(128, r)
        log_2 := or(log_2, shl(55, f))
        r := shr(f, r)
    }
    assembly {
        r := shr(127, mul(r, r))
        let f := shr(128, r)
        log_2 := or(log_2, shl(54, f))
        r := shr(f, r)
    }
    assembly {
        r := shr(127, mul(r, r))
        let f := shr(128, r)
        log_2 := or(log_2, shl(53, f))
        r := shr(f, r)
    }
    assembly {
        r := shr(127, mul(r, r))
        let f := shr(128, r)
        log_2 := or(log_2, shl(52, f))
        r := shr(f, r)
    }
    assembly {
        r := shr(127, mul(r, r))
        let f := shr(128, r)
        log_2 := or(log_2, shl(51, f))
        r := shr(f, r)
    }
    assembly {
        r := shr(127, mul(r, r))
        let f := shr(128, r)
        log_2 := or(log_2, shl(50, f))
    }
```

上述计算的log_2即为`Q192.64`表示的 $log_2{\sqrt{P}}$ ，精度为 $2^{-14}$ 。

```solidity
    int256 log_sqrt10001 = log_2 * 255738958999603826347141; // 128.128 number
```
因为：

$$
\log_{\sqrt{1.0001}} \sqrt{P} = \frac{log_2{\sqrt{P}}}{log_2{\sqrt{1.0001}}} = log_2{\sqrt{P}}  \cdot log_{\sqrt{1.0001}}{2}
$$

这里`255738958999603826347141`即为 $log_{\sqrt{1.0001}}{2} \cdot 2^{64}$ ，两个`Q192.64`乘以的结果为`Q128.64`（不会发生溢出）。

由于这里算出的 $log_2{\sqrt{P}}$ 精度为 $2^{-14}$ ，乘以`255738958999603826347141`后误差进一步放大，因此需要修正并确保结果是最接近给定价格的tick。

```solidity
    int24 tickLow = int24((log_sqrt10001 - 3402992956809132418596140100660247210) >> 128);
    int24 tickHi = int24((log_sqrt10001 + 291339464771989622907027621153398088495) >> 128);

tick = tickLow == tickHi ? tickLow : getSqrtRatioAtTick(tickHi) <= sqrtPriceX96 ? tickHi : tickLow;
```

其中，`3402992956809132418596140100660247210`表示`0.01000049749154292 << 128`，`291339464771989622907027621153398088495`表示`0.8561697375276566 << 128`。

参考[abdk的这篇文章](https://hackmd.io/@abdk/SkVJeHK9v)，当精度为 $2^{-14}$ 时，tick的最小误差为 $−
0.85617$ ，最大误差为 $0.0100005$ 。同时，这篇文章也从理论上证明了，只有当精度等于（或高于） $2^{-14}$ 时，只需要一次计算即可得出所需的tick值。

我们的目的是寻找满足当前条件的最大tick，使得tick对应的 $\sqrt{P}$ 小于等于传入的值。因此如果补偿后的tickHi满足要求，则优先使用tickHi；否则使用tickLow。

以下是本节参考文章，有兴趣的朋友请扩展阅读：

* [Solidity中的对数计算](https://liaoph.com/logarithm-in-solidity/)
* [Math in Solidity](https://medium.com/coinmonks/math-in-solidity-part-5-exponent-and-logarithm-9aef8515136e)
* [Logarithm Approximation Precision](https://hackmd.io/@abdk/SkVJeHK9v)

#### TickBitmap.sol

TickBitmap使用Bitmap（位图）保存Tick的初始化状态，提供以下几个方法：

* [position](#position)：根据tick返回位图索引数据
* [flipTick](#flipTick)：翻转tick状态
* [nextInitializedTickWithinOneWord](#nextInitializedTickWithinOneWord)：寻找同组内下一个初始化的tick

##### position

```solidity
/// @notice Computes the position in the mapping where the initialized bit for a tick lives
/// @param tick The tick for which to compute the position
/// @return wordPos The key in the mapping containing the word in which the bit is stored
/// @return bitPos The bit position in the word where the flag is stored
function position(int24 tick) private pure returns (int16 wordPos, uint8 bitPos) {
    wordPos = int16(tick >> 8);
    bitPos = uint8(tick % 256);
}
```

只有能被`tickSpacing`整除的`tick`才能记录在位图中，因此此处的参数： $tick = \frac{tick}{tickSpacing}$。

`tick`类型为`int24`，其二进制从右到左，从低位到高位，前8位表示`bitPos`，后16位表示`wordPos`，如下图所示：

$$
\overbrace{23,...,8}^{wordPos},\overbrace{7,...,0}^{bitPos}
$$

`tick`的bitmap表示为：`self[wordPos] ^= 1 << bitPos`。

##### flipTick

当`tick`翻转初始化状态时，如果其位图的值为0，则需要修改为1；否则，修改为0；即对该位“取反”。

```solidity
/// @notice Flips the initialized state for a given tick from false to true, or vice versa
/// @param self The mapping in which to flip the tick
/// @param tick The tick to flip
/// @param tickSpacing The spacing between usable ticks
function flipTick(
    mapping(int16 => uint256) storage self,
    int24 tick,
    int24 tickSpacing
) internal {
    require(tick % tickSpacing == 0); // ensure that the tick is spaced
    (int16 wordPos, uint8 bitPos) = position(tick / tickSpacing);
    uint256 mask = 1 << bitPos;
    self[wordPos] ^= mask;
}
```

首先获取`tick`对应的`wordPos`和`bitPos`，由于最后针对`bitPos`执行按位“异或”操作：

> 因为1与任何值b（0或1）异或等于~b；0与任何值b（0或1）异或等于b。

* 对于`tick`对应的位（bit），mask为1
    - 如果旧值为1，`1^1=0`，因此`tick`状态由“初始化”变成“未初始化”；
    - 如果旧值为0，`0^1=1`，因此`tick`状态由“未初始化”变成“初始化”
* 对于非`tick`对应的位，mask为0
    - 如果旧值为1，`1^0=1`，因此状态不变
    - 如果旧值为0，`0^0=0`，因此状态不变

所以，上述代码实现了`tick`位取反的效果。

##### nextInitializedTickWithinOneWord

根据参数`tick`，寻找位图上最近一个已初始化的`tick`，如未找到，返回本组最后一个未初始化的tick。

```solidity
/// @notice Returns the next initialized tick contained in the same word (or adjacent word) as the tick that is either
/// to the left (less than or equal to) or right (greater than) of the given tick
/// @param self The mapping in which to compute the next initialized tick
/// @param tick The starting tick
/// @param tickSpacing The spacing between usable ticks
/// @param lte Whether to search for the next initialized tick to the left (less than or equal to the starting tick)
/// @return next The next initialized or uninitialized tick up to 256 ticks away from the current tick
/// @return initialized Whether the next tick is initialized, as the function only searches within up to 256 ticks
function nextInitializedTickWithinOneWord(
    mapping(int16 => uint256) storage self,
    int24 tick,
    int24 tickSpacing,
    bool lte
) internal view returns (int24 next, bool initialized) {
    int24 compressed = tick / tickSpacing;
    if (tick < 0 && tick % tickSpacing != 0) compressed--; // round towards negative infinity

if (lte) {
        (int16 wordPos, uint8 bitPos) = position(compressed);
        // all the 1s at or to the right of the current bitPos
        uint256 mask = (1 << bitPos) - 1 + (1 << bitPos);
        uint256 masked = self[wordPos] & mask;

// if there are no initialized ticks to the right of or at the current tick, return rightmost in the word
        initialized = masked != 0;
        // overflow/underflow is possible, but prevented externally by limiting both tickSpacing and tick
        next = initialized
            ? (compressed - int24(bitPos - BitMath.mostSignificantBit(masked))) * tickSpacing
            : (compressed - int24(bitPos)) * tickSpacing;
    } else {
        // start from the word of the next tick, since the current tick state doesn't matter
        (int16 wordPos, uint8 bitPos) = position(compressed + 1);
        // all the 1s at or to the left of the bitPos
        uint256 mask = ~((1 << bitPos) - 1);
        uint256 masked = self[wordPos] & mask;

// if there are no initialized ticks to the left of the current tick, return leftmost in the word
        initialized = masked != 0;
        // overflow/underflow is possible, but prevented externally by limiting both tickSpacing and tick
        next = initialized
            ? (compressed + 1 + int24(BitMath.leastSignificantBit(masked) - bitPos)) * tickSpacing
            : (compressed + 1 + int24(type(uint8).max - bitPos)) * tickSpacing;
    }
}
```

如果`lte == true`，即寻找小于等于当前`tick`的值，因此问题转化为：寻找低比特位上是否有`1`。

```solidity
if (lte) {
    (int16 wordPos, uint8 bitPos) = position(compressed);
    // all the 1s at or to the right of the current bitPos
    uint256 mask = (1 << bitPos) - 1 + (1 << bitPos);
    uint256 masked = self[wordPos] & mask;
}
```

其中，mask的值为所有小于等于`bitPos`的位全部置1，比如`bitPos = 7`，则`mask二进制 = 1111111`；masked保留位图中小于等于`bitPos`的位值，比如`self[wordPos] = 110101011`，则`masked = 110101011 & 1111111 = 000101011`。

如果`masked != 0`，则表示当前方向（`lte`）该`wordPos`上有初始化的`tick`；否则，则表示该方向都是未初始化的tick。

```solidity
// if there are no initialized ticks to the right of or at the current tick, return rightmost in the word
initialized = masked != 0;
```

如果存在已初始化的`tick`，则需要定位到`masked`中最高位的1；如果不存在，则返回当前方向最后一个`tick`。

```solidity
// overflow/underflow is possible, but prevented externally by limiting both tickSpacing and tick
next = initialized
    ? (compressed - int24(bitPos - BitMath.mostSignificantBit(masked))) * tickSpacing
    : (compressed - int24(bitPos)) * tickSpacing;
```

`BitMath.mostSignificantBit(masked)`通过二分查找法找到`masked`最高位的1，关于该算法的具体说明，可参考本文[TickMath对数计算部分](#整数部分)。简单而言，`mostSignificantBit(masked)`会返回一个数`n`，使得：

$$
2^n \leq masked < 2^{n+1}
$$

比如，如果`masked = 000101011`，`mostSignificantBit`将返回5，因为最高位的1在（从0开始）第5位：000`1`01011。

`compressed - int24(bitPos)`表示当前`wordPos`第一个`tick`，`compressed - int24(bitPos - BitMath.mostSignificantBit(masked))`表示该最高位`bitPos`对应的`tick`；`* tickSpacing`将恢复到原始`tick`值，因为保存位图时，`tick`需要先除以`tickSpacing`。

同样，如果向大于当前`tick`方向寻找第一个已初始化的`tick`时，方法和上述类似，只是`mostSignificantBit`需要换成`leastSignificantBit`，即从`tick`比特位开始（不含），往高位寻找第一个`bitPos`位为1的`tick`。

```solidity
else {
    // start from the word of the next tick, since the current tick state doesn't matter
    (int16 wordPos, uint8 bitPos) = position(compressed + 1);
    // all the 1s at or to the left of the bitPos
    uint256 mask = ~((1 << bitPos) - 1);
    uint256 masked = self[wordPos] & mask;

// if there are no initialized ticks to the left of the current tick, return leftmost in the word
    initialized = masked != 0;
    // overflow/underflow is possible, but prevented externally by limiting both tickSpacing and tick
    next = initialized
        ? (compressed + 1 + int24(BitMath.leastSignificantBit(masked) - bitPos)) * tickSpacing
        : (compressed + 1 + int24(type(uint8).max - bitPos)) * tickSpacing;
}
```

#### Position.sol

`Position.sol`管理头寸相关信息，包括以下方法：

* [get](#get)：获取头寸对象
* [update](#update1)：更新头寸

一个头寸（Position）可由“所有者”`owner`、“区间低点”`tickLower`和“区间高点”`tickUpper`唯一确定，对于同一个交易对池子，每个用户可以创建多个不同价格区间的头寸，但只能创建一个相同价格区间的头寸；不同用户可以创建相同价格区间的头寸。

##### get

根据`owner`、`tickLower`和`tickUpper`返回一个头寸对象：

```solidity
/// @notice Returns the Info struct of a position, given an owner and position boundaries
/// @param self The mapping containing all user positions
/// @param owner The address of the position owner
/// @param tickLower The lower tick boundary of the position
/// @param tickUpper The upper tick boundary of the position
/// @return position The position info struct of the given owners' position
function get(
    mapping(bytes32 => Info) storage self,
    address owner,
    int24 tickLower,
    int24 tickUpper
) internal view returns (Position.Info storage position) {
    position = self[keccak256(abi.encodePacked(owner, tickLower, tickUpper))];
}
```

##### update

更新头寸的流动性和可取回代币，注意，该方法只会在`mint`和`burn`时被触发，`swap`并不会更新头寸信息。

```solidity
/// @notice Credits accumulated fees to a user's position
/// @param self The individual position to update
/// @param liquidityDelta The change in pool liquidity as a result of the position update
/// @param feeGrowthInside0X128 The all-time fee growth in token0, per unit of liquidity, inside the position's tick boundaries
/// @param feeGrowthInside1X128 The all-time fee growth in token1, per unit of liquidity, inside the position's tick boundaries
function update(
    Info storage self,
    int128 liquidityDelta,
    uint256 feeGrowthInside0X128,
    uint256 feeGrowthInside1X128
) internal {
    Info memory _self = self;

uint128 liquidityNext;
    if (liquidityDelta == 0) {
        require(_self.liquidity > 0, 'NP'); // disallow pokes for 0 liquidity positions
        liquidityNext = _self.liquidity;
    } else {
        liquidityNext = LiquidityMath.addDelta(_self.liquidity, liquidityDelta);
    }
```

更新流动性：

* 如果是`mint`，则`liquidityDelta > 0`
* 如果是`burn`，则`liquidityDelta < 0`

```solidity
    // calculate accumulated fees
    uint128 tokensOwed0 =
        uint128(
            FullMath.mulDiv(
                feeGrowthInside0X128 - _self.feeGrowthInside0LastX128,
                _self.liquidity,
                FixedPoint128.Q128
            )
        );
    uint128 tokensOwed1 =
        uint128(
            FullMath.mulDiv(
                feeGrowthInside1X128 - _self.feeGrowthInside1LastX128,
                _self.liquidity,
                FixedPoint128.Q128
            )
        );
```

根据头寸区间自上一次更新后的每流动性手续费增长值，分别计算`token0`和`token1`的应收手续费。

```solidity
    // update the position
    if (liquidityDelta != 0) self.liquidity = liquidityNext;
    self.feeGrowthInside0LastX128 = feeGrowthInside0X128;
    self.feeGrowthInside1LastX128 = feeGrowthInside1X128;
    if (tokensOwed0 > 0 || tokensOwed1 > 0) {
        // overflow is acceptable, have to withdraw before you hit type(uint128).max fees
        self.tokensOwed0 += tokensOwed0;
        self.tokensOwed1 += tokensOwed1;
    }
}
```

更新头寸流动性、本次每流动性手续费和可取回代币数。

#### SwapMath.sol

##### computeSwapStep

SwapMath只有一个方法，即`computeSwapStep`，计算单步交换的输入输出。

该方法实际上实现的是白皮书6.2.3小节：在一个Tick内交易。在该场景下，流动性 $L$ 不变，仅价格 $\sqrt{P}$ 发生变化。

参数如下：

* `sqrtRatioCurrentX96`：当前价格
* `sqrtRatioTargetX96`：目标价格
* `liquidity`：可用流动性
* `amountRemaining`：剩余输入代币
* `feePips`：手续费

返回值：

* `sqrtRatioNextX96`：交换后价格
* `amountIn`：本次交换消耗的输入代币
* `amountOut`：本次交换消耗的输出代币
* `feeAmount`：交易手续费（含协议手续费）

```solidity
function computeSwapStep(
    uint160 sqrtRatioCurrentX96,
    uint160 sqrtRatioTargetX96,
    uint128 liquidity,
    int256 amountRemaining,
    uint24 feePips
)
    internal
    pure
    returns (
        uint160 sqrtRatioNextX96,
        uint256 amountIn,
        uint256 amountOut,
        uint256 feeAmount
    )
{
    bool zeroForOne = sqrtRatioCurrentX96 >= sqrtRatioTargetX96;
    bool exactIn = amountRemaining >= 0;
```

我们在[swap](#swap)章节提到，对于一个交换操作，根据`zeroForOne`和`exactIn`的值，有四种组合：

```solidity
if (exactIn) {
    uint256 amountRemainingLessFee = FullMath.mulDiv(uint256(amountRemaining), 1e6 - feePips, 1e6);
    amountIn = zeroForOne
        ? SqrtPriceMath.getAmount0Delta(sqrtRatioTargetX96, sqrtRatioCurrentX96, liquidity, true)
        : SqrtPriceMath.getAmount1Delta(sqrtRatioCurrentX96, sqrtRatioTargetX96, liquidity, true);
    if (amountRemainingLessFee >= amountIn) sqrtRatioNextX96 = sqrtRatioTargetX96;
    else
        sqrtRatioNextX96 = SqrtPriceMath.getNextSqrtPriceFromInput(
            sqrtRatioCurrentX96,
            liquidity,
            amountRemainingLessFee,
            zeroForOne
        );
}
```

如果是输入固定数量，则交易手续费需要从输入代币中扣除。

因为`feePips`的单位是百分之一基点，即 $\frac{1}{10^6}$ ，因此，按照如下公式扣除手续费：

$$
amountRemaining \cdot (1 - \frac{feePips}{10^6})
$$

使用[getAmount0Delta](#getAmount0Delta)或[getAmount1Delta](#getAmount1Delta)，根据当前价格、目标价格和可用流动性计算所需的输入代币数量`amountIn`：

* 如果从`token0`交换`token1`，则输入代币为`token0`，因此计算`amount0`
* 如果从`token1`交换`token0`，则输入代币为`token1`，因此计算`amount1`

注意，方法的最后一个参数`roundUp`，当计算`amountIn`时需要设置`roundUp = true`，也就是合约只能多收钱，而不能少收，否则合约资金将出现损失；同样，当计算`amountOut`时设置`roundUp = false`，也就是合约可以少付钱，而不能多付。

* 如果可用代币数量大于`amountIn`，则表示该步交易可以完成，因此交换后的价格等于目标价格
* 否则，根据当前价格，可用流动性和可用代币`amountRemainingLessFee`，计算交换后价格，我们在[SqrtPriceMath.getNextSqrtPriceFromInput](#getNextSqrtPriceFromInput)会具体介绍计算方法

```solidity
else {
    amountOut = zeroForOne
        ? SqrtPriceMath.getAmount1Delta(sqrtRatioTargetX96, sqrtRatioCurrentX96, liquidity, false)
        : SqrtPriceMath.getAmount0Delta(sqrtRatioCurrentX96, sqrtRatioTargetX96, liquidity, false);
    if (uint256(-amountRemaining) >= amountOut) sqrtRatioNextX96 = sqrtRatioTargetX96;
    else
        sqrtRatioNextX96 = SqrtPriceMath.getNextSqrtPriceFromOutput(
            sqrtRatioCurrentX96,
            liquidity,
            uint256(-amountRemaining),
            zeroForOne
        );
}
```

如果不是输入固定数量，也就是输出固定数量，需要将`amountRemaining`作为输出代币数量使用：

* 如果从`token0`交换`token1`，则输出代币为`token1`，因此计算`amount1`
* 如果从`token1`交换`token0`，则输出代币为`token0`，因此计算`amount0`

首先根据当前价格、目标价格和可用流动性，计算可产生的输出代币数量`amountOut`，注意，这里与上面相反，如果是`token0`交换`token1`，需要计算`token1`的数量，需使用`SqrtPriceMath.getAmount1Delta`方法。

当表示固定输出代币时，传入的参数`amountRemaining`是负值：

* 如果`amountOut`的绝对值小于应输出代币数量，则表示可完全交换，交换后价格等于目标价格
* 否则，根据可用输出`amountRemaining`，计算交换后价格

```solidity
    bool max = sqrtRatioTargetX96 == sqrtRatioNextX96;
```

如果交换后价格等于目标价格，则表示完全交换。

```solidity
// get the input/output amounts
if (zeroForOne) {
    amountIn = max && exactIn
        ? amountIn
        : SqrtPriceMath.getAmount0Delta(sqrtRatioNextX96, sqrtRatioCurrentX96, liquidity, true);
    amountOut = max && !exactIn
        ? amountOut
        : SqrtPriceMath.getAmount1Delta(sqrtRatioNextX96, sqrtRatioCurrentX96, liquidity, false);
} else {
    amountIn = max && exactIn
        ? amountIn
        : SqrtPriceMath.getAmount1Delta(sqrtRatioCurrentX96, sqrtRatioNextX96, liquidity, true);
    amountOut = max && !exactIn
        ? amountOut
        : SqrtPriceMath.getAmount0Delta(sqrtRatioCurrentX96, sqrtRatioNextX96, liquidity, false);
}
```

计算本次交换所需输入`amountIn`和所得输出`amountOut`：

* 如果从`token0`交换`token1`
    - 所需输入`amountIn`
        - 如果完全交换，并且固定输入，则所需输入即为`amountIn`
        - 否则（非完全交换，或者固定输出），则使用`getAmount0Delta`，根据价格区间和流动性计算所需输入
    - 所得输出`amountOut`
        - 如果完全交换，并且固定输出，则所得输出即为`amountOut`
        - 否则（非完全交换，或者固定输入），则使用`getAmount1Delta`，根据价格区间和流动性计算所得输出
* 如果从`token1`交换`token1`
    - 所需输入`amountIn`
        - 如果完全交换，并且固定输入，则所需输入即为`amountIn`
        - 否则（非完全交换，或者固定输出），则使用`getAmount1Delta`，根据价格区间和流动性计算所需输入
    - 所得输出`amountOut`
        - 如果完全交换，并且固定输出，则所得输出即为`amountOut`
        - 否则（非完全交换，或者固定输入），则使用`getAmount0Delta`，根据价格区间和流动性计算所得输出

```solidity
// cap the output amount to not exceed the remaining output amount
if (!exactIn && amountOut > uint256(-amountRemaining)) {
    amountOut = uint256(-amountRemaining);
}
```

确认所得输出没有超过指定输出。

```solidity
if (exactIn && sqrtRatioNextX96 != sqrtRatioTargetX96) {
    // we didn't reach the target, so take the remainder of the maximum input as fee
    feeAmount = uint256(amountRemaining) - amountIn;
} else {
    feeAmount = FullMath.mulDivRoundingUp(amountIn, feePips, 1e6 - feePips);
}
```

计算交易手续费（包括协议手续费）：

* 如果固定输入，并且没有达到目标价格，相当于是最后一次交换，则原始输入代币`amountRemaining`扣除本次交换所需输入`amountIn`，即为手续费；
* 否则，需要根据`amountIn`计算手续费；注意，除了上述最后一次交换外，这里的`amountIn`和`amountOut`都是不包括手续费的，因此计算手续费需要除以`1e6 - feePips`，而非`1e6`。

#### SqrtPriceMath.sol

##### getNextSqrtPriceFromAmount0RoundingUp

根据当前价格、`liquidity`和 $\Delta{x}$ ，计算目标价格。

根据白皮书公式6.16:

$$
\Delta{x} = \Delta{\frac{1}{\sqrt{P}}} \cdot L
$$

假设 $\sqrt{P_a} > \sqrt{P_b}$ ，则 $x_a < x_b$ ：

$$
\Delta{x} = x_b - x_a
$$

如果已知 $\sqrt{P_a}$ 计算 $\sqrt{P_b}$ ，则：

$$
\frac{1}{\sqrt{P_b}} = \frac{\Delta{x}}{L} + \frac{1}{\sqrt{P_a}} = \frac{1}{L} \cdot (\Delta{x} + \frac{L}{\sqrt{P_a}}) \quad \text{(1.1)}
$$

$$
\sqrt{P_b} = \frac{L}{\Delta{x} + \frac{L}{\sqrt{P_a}}} \quad \text{(1.2)}
$$

$$
{\sqrt{P_b}} = \frac{L \cdot \sqrt{P_a}}{L + \Delta{x} \cdot \sqrt{P_a}}  \quad \text{(1.3)}
$$

如果已知 $\sqrt{P_b}$ 计算 $\sqrt{P_a}$ ，则：

$$
\frac{1}{\sqrt{P_a}} = \frac{1}{\sqrt{P_b}} - \frac{\Delta{x}}{L}  \quad \text{(1.4)}
$$

$$
{\sqrt{P_a}} = \frac{L \cdot \sqrt{P_b}}{L - \Delta{x} \cdot \sqrt{P_b}}  \quad \text{(1.5)}
$$

```solidity
/// @notice Gets the next sqrt price given a delta of token0
/// @dev Always rounds up, because in the exact output case (increasing price) we need to move the price at least
/// far enough to get the desired output amount, and in the exact input case (decreasing price) we need to move the
/// price less in order to not send too much output.
/// The most precise formula for this is liquidity * sqrtPX96 / (liquidity +- amount * sqrtPX96),
/// if this is impossible because of overflow, we calculate liquidity / (liquidity / sqrtPX96 +- amount).
/// @param sqrtPX96 The starting price, i.e. before accounting for the token0 delta
/// @param liquidity The amount of usable liquidity
/// @param amount How much of token0 to add or remove from virtual reserves
/// @param add Whether to add or remove the amount of token0
/// @return The price after adding or removing amount, depending on add
function getNextSqrtPriceFromAmount0RoundingUp(
    uint160 sqrtPX96,
    uint128 liquidity,
    uint256 amount,
    bool add
) internal pure returns (uint160) {
    // we short circuit amount == 0 because the result is otherwise not guaranteed to equal the input price
    if (amount == 0) return sqrtPX96;
    uint256 numerator1 = uint256(liquidity) << FixedPoint96.RESOLUTION;

if (add) {
        uint256 product;
        if ((product = amount * sqrtPX96) / amount == sqrtPX96) {
            uint256 denominator = numerator1 + product;
            if (denominator >= numerator1)
                // always fits in 160 bits
                return uint160(FullMath.mulDivRoundingUp(numerator1, sqrtPX96, denominator));
        }

return uint160(UnsafeMath.divRoundingUp(numerator1, (numerator1 / sqrtPX96).add(amount)));
    } else {
        uint256 product;
        // if the product overflows, we know the denominator underflows
        // in addition, we must check that the denominator does not underflow
        require((product = amount * sqrtPX96) / amount == sqrtPX96 && numerator1 > product);
        uint256 denominator = numerator1 - product;
        return FullMath.mulDivRoundingUp(numerator1, sqrtPX96, denominator).toUint160();
    }
}
```

* 当`add = true`时，即已知 $\sqrt{P_a}$ 计算 $\sqrt{P_b}$
    - 如果 $\Delta{x} \cdot \sqrt{P_a}$ 没有发生溢出，则使用公式1.3计算 $\sqrt{P_b}$ ，因为这种方式精度最高
    - 否则，使用公式1.2计算 $\sqrt{P_b}$
* 当`add = false`时，即已知 $\sqrt{P_b}$ ，根据上述公式1.5计算 $\sqrt{P_a}$

##### getNextSqrtPriceFromAmount1RoundingDown

根据当前价格、`liquidity`和 $\Delta{y}$ ，计算目标价格。

根据白皮书公式6.13:

$$
\Delta{\sqrt{P}} = \frac{\Delta{y}}{L} \quad \text{(6.13)}
$$

假设 $\sqrt{P_a} > \sqrt{P_b}$ ，则 $y_a > y_b$：

$$
\Delta{y} = y_a - y_b
$$

如果已知 $\sqrt{P_b}$ 计算 $\sqrt{P_a}$ ，则：

$$
\sqrt{P_a} = \sqrt{P_b} + \frac{\Delta{y}}{L} \quad \text{(1.6)}
$$

如果已知 $\sqrt{P_a}$ 计算 $\sqrt{P_b}$ ，则：

$$
\sqrt{P_b} = \sqrt{P_a} - \frac{\Delta{y}}{L} \quad \text{(1.7)}
$$

```solidity
/// @notice Gets the next sqrt price given a delta of token1
/// @dev Always rounds down, because in the exact output case (decreasing price) we need to move the price at least
/// far enough to get the desired output amount, and in the exact input case (increasing price) we need to move the
/// price less in order to not send too much output.
/// The formula we compute is within <1 wei of the lossless version: sqrtPX96 +- amount / liquidity
/// @param sqrtPX96 The starting price, i.e., before accounting for the token1 delta
/// @param liquidity The amount of usable liquidity
/// @param amount How much of token1 to add, or remove, from virtual reserves
/// @param add Whether to add, or remove, the amount of token1
/// @return The price after adding or removing `amount`
function getNextSqrtPriceFromAmount1RoundingDown(
    uint160 sqrtPX96,
    uint128 liquidity,
    uint256 amount,
    bool add
) internal pure returns (uint160) {
    // if we're adding (subtracting), rounding down requires rounding the quotient down (up)
    // in both cases, avoid a mulDiv for most inputs
    if (add) {
        uint256 quotient =
            (
                amount <= type(uint160).max
                    ? (amount << FixedPoint96.RESOLUTION) / liquidity
                    : FullMath.mulDiv(amount, FixedPoint96.Q96, liquidity)
            );

return uint256(sqrtPX96).add(quotient).toUint160();
    } else {
        uint256 quotient =
            (
                amount <= type(uint160).max
                    ? UnsafeMath.divRoundingUp(amount << FixedPoint96.RESOLUTION, liquidity)
                    : FullMath.mulDivRoundingUp(amount, FixedPoint96.Q96, liquidity)
            );

require(sqrtPX96 > quotient);
        // always fits 160 bits
        return uint160(sqrtPX96 - quotient);
    }
}
```

* 当`add = true`时，即按照公式1.6，根据 $\sqrt{P_b}$ 计算 $\sqrt{P_a}$
* 当`add = false`时，按照公式1.7，根据 $\sqrt{P_a}$ 计算 $\sqrt{P_b}$

##### getNextSqrtPriceFromInput

根据输入代币计算下一个价格，即添加`amountIn`数量的代币后的价格：

```solidity
/// @notice Gets the next sqrt price given an input amount of token0 or token1
/// @dev Throws if price or liquidity are 0, or if the next price is out of bounds
/// @param sqrtPX96 The starting price, i.e., before accounting for the input amount
/// @param liquidity The amount of usable liquidity
/// @param amountIn How much of token0, or token1, is being swapped in
/// @param zeroForOne Whether the amount in is token0 or token1
/// @return sqrtQX96 The price after adding the input amount to token0 or token1
function getNextSqrtPriceFromInput(
    uint160 sqrtPX96,
    uint128 liquidity,
    uint256 amountIn,
    bool zeroForOne
) internal pure returns (uint160 sqrtQX96) {
    require(sqrtPX96 > 0);
    require(liquidity > 0);

// round to make sure that we don't pass the target price
    return
        zeroForOne
            ? getNextSqrtPriceFromAmount0RoundingUp(sqrtPX96, liquidity, amountIn, true)
            : getNextSqrtPriceFromAmount1RoundingDown(sqrtPX96, liquidity, amountIn, true);
}
```

##### getNextSqrtPriceFromOutput

根据输出代币计算下一个价格，即移除`amountOut`数量的代币后的价格：

```solidity
/// @notice Gets the next sqrt price given an output amount of token0 or token1
/// @dev Throws if price or liquidity are 0 or the next price is out of bounds
/// @param sqrtPX96 The starting price before accounting for the output amount
/// @param liquidity The amount of usable liquidity
/// @param amountOut How much of token0, or token1, is being swapped out
/// @param zeroForOne Whether the amount out is token0 or token1
/// @return sqrtQX96 The price after removing the output amount of token0 or token1
function getNextSqrtPriceFromOutput(
    uint160 sqrtPX96,
    uint128 liquidity,
    uint256 amountOut,
    bool zeroForOne
) internal pure returns (uint160 sqrtQX96) {
    require(sqrtPX96 > 0);
    require(liquidity > 0);

// round to make sure that we pass the target price
    return
        zeroForOne
            ? getNextSqrtPriceFromAmount1RoundingDown(sqrtPX96, liquidity, amountOut, false)
            : getNextSqrtPriceFromAmount0RoundingUp(sqrtPX96, liquidity, amountOut, false);
}
```

##### getAmount0Delta

该方法计算白皮书中的公式6.16：

$$
\Delta{x} = \Delta{\frac{1}{\sqrt{P}}} \cdot L
$$

展开公式为：

$$
amount0 = x_b - x_a = L \cdot (\frac{1}{\sqrt{P_b}} - \frac{1}{\sqrt{P_a}}) = L \cdot (\frac{\sqrt{P_a} - \sqrt{P_b}}{\sqrt{P_a} \cdot \sqrt{P_b}})
$$

```solidity
/// @notice Gets the amount0 delta between two prices
/// @dev Calculates liquidity / sqrt(lower) - liquidity / sqrt(upper),
/// i.e. liquidity * (sqrt(upper) - sqrt(lower)) / (sqrt(upper) * sqrt(lower))
/// @param sqrtRatioAX96 A sqrt price
/// @param sqrtRatioBX96 Another sqrt price
/// @param liquidity The amount of usable liquidity
/// @param roundUp Whether to round the amount up or down
/// @return amount0 Amount of token0 required to cover a position of size liquidity between the two passed prices
function getAmount0Delta(
    uint160 sqrtRatioAX96,
    uint160 sqrtRatioBX96,
    uint128 liquidity,
    bool roundUp
) internal pure returns (uint256 amount0) {
    if (sqrtRatioAX96 > sqrtRatioBX96) (sqrtRatioAX96, sqrtRatioBX96) = (sqrtRatioBX96, sqrtRatioAX96);

uint256 numerator1 = uint256(liquidity) << FixedPoint96.RESOLUTION;
    uint256 numerator2 = sqrtRatioBX96 - sqrtRatioAX96;

require(sqrtRatioAX96 > 0);

return
        roundUp
            ? UnsafeMath.divRoundingUp(
                FullMath.mulDivRoundingUp(numerator1, numerator2, sqrtRatioBX96),
                sqrtRatioAX96
            )
            : FullMath.mulDiv(numerator1, numerator2, sqrtRatioBX96) / sqrtRatioAX96;
}
```

##### getAmount1Delta

同样，该方法计算白皮书公式6.7：

$$
L = \frac{\Delta{Y}}{\Delta{\sqrt{P}}}
$$

展开公式为：

$$
amount1 = y_b - y_a = L \cdot \Delta{\sqrt{P}} = L \cdot (\sqrt{P_b} - \sqrt{P_a})
$$

```solidity
/// @notice Gets the amount1 delta between two prices
/// @dev Calculates liquidity * (sqrt(upper) - sqrt(lower))
/// @param sqrtRatioAX96 A sqrt price
/// @param sqrtRatioBX96 Another sqrt price
/// @param liquidity The amount of usable liquidity
/// @param roundUp Whether to round the amount up, or down
/// @return amount1 Amount of token1 required to cover a position of size liquidity between the two passed prices
function getAmount1Delta(
    uint160 sqrtRatioAX96,
    uint160 sqrtRatioBX96,
    uint128 liquidity,
    bool roundUp
) internal pure returns (uint256 amount1) {
    if (sqrtRatioAX96 > sqrtRatioBX96) (sqrtRatioAX96, sqrtRatioBX96) = (sqrtRatioBX96, sqrtRatioAX96);

return
        roundUp
            ? FullMath.mulDivRoundingUp(liquidity, sqrtRatioBX96 - sqrtRatioAX96, FixedPoint96.Q96)
            : FullMath.mulDiv(liquidity, sqrtRatioBX96 - sqrtRatioAX96, FixedPoint96.Q96);
```

#### Oracle.sol

本合约定义预言机相关方法。

在交易对合约创建时，默认初始化长度为1的数组用于存储观测点数据；任何用户都可以调用`UniswapV3Pool.sol`的[increaseObservationCardinalityNext](#increaseObservationCardinalityNext)方法扩展预言机的观测点空间，但需要承担扩展空间带来的手续费。

一个预言机观测点包括以下内容：

* `blockTimestamp`：该观测点写入时的时间
* `tickCumulative`：截止该观测点时间的累计`tick`
* `secondsPerLiquidityCumulativeX128`：截止该观测点的累计每流动性持续时间
* `initialized`：观测点是否初始化

观测点结构定义如下：

```solidity
struct Observation {
    // the block timestamp of the observation
    uint32 blockTimestamp;
    // the tick accumulator, i.e. tick * time elapsed since the pool was first initialized
    int56 tickCumulative;
    // the seconds per liquidity, i.e. seconds elapsed / max(1, liquidity) since the pool was first initialized
    uint160 secondsPerLiquidityCumulativeX128;
    // whether or not the observation is initialized
    bool initialized;
}
```

本合约包含以下方法：

* [transform](#transform)：基于上一个观测点，返回新观测点对象（但是不写入观测点）
* [initialize](#initialize2)：初始化观测点数组
* [write](#write)：写入一个观测点
* [grow](#grow)：扩展预言机观测点空间
* [lte](#lte)：判断两个timestamp大小
* [binarySearch](#binarySearch)：二分查找指定时间的观测点边界
* [getSurroundingObservations](#getSurroundingObservations)：获取指定时间的观测点边界
* [observeSingle](#observeSingle)：获取指定时间的观测点数据
* [observe](#observe)：批量获取指定时间的观测点数据

##### transform

基于上一个观测点，返回临时观测点对象，但是不写入观测点。

```solidity
function transform(
    Observation memory last,
    uint32 blockTimestamp,
    int24 tick,
    uint128 liquidity
) private pure returns (Observation memory) {
    uint32 delta = blockTimestamp - last.blockTimestamp;
    return
        Observation({
            blockTimestamp: blockTimestamp,
            tickCumulative: last.tickCumulative + int56(tick) * delta,
            secondsPerLiquidityCumulativeX128: last.secondsPerLiquidityCumulativeX128 +
                ((uint160(delta) << 128) / (liquidity > 0 ? liquidity : 1)),
            initialized: true
        });
}
```

首先，计算当前时间与上一个观测点的时间差：`delta = blockTimestamp - last.blockTimestamp`。

接着主要计算`tickCumulative`和`secondsPerLiquidityCumulativeX128`,

其中：`tickCumulative: last.tickCumulative + int56(tick) * delta`。

根据白皮书公式5.3-5.5：

$$
\log_{1.0001}(P_{t_1,t_2}) = \frac{\sum_{i=t_1}^{t_2} \log_{1.0001}(P_i)}{t_2 - t_1} \quad \text{(5.3)}
$$

$$
\log_{1.0001}(P_{t_1,t_2}) = \frac{a_{t_2} - a_{t_1}}{t_2 - t_1} \quad \text{(5.4)}
$$

$$
P_{t_1,t_2} = 1.0001^{\frac{a_{t_2} - a_{t_1}}{t_2 - t_1}} \quad \text{(5.5)}
$$

这里保存的`tickCumulative`即为 $a_{t_n}$ ，其对应的公式为：

$$
tickCumulative = \sum_{i=0}^{t_n} \log_{1.0001}(P_i)
$$

同样，每流动性持续时间`secondsPerLiquidityCumulative`为：

$$
secondsPerLiquidityCumulative = \sum_{i=0}^{n} \frac{t_i}{L_i}
$$

##### initialize

初始化预言机存储空间，设置第一个`slot`。

```solidity
/// @notice Initialize the oracle array by writing the first slot. Called once for the lifecycle of the observations array
/// @param self The stored oracle array
/// @param time The time of the oracle initialization, via block.timestamp truncated to uint32
/// @return cardinality The number of populated elements in the oracle array
/// @return cardinalityNext The new length of the oracle array, independent of population
function initialize(Observation[65535] storage self, uint32 time)
    internal
    returns (uint16 cardinality, uint16 cardinalityNext)
{
    self[0] = Observation({
        blockTimestamp: time,
        tickCumulative: 0,
        secondsPerLiquidityCumulativeX128: 0,
        initialized: true
    });
    return (1, 1);
}
```

默认返回的`cardinality`和`cardinalityNext`都为1，即仅能存放一个观测点数据。

该方法只能被调用一次，实际上是在`UniswapV3Pool.sol`的[initialize](#initialize)方法中调用：

```solidity
(uint16 cardinality, uint16 cardinalityNext) = observations.initialize(_blockTimestamp());
```

##### write

写入一次观测点数据。根据之前的描述，只有在`UniswapV3Pool`发生`mint`、`burn`和`swap`操作时，才可能触发写入操作。

可以将预言机观测点数组看作一个循环列表，可写入空间由`cardinality`和`cardinalityNext`决定；当数组空间写满以后，会继续从0位置开始覆盖写。

本方法的参数如下：

* `self`：预言机观测点数组
* `index`：最后一次写入的观测点索引，从0开始
* `blockTimestamp`：待添加观测点的时间
* `tick`：待添加观测点的`tick`
* `liquidity`：待添加观测点时间的全局可用流动性
* `cardinality`：观测点数组当前长度（可写入空间）
* `cardinalityNext`：观测点数组（扩展后的）最新长度

```solidity
function write(
    Observation[65535] storage self,
    uint16 index,
    uint32 blockTimestamp,
    int24 tick,
    uint128 liquidity,
    uint16 cardinality,
    uint16 cardinalityNext
) internal returns (uint16 indexUpdated, uint16 cardinalityUpdated) {
    Observation memory last = self[index];

// early return if we've already written an observation this block
    if (last.blockTimestamp == blockTimestamp) return (index, cardinality);
```

如果新观测点时间与最后一次观测点时间相同，则直接返回，确保每个区块最多只能写入一次观测点。

```solidity
    // if the conditions are right, we can bump the cardinality
    if (cardinalityNext > cardinality && index == (cardinality - 1)) {
        cardinalityUpdated = cardinalityNext;
    } else {
        cardinalityUpdated = cardinality;
    }
```

* 如果`cardinalityNext > cardinality`，则表示预言机数组被扩容过；如果`index == (cardinality - 1)`即上一次写入的位置是最后一个观测点，则本次需要继续写入扩容后的空间，因此`cardinalityUpdated`使用扩容后的数组长度`cardinalityNext`；
* 否则继续使用旧的数组长度`cardinality`，因为还未写满。

```solidity
    indexUpdated = (index + 1) % cardinalityUpdated;
```

更新本次写入观测点数组的索引`indexUpdated`，`% cardinalityUpdated`是为了计算循环写的索引。

```solidity
    self[indexUpdated] = transform(last, blockTimestamp, tick, liquidity);
```

调用[transform](#transform)方法计算最新的观测点数据，并写入观测点数组的`indexUpdated`位置。

##### grow

扩容观测点数组，增加可写入的观测点数量。由于合约默认只能保存1个观测点，为了能够支持更多观测点，任何用户都可以手动调用合约以扩容观测点数组。

> 注意，`grow`方法使用`internal`修饰，用户实际上是通过`UniswapV3Pool.sol`的[increaseObservationCardinalityNext](#increaseObservationCardinalityNext)方法进行扩容。

扩容方法会触发`SSTORE`操作，因此调用该方法的用户需要支付由此带来的gas开销。

```solidity
/// @notice Prepares the oracle array to store up to `next` observations
/// @param self The stored oracle array
/// @param current The current next cardinality of the oracle array
/// @param next The proposed next cardinality which will be populated in the oracle array
/// @return next The next cardinality which will be populated in the oracle array
function grow(
    Observation[65535] storage self,
    uint16 current,
    uint16 next
) internal returns (uint16) {
    require(current > 0, 'I');
    // no-op if the passed next value isn't greater than the current next value
    if (next <= current) return current;
    // store in each slot to prevent fresh SSTOREs in swaps
    // this data will not be used because the initialized boolean is still false
    for (uint16 i = current; i < next; i++) self[i].blockTimestamp = 1;
    return next;
}
```

##### lte

比较两个时间戳的大小。

由于合约使用`uint32`类型表示时间戳，其最大值 $2^{32}-1 = 4294967295$，对应的时间为`February 7, 2106 6:28:15 AM GMT+00:00`，如果两个时间`a`和`b`（ $a < b$ ）正好位于`uint32`最大值的两边，`b`由于溢出，因此 $a > b$，直接比较数值会导致错误的结果，因此需要统一时间。

> 注：实际上每136年左右就会有溢出问题。

方法接受3个参数：`time`、`a`和`b`；其中，`time`为基准时间，`a`和`b`在逻辑（时间）上小于等于`time`；方法返回一个`bool`，表示`a`是否在逻辑时间点上小于等于`b`。

```solidity
/// @notice comparator for 32-bit timestamps
/// @dev safe for 0 or 1 overflows, a and b _must_ be chronologically before or equal to time
/// @param time A timestamp truncated to 32 bits
/// @param a A comparison timestamp from which to determine the relative position of `time`
/// @param b From which to determine the relative position of `time`
/// @return bool Whether `a` is chronologically <= `b`
function lte(
    uint32 time,
    uint32 a,
    uint32 b
) private pure returns (bool) {
    // if there hasn't been overflow, no need to adjust
    if (a <= time && b <= time) return a <= b;

uint256 aAdjusted = a > time ? a : a + 2**32;
    uint256 bAdjusted = b > time ? b : b + 2**32;

return aAdjusted <= bAdjusted;
}
```

* 如果`a`和`b`都小于等于`time`，则表示没有发生溢出，因此直接返回`a <= b`
* 由于`a`和`b`在时间线上均小于等于`time`
  * 如果 $a > time$ ，则表示发生溢出，`aAdjusted`为补齐`2**32`后的时间`a`
  * 同理，`bAdjusted`为补齐后的时间`b`
  * 最后比较两个补齐后的时间`aAdjusted`和`bAdjusted`

因此，该方法在`a`、`b`与`time`存在0到1个 $2^{32}$ 的时间差时是溢出安全的。

> 不能跨越两个 $2^{32} - 1$ 。

##### binarySearch

二分查找指定目标时间的观测点。

参数如下：

* `self`：观测点数组
* `time`：当前区块时间
* `target`：目标时间
* `index`：最后一次写入的观测点索引
* `cardinality`：观测点数组当前长度（可写入空间）

```solidity
/// @notice Fetches the observations beforeOrAt and atOrAfter a target, i.e. where [beforeOrAt, atOrAfter] is satisfied.
/// The result may be the same observation, or adjacent observations.
/// @dev The answer must be contained in the array, used when the target is located within the stored observation
/// boundaries: older than the most recent observation and younger, or the same age as, the oldest observation
/// @param self The stored oracle array
/// @param time The current block.timestamp
/// @param target The timestamp at which the reserved observation should be for
/// @param index The index of the observation that was most recently written to the observations array
/// @param cardinality The number of populated elements in the oracle array
/// @return beforeOrAt The observation recorded before, or at, the target
/// @return atOrAfter The observation recorded at, or after, the target
function binarySearch(
    Observation[65535] storage self,
    uint32 time,
    uint32 target,
    uint16 index,
    uint16 cardinality
) private view returns (Observation memory beforeOrAt, Observation memory atOrAfter) {
```

对于二分查找算法，首先需要确认左右边界点。

```solidity
    uint256 l = (index + 1) % cardinality; // oldest observation
    uint256 r = l + cardinality - 1; // newest observation
```

由于最后一次索引为`index`，因此循环继续右移一位（并取模）即为左边界点`l`（最老的索引，如果已写满一轮的话）；右边界点为`l + cardinality - 1`，注意，右边界点`r`没有取模，因为在二分查找中右边节点一定不能小于左边节点。

```solidity
    uint256 i;
    while (true) {
        i = (l + r) / 2;

beforeOrAt = self[i % cardinality];

// we've landed on an uninitialized tick, keep searching higher (more recently)
        if (!beforeOrAt.initialized) {
            l = i + 1;
            continue;
        }

atOrAfter = self[(i + 1) % cardinality];
```

如果计算的中点未初始化（即此时数组空间未写满），则使用右半部分区间（往时间点更近的方向）继续进行二分查找。

`atOrAfter`为右侧紧邻的观测点。

```solidity
        bool targetAtOrAfter = lte(time, beforeOrAt.blockTimestamp, target);

// check if we've found the answer!
        if (targetAtOrAfter && lte(time, target, atOrAfter.blockTimestamp)) break;

if (!targetAtOrAfter) r = i - 1;
        else l = i + 1;
    }
}
```

* 如果目标时间`target`位于`beforeOrAt`与`atOrAfter`时间之间，则退出二分查找，返回`beforeOrAt`与`atOrAfter`两个观测点
* 否则：
  * 如果`beforeOrAt`时间大于目标时间`target`，则继续在左半部分（往更小的时间）进行二分查找
  * 如果目标时间`target`大于`atOrAfter`时间，则继续往右半部分（往更大的时间）进行二分查找

假设`cardinality`为10，`index`为5，我们可以画出几个变量的逻辑关系如下：

![](https://img.learnblockchain.cn/2025/04/04/binarySearch.jpg)

##### getSurroundingObservations

获取目标时间`target`的观测点数据`beforeOrAt`和`atOrAfter`，满足`target`位于`[beforeOrAt, atOrAfter]`之间。

```solidity
/// @notice Fetches the observations beforeOrAt and atOrAfter a given target, i.e. where [beforeOrAt, atOrAfter] is satisfied
/// @dev Assumes there is at least 1 initialized observation.
/// Used by observeSingle() to compute the counterfactual accumulator values as of a given block timestamp.
/// @param self The stored oracle array
/// @param time The current block.timestamp
/// @param target The timestamp at which the reserved observation should be for
/// @param tick The active tick at the time of the returned or simulated observation
/// @param index The index of the observation that was most recently written to the observations array
/// @param liquidity The total pool liquidity at the time of the call
/// @param cardinality The number of populated elements in the oracle array
/// @return beforeOrAt The observation which occurred at, or before, the given timestamp
/// @return atOrAfter The observation which occurred at, or after, the given timestamp
function getSurroundingObservations(
    Observation[65535] storage self,
    uint32 time,
    uint32 target,
    int24 tick,
    uint16 index,
    uint128 liquidity,
    uint16 cardinality
) private view returns (Observation memory beforeOrAt, Observation memory atOrAfter) {
    // optimistically set before to the newest observation
    beforeOrAt = self[index];

// if the target is chronologically at or after the newest observation, we can early return
    if (lte(time, beforeOrAt.blockTimestamp, target)) {
        if (beforeOrAt.blockTimestamp == target) {
            // if newest observation equals target, we're in the same block, so we can ignore atOrAfter
            return (beforeOrAt, atOrAfter);
        } else {
            // otherwise, we need to transform
            return (beforeOrAt, transform(beforeOrAt, target, tick, liquidity));
        }
    }
```

首先将`beforeOrAt`设置为最近一次的观测点：

如果`beforeOrAt`时间小于等于目标时间`target`：
  * 如果等于目标时间，则直接返回`beforeOrAt`，忽略`atOrAfter`
  * 如果小于目标时间，则使用[transform](#transform)基于`beforeOrAt`和`target`临时生成一个观测点作为`atOrAfter`

```solidity
    // now, set before to the oldest observation
    beforeOrAt = self[(index + 1) % cardinality];
    if (!beforeOrAt.initialized) beforeOrAt = self[0];
```

否则，可以确认最后（最晚）一个观测点的时间大于`target`。

设置`beforeOrAt`为循环右移一个位置的观测点，即最老的观测点；如果该观测点未初始化，则表示数组没有写满，因此最早的一定是索引为0的观测点。

```solidity
    // ensure that the target is chronologically at or after the oldest observation
    require(lte(time, beforeOrAt.blockTimestamp, target), 'OLD');
```

确认`target`时间大于等于最早的观测点时间，因此此时`target`一定位于最早的和最晚观测点的时间区间内，可以使用[binarySearch](#binarySearch)进行二分查找。

```solidity
    // if we've reached this point, we have to binary search
    return binarySearch(self, time, target, index, cardinality);
}
```

##### observeSingle

获取指定时间的观测点数据。

方法的参数如下：

* `self`：预言机观测点数组
* `time`：当前区块时间
* `secondsAgo`：距离当前时间多少秒以前的指定时间，根据该时间寻找观测点
* `tick`：目标时间的`tick`（价格）
* `index`：最后一次写入的观测点索引
* `liquidity`：当前全局可用流动性
* `cardinality`：预言机数组当前长度（可写入空间）

* `tickCumulative`：从交易对池子创建以来，截止到`secondsAgo`的累计`tick`
* `secondsPerLiquidityCumulativeX128`：从交易对池子创建以来，截止到`secondsAgo`的累计每流动性持续时间

```solidity
/// @dev Reverts if an observation at or before the desired observation timestamp does not exist.
/// 0 may be passed as `secondsAgo' to return the current cumulative values.
/// If called with a timestamp falling between two observations, returns the counterfactual accumulator values
/// at exactly the timestamp between the two observations.
/// @param self The stored oracle array
/// @param time The current block timestamp
/// @param secondsAgo The amount of time to look back, in seconds, at which point to return an observation
/// @param tick The current tick
/// @param index The index of the observation that was most recently written to the observations array
/// @param liquidity The current in-range pool liquidity
/// @param cardinality The number of populated elements in the oracle array
/// @return tickCumulative The tick * time elapsed since the pool was first initialized, as of `secondsAgo`
/// @return secondsPerLiquidityCumulativeX128 The time elapsed / max(1, liquidity) since the pool was first initialized, as of `secondsAgo`
function observeSingle(
    Observation[65535] storage self,
    uint32 time,
    uint32 secondsAgo,
    int24 tick,
    uint16 index,
    uint128 liquidity,
    uint16 cardinality
) internal view returns (int56 tickCumulative, uint160 secondsPerLiquidityCumulativeX128) {
    if (secondsAgo == 0) {
        Observation memory last = self[index];
        if (last.blockTimestamp != time) last = transform(last, time, tick, liquidity);
        return (last.tickCumulative, last.secondsPerLiquidityCumulativeX128);
    }
```

如果`secondsAgo == 0`，表示取当前时间的观测点。如果当前时间不等于最后一次写入的观测点时间，则使用[transform](#transform)生成一个临时观测点（注意，这里没有写入该观测点），并返回相关数据。

```solidity
    uint32 target = time - secondsAgo;

(Observation memory beforeOrAt, Observation memory atOrAfter) =
        getSurroundingObservations(self, time, target, tick, index, liquidity, cardinality);
```

根据`secondsAgo`计算目标时间`target`，使用[getSurroundingObservations](#getSurroundingObservations)方法寻找距离目标时间最近的观测点边界`beforeOrAt`和`atOrAfter`。

```solidity
    if (target == beforeOrAt.blockTimestamp) {
        // we're at the left boundary
        return (beforeOrAt.tickCumulative, beforeOrAt.secondsPerLiquidityCumulativeX128);
    } else if (target == atOrAfter.blockTimestamp) {
        // we're at the right boundary
        return (atOrAfter.tickCumulative, atOrAfter.secondsPerLiquidityCumulativeX128);
    } else {
        // we're in the middle
        uint32 observationTimeDelta = atOrAfter.blockTimestamp - beforeOrAt.blockTimestamp;
        uint32 targetDelta = target - beforeOrAt.blockTimestamp;
        return (
            beforeOrAt.tickCumulative +
                ((atOrAfter.tickCumulative - beforeOrAt.tickCumulative) / observationTimeDelta) *
                targetDelta,
            beforeOrAt.secondsPerLiquidityCumulativeX128 +
                uint160(
                    (uint256(
                        atOrAfter.secondsPerLiquidityCumulativeX128 - beforeOrAt.secondsPerLiquidityCumulativeX128
                    ) * targetDelta) / observationTimeDelta
                )
        );
    }
```

根据[getSurroundingObservations](#getSurroundingObservations)方法，需要优先使用左边界点`beforeOrAt`：

* 如果目标时间等于`beforeOrAt`时间，则直接返回该观测点的相关数据
* 如果目标时间等于`atOrAfter`时间，则也返回相关数据
* 如果目标时间介于`beforeOrAt`和`atOrAfter`之间，则需要根据时间比例计算相关值：
  * `observationTimeDelta`为`beforeOrAt`和`atOrAfter`的时间差值（下面用 $\Delta{t}$ 表示），`targetDelta`为`beforeOrAt`和`target`的时间差值
  * 因为 $\Delta{tickCumulative} = tick \cdot \Delta{t}$，则截止到`target`的值应为： $\frac{\Delta{tickCumulative}}{\Delta{t}} \cdot targetDelta$
  * 同样， $\Delta{secondsPerLiquidityCumulativeX128} = \frac{\Delta{t}}{liquidity}$，则截止到`target`的值应为： $\frac{\Delta{secondsPerLiquidityCumulativeX128}}{\Delta{t}} \cdot targetDelta$

##### observe

批量获取指定时间的观测点数据。

该方法主要调用[observeSingle](#observeSingle)获取单个指定时间的观测点数据，然后批量返回。

```solidity
/// @notice Returns the accumulator values as of each time seconds ago from the given time in the array of `secondsAgos`
/// @dev Reverts if `secondsAgos` > oldest observation
/// @param self The stored oracle array
/// @param time The current block.timestamp
/// @param secondsAgos Each amount of time to look back, in seconds, at which point to return an observation
/// @param tick The current tick
/// @param index The index of the observation that was most recently written to the observations array
/// @param liquidity The current in-range pool liquidity
/// @param cardinality The number of populated elements in the oracle array
/// @return tickCumulatives The tick * time elapsed since the pool was first initialized, as of each `secondsAgo`
/// @return secondsPerLiquidityCumulativeX128s The cumulative seconds / max(1, liquidity) since the pool was first initialized, as of each `secondsAgo`
function observe(
    Observation[65535] storage self,
    uint32 time,
    uint32[] memory secondsAgos,
    int24 tick,
    uint16 index,
    uint128 liquidity,
    uint16 cardinality
) internal view returns (int56[] memory tickCumulatives, uint160[] memory secondsPerLiquidityCumulativeX128s) {
    require(cardinality > 0, 'I');

tickCumulatives = new int56[](secondsAgos.length);
    secondsPerLiquidityCumulativeX128s = new uint160[](secondsAgos.length);
    for (uint256 i = 0; i < secondsAgos.length; i++) {
        (tickCumulatives[i], secondsPerLiquidityCumulativeX128s[i]) = observeSingle(
            self,
            time,
            secondsAgos[i],
            tick,
            index,
            liquidity,
            cardinality
        );
    }
}
```

>- 本文转载自： [github.com/adshao/public...](https://github.com/adshao/publications/blob/master/uniswap/dive-into-uniswap-v3-contracts/README_zh.md)

深入理解 Uniswap v3 智能合约（一）

概述

与Uniswap v2一样，Uniswap v3的合约也分为两类：

Uniswap-v3-core
- Uniswap v3的核心代码，实现了协议定义的所有功能，外部合约可直接与core合约交互
Uniswap-v3-periphery
- 基于使用场景封装接口，如头寸（Position）管理、多路径代币交换等功能，Uniswap界面即与periphery合约交互

如果你希望以用户场景角度阅读本文，请直接从Uniswap-v3-periphery开始，它包含了创建头寸、修改头寸流动性、交换代币等常用功能。

如果你希望从底层核心模块开始阅读，请从Uniswap-v3-core开始。

Uniswap-v3-core

UniswapV3Factory.sol

工厂合约主要包含三个功能：

createPool：创建交易对池子
setOwner：设置工厂合约Owner
enableFeeAmount：添加手续费等级

createPool

创建一个Uniswap v3交易对池子，注意，由于Uniswap v3支持不同手续费等级，如0.05%、0.30%、1.00%等，因此一个交易对合约由tokenA、tokenB和fee（手续费）唯一确定。

计算交易对合约还需要：factory工厂合约地址、合约初始化代码的hash。

/// @inheritdoc IUniswapV3Factory
function createPool(
    address tokenA,
    address tokenB,
    uint24 fee
) external override noDelegateCall returns (address pool) {
    require(tokenA != tokenB);
    (address token0, address token1) = tokenA &lt; tokenB ? (tokenA, tokenB) : (tokenB, tokenA);
    require(token0 != address(0));
    int24 tickSpacing = feeAmountTickSpacing[fee];
    require(tickSpacing != 0);
    require(getPool[token0][token1][fee] == address(0));
    pool = deploy(address(this), token0, token1, fee, tickSpacing);
    getPool[token0][token1][fee] = pool;
    // populate mapping in the reverse direction, deliberate choice to avoid the cost of comparing addresses
    getPool[token1][token0][fee] = pool;
    emit PoolCreated(token0, token1, fee, tickSpacing, pool);
}

因为传入的tokenA和tokenB是无序的，首先对tokenA、tokenB排序，确保tokenA < tokenB。

通过手续费等级获取对应的tickSpacing：

int24 tickSpacing = feeAmountTickSpacing[fee];

我们在《深入理解Uniswap v3白皮书》中介绍过tickSpacing的作用，每个手续费等级对应一个tickSpacing，只有被tickSpacing整除的tick才允许被初始化，tickSpacing越大，每个tick流动性越多，tick之间滑点越大，但会节省跨tick操作的gas。这里作为Pool的参数保存起来。

确认该交易对对应的手续费等级没有创建过：

require(getPool[token0][token1][fee] == address(0));

创建（部署）交易对合约：

pool = deploy(address(this), token0, token1, fee, tickSpacing);

deploy代码如下：

/// @dev Deploys a pool with the given parameters by transiently setting the parameters storage slot and then
/// clearing it after deploying the pool.
/// @param factory The contract address of the Uniswap V3 factory
/// @param token0 The first token of the pool by address sort order
/// @param token1 The second token of the pool by address sort order
/// @param fee The fee collected upon every swap in the pool, denominated in hundredths of a bip
/// @param tickSpacing The spacing between usable ticks
function deploy(
    address factory,
    address token0,
    address token1,
    uint24 fee,
    int24 tickSpacing
) internal returns (address pool) {
    parameters = Parameters({factory: factory, token0: token0, token1: token1, fee: fee, tickSpacing: tickSpacing});
    pool = address(new UniswapV3Pool{salt: keccak256(abi.encode(token0, token1, fee))}());
    delete parameters;
}

我们在Uniswap v2中提到，为了确保交易对合约地址的可计算性和唯一性，Uniswap v2使用CREATE2操作码创建交易对合约；从Solidity 0.6.2版本开始（Github PR），支持在new方法中传递salt参数实现CREATE2功能；salt参数确保了合约地址的唯一性和可计算性。从代码可知，Uniswap v3交易对合约使用token0、token1、fee唯一确定一个交易对合约，比如，根据ETH-USDC 0.05%手续费（以及工厂合约地址、初始化代码hash）等信息，可计算交易对合约地址。

最后，保存交易对合约地址到getPool变量中：

getPool[token0][token1][fee] = pool;
// populate mapping in the reverse direction, deliberate choice to avoid the cost of comparing addresses
getPool[token1][token0][fee] = pool;

setOwner

设置工厂合约owner，owner具有以下权限：

setOwner：修改owner
enableFeeAmount：添加手续费等级
setFeeProtocol：修改某个交易对的协议手续费比例
collectProtocol：收集某个交易对的协议手续费

首先判断请求由当前owner发起，确认后修改owner：

/// @inheritdoc IUniswapV3Factory
function setOwner(address _owner) external override {
    require(msg.sender == owner);
    emit OwnerChanged(owner, _owner);
    owner = _owner;
}

enableFeeAmount

Uniswap v3默认支持三种手续费等级：0.05%、0.30%和1.00%，对应的fee值分别为500、3000和10000；fee的基本单位是百分之一基点，即0.01 bp = $10^{-6}$。

手续费百分比计算公式为：

$$ f_{ratio} = \frac{fee}{1,000,000} $$

/// @inheritdoc IUniswapV3Factory
function enableFeeAmount(uint24 fee, int24 tickSpacing) public override {
    require(msg.sender == owner);
    require(fee &lt; 1000000);
    // tick spacing is capped at 16384 to prevent the situation where tickSpacing is so large that
    // TickBitmap#nextInitializedTickWithinOneWord overflows int24 container from a valid tick
    // 16384 ticks represents a >5x price change with ticks of 1 bips
    require(tickSpacing > 0 && tickSpacing &lt; 16384);
    require(feeAmountTickSpacing[fee] == 0);

    feeAmountTickSpacing[fee] = tickSpacing;
    emit FeeAmountEnabled(fee, tickSpacing);
}

UniswapV3Pool.sol

这是Uniswap v3的主要代码，定义了交易对池子的功能：

initialize：初始化交易对
mint：添加流动性
burn：移除流动性
swap：交换代币
flash：闪电贷
collect：取回代币
increaseObservationCardinalityNext：扩展预言机空间
observe：获取预言机数据

此外，factory（工厂合约）owner还可以调用以下两个方法：

setFeeProtocol：修改某个交易对的协议手续费比例
collectProtocol：收集某个交易对的协议手续费

initialize

创建完交易对后，需要调用initialize方法初始化合约，才能正常使用交易对功能。

该方法初始化slot0变量：

/// @inheritdoc IUniswapV3PoolActions
/// @dev not locked because it initializes unlocked
function initialize(uint160 sqrtPriceX96) external override {
    require(slot0.sqrtPriceX96 == 0, 'AI');

    int24 tick = TickMath.getTickAtSqrtRatio(sqrtPriceX96);

    (uint16 cardinality, uint16 cardinalityNext) = observations.initialize(_blockTimestamp());

    slot0 = Slot0({
        sqrtPriceX96: sqrtPriceX96,
        tick: tick,
        observationIndex: 0,
        observationCardinality: cardinality,
        observationCardinalityNext: cardinalityNext,
        feeProtocol: 0,
        unlocked: true
    });

    emit Initialize(sqrtPriceX96, tick);
}

slot0定义如下：

sqrtPriceX96：交易对当前的开根号价格 $\sqrt{P}$
tick：当前 $\sqrt{P}$ 对应的tick，使用getTickAtSqrtRatio计算得出
observationIndex：最近更新的（预言机）观测点数组序号
observationCardinality：（预言机）观测点数组容量，最大65536，初始时为1
observationCardinalityNext：下一个（预言机）观测点数组容量，如果手动扩容容量，会更新这个值，初始时为1
feeProtocol：协议手续费比例，可以分别为token0和token1设置交易手续费中分给协议的比例
unlocked：当前交易对合约是否非锁定状态

mint

该方法实现添加流动性功能。实际上，首次添加流动性和后续增加流动性，都会使用该方法。

mint方法参数如下：

recipient：头寸接收者（owner）
tickLower：流动性区间低点
tickUpper：流动性区间高点
amount：流动性数量
data：回调参数

/// @inheritdoc IUniswapV3PoolActions
/// @dev noDelegateCall is applied indirectly via _modifyPosition
function mint(
    address recipient,
    int24 tickLower,
    int24 tickUpper,
    uint128 amount,
    bytes calldata data
) external override lock returns (uint256 amount0, uint256 amount1) {
    require(amount > 0);
    (, int256 amount0Int, int256 amount1Int) =
        _modifyPosition(
            ModifyPositionParams({
                owner: recipient,
                tickLower: tickLower,
                tickUpper: tickUpper,
                liquidityDelta: int256(amount).toInt128()
            })
        );

    amount0 = uint256(amount0Int);
    amount1 = uint256(amount1Int);

    uint256 balance0Before;
    uint256 balance1Before;
    if (amount0 > 0) balance0Before = balance0();
    if (amount1 > 0) balance1Before = balance1();
    IUniswapV3MintCallback(msg.sender).uniswapV3MintCallback(amount0, amount1, data);
    if (amount0 > 0) require(balance0Before.add(amount0) &lt;= balance0(), 'M0');
    if (amount1 > 0) require(balance1Before.add(amount1) &lt;= balance1(), 'M1');

    emit Mint(msg.sender, recipient, tickLower, tickUpper, amount, amount0, amount1);
}

mint方法主要的逻辑都在_modifyPosition中，其返回的amount0Int和amount1Int表示:如果添加amount数量的流动性，则需要分别向交易对合约转入的token0和token1的代币数量。

调用方需在uniswapV3MintCallback完成代币的转入操作；调用mint方法的合约需要实现IUniswapV3MintCallback接口，Uniswap v3在periphery合约的NonfungiblePositionManager.sol实现该接口。

因为mint调用方需要实现接口方法，因此个人ETH账户（EOA）无法调用该方法。

IUniswapV3MintCallback(msg.sender).uniswapV3MintCallback(amount0, amount1, data);

_modifyPosition

继续来看_modifyPosition：

/// @dev Effect some changes to a position
/// @param params the position details and the change to the position's liquidity to effect
/// @return position a storage pointer referencing the position with the given owner and tick range
/// @return amount0 the amount of token0 owed to the pool, negative if the pool should pay the recipient
/// @return amount1 the amount of token1 owed to the pool, negative if the pool should pay the recipient
function _modifyPosition(ModifyPositionParams memory params)
    private
    noDelegateCall
    returns (
        Position.Info storage position,
        int256 amount0,
        int256 amount1
    )
{
    checkTicks(params.tickLower, params.tickUpper);

    Slot0 memory _slot0 = slot0; // SLOAD for gas optimization

    position = _updatePosition(
        params.owner,
        params.tickLower,
        params.tickUpper,
        params.liquidityDelta,
        _slot0.tick
    );

先通过_updatePosition更新头寸信息，我们在下一节会具体介绍。

    if (params.liquidityDelta != 0) {
        if (_slot0.tick &lt; params.tickLower) {
            // current tick is below the passed range; liquidity can only become in range by crossing from left to
            // right, when we'll need _more_ token0 (it's becoming more valuable) so user must provide it
            amount0 = SqrtPriceMath.getAmount0Delta(
                TickMath.getSqrtRatioAtTick(params.tickLower),
                TickMath.getSqrtRatioAtTick(params.tickUpper),
                params.liquidityDelta
            );
        } else if (_slot0.tick &lt; params.tickUpper) {
            // current tick is inside the passed range
            uint128 liquidityBefore = liquidity; // SLOAD for gas optimization

            // write an oracle entry
            (slot0.observationIndex, slot0.observationCardinality) = observations.write(
                _slot0.observationIndex,
                _blockTimestamp(),
                _slot0.tick,
                liquidityBefore,
                _slot0.observationCardinality,
                _slot0.observationCardinalityNext
            );

            amount0 = SqrtPriceMath.getAmount0Delta(
                _slot0.sqrtPriceX96,
                TickMath.getSqrtRatioAtTick(params.tickUpper),
                params.liquidityDelta
            );
            amount1 = SqrtPriceMath.getAmount1Delta(
                TickMath.getSqrtRatioAtTick(params.tickLower),
                _slot0.sqrtPriceX96,
                params.liquidityDelta
            );

            liquidity = LiquidityMath.addDelta(liquidityBefore, params.liquidityDelta);
        } else {
            // current tick is above the passed range; liquidity can only become in range by crossing from right to
            // left, when we'll need _more_ token1 (it's becoming more valuable) so user must provide it
            amount1 = SqrtPriceMath.getAmount1Delta(
                TickMath.getSqrtRatioAtTick(params.tickLower),
                TickMath.getSqrtRatioAtTick(params.tickUpper),
                params.liquidityDelta
            );
        }
    }
}

代码的下半部分则主要通过getAmount0Delta和getAmount1Delta计算该流动性需要分别提供的token0和token1的数量，即amount0和amount1。

具体地，当你提供流动性的区间大于当前tick $i_c$ 时，因为tick大小与 $\sqrt{P}$（即 $\sqrt{\frac{y}{x}}$ ）成正比，意味着在大于 $i_c$ 的区间， $x$ 的价值更高（需要更少的 $x$ ），因此添加流动性时需在该部分提供 $x$ 代币，即amount0数量的token0；反之，则提供 $y$ 代币，即amount1的token1。

如下所示：

$$ \begin{cases}i_c, ..., \overbrace{i_l, ..., i_u}^{amount0} & \text{$i_c < i_l$}\ \overbrace{i_l, ...}^{amount1}, i_c, \overbrace{..., i_u}^{amount0} & \text{$i_l \leq i_c < i_u$}\ \overbrace{i_l, ..., i_u}^{amount1}, ..., i_c & \text{$i_u \leq i_c$}\end{cases} $$

其中， $i_l$ , $i_u$ 为提供流动性价格区间的边界， $i_c$ 为当前价格对应的tick。

如果当前价格在区间中，即 $i_l \leq i_c < i_u$ 时，_modifyPosition会记录一次（预言机）观测点数据，因为此时区间的流动性发生了变化，需要记录每流动性的持续时间secondsPerLiquidityCumulativeX128：

// write an oracle entry
(slot0.observationIndex, slot0.observationCardinality) = observations.write(
    _slot0.observationIndex,
    _blockTimestamp(),
    _slot0.tick,
    liquidityBefore,
    _slot0.observationCardinality,
    _slot0.observationCardinalityNext
);

在计算amount0和amount1后，更新当前交易对的全局活跃流动性liquidity：

liquidity = LiquidityMath.addDelta(liquidityBefore, params.liquidityDelta);

这个全局流动性会在swap时用到。

_updatePosition

_modifyPosition中的_updatePosition代码如下：

/// @dev Gets and updates a position with the given liquidity delta
/// @param owner the owner of the position
/// @param tickLower the lower tick of the position's tick range
/// @param tickUpper the upper tick of the position's tick range
/// @param tick the current tick, passed to avoid sloads
function _updatePosition(
    address owner,
    int24 tickLower,
    int24 tickUpper,
    int128 liquidityDelta,
    int24 tick
) private returns (Position.Info storage position) {
    position = positions.get(owner, tickLower, tickUpper);

    uint256 _feeGrowthGlobal0X128 = feeGrowthGlobal0X128; // SLOAD for gas optimization
    uint256 _feeGrowthGlobal1X128 = feeGrowthGlobal1X128; // SLOAD for gas optimization

    // if we need to update the ticks, do it
    bool flippedLower;
    bool flippedUpper;
    if (liquidityDelta != 0) {
        uint32 time = _blockTimestamp();
        (int56 tickCumulative, uint160 secondsPerLiquidityCumulativeX128) =
            observations.observeSingle(
                time,
                0,
                slot0.tick,
                slot0.observationIndex,
                liquidity,
                slot0.observationCardinality
            );

observations.observeSingle计算从最后一次观测点到现在的累积ticktickCumulative和累积每份流动性的持续时间secondsPerLiquidityCumulativeX128。

        flippedLower = ticks.update(
            tickLower,
            tick,
            liquidityDelta,
            _feeGrowthGlobal0X128,
            _feeGrowthGlobal1X128,
            secondsPerLiquidityCumulativeX128,
            tickCumulative,
            time,
            false,
            maxLiquidityPerTick
        );
        flippedUpper = ticks.update(
            tickUpper,
            tick,
            liquidityDelta,
            _feeGrowthGlobal0X128,
            _feeGrowthGlobal1X128,
            secondsPerLiquidityCumulativeX128,
            tickCumulative,
            time,
            true,
            maxLiquidityPerTick
        );
        if (flippedLower) {
            tickBitmap.flipTick(tickLower, tickSpacing);
        }
        if (flippedUpper) {
            tickBitmap.flipTick(tickUpper, tickSpacing);
        }
    }

接着使用ticks.update分别更新tickLower（价格区间低点）和tickUpper（价格区间高点）的状态，具体请参考Tick.update。

如果对应tick的流动性从从0到有，或从有到0，则表示该tick需要被翻转。如果该tick未被标记为初始化，则标记为初始化；否则，将其取消初始化；这里用到tickBitmap.flipTick方法，请参考TickBitmap.flipTick。

    (uint256 feeGrowthInside0X128, uint256 feeGrowthInside1X128) =
        ticks.getFeeGrowthInside(tickLower, tickUpper, tick, _feeGrowthGlobal0X128, _feeGrowthGlobal1X128);

接着，计算该价格区间的累积每流动性手续费。

    position.update(liquidityDelta, feeGrowthInside0X128, feeGrowthInside1X128);

更新头寸（Position）信息，这里主要更新了头寸的应收手续费tokensOwed0和tokensOwed1，以及头寸流动性liquidity，请参考Position.update。

    // clear any tick data that is no longer needed
    if (liquidityDelta &lt; 0) {
        if (flippedLower) {
            ticks.clear(tickLower);
        }
        if (flippedUpper) {
            ticks.clear(tickUpper);
        }
    }
}

如果是移除流动性，并且tick被翻转，则调用clear清空tick状态。

最后，回到mint方法，调用者需要确保在uniswapV3MintCallback方法中，将这里计算出的amount0和amount1数量的token0和token1代币转入交易对合约。

总结mint方法的主要工作如下：

更新价格区间端点（lower, upper）的信息：ticks.update
如果Tick状态翻转，则更新位图的标识位，设置为“已初始化”或“未初始化”：tickBitmap.flipTick
更新头寸（Position）信息：positions.update
如果当前tick位于价格区间中，则：
- 写入一次预言机观测点：observations.write
- 更新全局活跃流动性：liquidity
调用方向交易对合约转账：uniswapV3MintCallback

burn

销毁流动性（burn）的逻辑与添加流动性（mint）几乎完全一样，唯一的区别是liquidityDelta是负的。

/// @inheritdoc IUniswapV3PoolActions
/// @dev noDelegateCall is applied indirectly via _modifyPosition
function burn(
    int24 tickLower,
    int24 tickUpper,
    uint128 amount
) external override lock returns (uint256 amount0, uint256 amount1) {
    (Position.Info storage position, int256 amount0Int, int256 amount1Int) =
        _modifyPosition(
            ModifyPositionParams({
                owner: msg.sender,
                tickLower: tickLower,
                tickUpper: tickUpper,
                liquidityDelta: -int256(amount).toInt128()
            })
        );

    amount0 = uint256(-amount0Int);
    amount1 = uint256(-amount1Int);

    if (amount0 > 0 || amount1 > 0) {
        (position.tokensOwed0, position.tokensOwed1) = (
            position.tokensOwed0 + uint128(amount0),
            position.tokensOwed1 + uint128(amount1)
        );
    }

    emit Burn(msg.sender, tickLower, tickUpper, amount, amount0, amount1);
}

这里与mint使用同一个_modifyPosition方法。

需注意，当销毁（部分）流动性后，代币并没有转回到调用方，而是以未领取代币的形式记在头寸（Position）上。

swap

swap方法是Uniswap v3代码的核心，该方法实现两个代币的交换，从token0交换到token1，或者相反。

相比Uniswap v2的同质化流动性，我们重点关注在swap过程中，价格如何变化，以及如何影响流动性的。

先来看swap方法的几个参数：

recipient：交易后的代币接收者
zeroForOne：如果从token0交换token1则为true，从token1交换token0则为false
amountSpecified: 指定的代币数量，如果为正，表示希望输入的代币数量；如果为负，则表示希望输出的代币数量
sqrtPriceLimitX96：能够承受的价格上限（或下限），格式为Q64.96；如果从token0到token1，则表示swap过程中的价格下限；如果从token1到token0，则表示价格上限；如果价格超过该值，则swap失败
data：回调参数

/// @inheritdoc IUniswapV3PoolActions
function swap(
    address recipient,
    bool zeroForOne,
    int256 amountSpecified,
    uint160 sqrtPriceLimitX96,
    bytes calldata data
) external override noDelegateCall returns (int256 amount0, int256 amount1) {
    require(amountSpecified != 0, 'AS');

    Slot0 memory slot0Start = slot0;

    require(slot0Start.unlocked, 'LOK');
    require(
        zeroForOne
            ? sqrtPriceLimitX96 &lt; slot0Start.sqrtPriceX96 && sqrtPriceLimitX96 > TickMath.MIN_SQRT_RATIO
            : sqrtPriceLimitX96 > slot0Start.sqrtPriceX96 && sqrtPriceLimitX96 &lt; TickMath.MAX_SQRT_RATIO,
        'SPL'
    );

    slot0.unlocked = false;

    SwapCache memory cache =
        SwapCache({
            liquidityStart: liquidity,
            blockTimestamp: _blockTimestamp(),
            feeProtocol: zeroForOne ? (slot0Start.feeProtocol % 16) : (slot0Start.feeProtocol >> 4),
            secondsPerLiquidityCumulativeX128: 0,
            tickCumulative: 0,
            computedLatestObservation: false
        });

    bool exactInput = amountSpecified > 0;

    SwapState memory state =
        SwapState({
            amountSpecifiedRemaining: amountSpecified,
            amountCalculated: 0,
            sqrtPriceX96: slot0Start.sqrtPriceX96,
            tick: slot0Start.tick,
            feeGrowthGlobalX128: zeroForOne ? feeGrowthGlobal0X128 : feeGrowthGlobal1X128,
            protocolFee: 0,
            liquidity: cache.liquidityStart
        });

上面代码主要是初始化状态相关的。

因为 $\sqrt{P} = \sqrt{\frac{y}{x}}$ ，当zeroForOne = true，即从token0到token1时，swap过程中 Pool 的 $x$ 变多， $y$ 变少，因此 $\sqrt{P}$ 逐渐减小，所以指定的价格极限sqrtPriceLimitX96需要小于当前市场价格sqrtPriceX96。

另外，需要注意几个关键数据：

初始交易价格state.sqrtPriceX96为：slot0.sqrtPriceX96
- 这里并没有使用slot0.tick计算初始价格，因为计算出来的值与slot0.sqrtPriceX96可能不一致，我们在后面代码会看到，slot0.tick不能作为当前价格
初始可用流动性state.liquidity为：liquidity，也就是我们在mint或burn时更新的全局可用流动性

根据zeroForOne和exactInput，可以有四种swap组合：

zeroForOne	exactInput	swap
true	true	输入固定数量`token0`，输出最大数量`token1`
true	false	输入最小数量`token0`，输出固定数量`token1`
false	true	输入固定数量`token1`，输出最大数量`token0`
false	false	输入最小数量`token1`，输出固定数量`token0`

一个完整的swap可以由多个step组成，代码如下：

// continue swapping as long as we haven't used the entire input/output and haven't reached the price limit
while (state.amountSpecifiedRemaining != 0 && state.sqrtPriceX96 != sqrtPriceLimitX96) {
    StepComputations memory step;

    step.sqrtPriceStartX96 = state.sqrtPriceX96;

    (step.tickNext, step.initialized) = tickBitmap.nextInitializedTickWithinOneWord(
        state.tick,
        tickSpacing,
        zeroForOne
    );

    // ensure that we do not overshoot the min/max tick, as the tick bitmap is not aware of these bounds
    if (step.tickNext &lt; TickMath.MIN_TICK) {
        step.tickNext = TickMath.MIN_TICK;
    } else if (step.tickNext > TickMath.MAX_TICK) {
        step.tickNext = TickMath.MAX_TICK;
    }

    // get the price for the next tick
    step.sqrtPriceNextX96 = TickMath.getSqrtRatioAtTick(step.tickNext);

    // compute values to swap to the target tick, price limit, or point where input/output amount is exhausted
    (state.sqrtPriceX96, step.amountIn, step.amountOut, step.feeAmount) = SwapMath.computeSwapStep(
        state.sqrtPriceX96,
        (zeroForOne ? step.sqrtPriceNextX96 &lt; sqrtPriceLimitX96 : step.sqrtPriceNextX96 > sqrtPriceLimitX96)
            ? sqrtPriceLimitX96
            : step.sqrtPriceNextX96,
        state.liquidity,
        state.amountSpecifiedRemaining,
        fee
    );

    if (exactInput) {
        state.amountSpecifiedRemaining -= (step.amountIn + step.feeAmount).toInt256();
        state.amountCalculated = state.amountCalculated.sub(step.amountOut.toInt256());
    } else {
        state.amountSpecifiedRemaining += step.amountOut.toInt256();
        state.amountCalculated = state.amountCalculated.add((step.amountIn + step.feeAmount).toInt256());
    }

    // if the protocol fee is on, calculate how much is owed, decrement feeAmount, and increment protocolFee
    if (cache.feeProtocol > 0) {
        uint256 delta = step.feeAmount / cache.feeProtocol;
        step.feeAmount -= delta;
        state.protocolFee += uint128(delta);
    }

    // update global fee tracker
    if (state.liquidity > 0)
        state.feeGrowthGlobalX128 += FullMath.mulDiv(step.feeAmount, FixedPoint128.Q128, state.liquidity);

    // shift tick if we reached the next price
    if (state.sqrtPriceX96 == step.sqrtPriceNextX96) {
        // if the tick is initialized, run the tick transition
        if (step.initialized) {
            // check for the placeholder value, which we replace with the actual value the first time the swap
            // crosses an initialized tick
            if (!cache.computedLatestObservation) {
                (cache.tickCumulative, cache.secondsPerLiquidityCumulativeX128) = observations.observeSingle(
                    cache.blockTimestamp,
                    0,
                    slot0Start.tick,
                    slot0Start.observationIndex,
                    cache.liquidityStart,
                    slot0Start.observationCardinality
                );
                cache.computedLatestObservation = true;
            }
            int128 liquidityNet =
                ticks.cross(
                    step.tickNext,
                    (zeroForOne ? state.feeGrowthGlobalX128 : feeGrowthGlobal0X128),
                    (zeroForOne ? feeGrowthGlobal1X128 : state.feeGrowthGlobalX128),
                    cache.secondsPerLiquidityCumulativeX128,
                    cache.tickCumulative,
                    cache.blockTimestamp
                );
            // if we're moving leftward, we interpret liquidityNet as the opposite sign
            // safe because liquidityNet cannot be type(int128).min
            if (zeroForOne) liquidityNet = -liquidityNet;

            state.liquidity = LiquidityMath.addDelta(state.liquidity, liquidityNet);
        }

        state.tick = zeroForOne ? step.tickNext - 1 : step.tickNext;
    } else if (state.sqrtPriceX96 != step.sqrtPriceStartX96) {
        // recompute unless we're on a lower tick boundary (i.e. already transitioned ticks), and haven't moved
        state.tick = TickMath.getTickAtSqrtRatio(state.sqrtPriceX96);
    }
}

整理成伪代码（pseudo code）如下：

loop if 剩余代币 != 0 and 当前价格 != 最小（或最大）价格:
    // step
    初始价格 := 上一个step的价格
    下一个tick := 根据当前tick，寻找最近的已初始化的tick，或者本组最后一个未初始化的tick
    目标价格 := 根据下一个tick计算的价格
    交换后的价格, 消耗的输入代币数量, 得到的输出代币数量, 交易手续费 := 完成一步交换(初始价格, 目标价格, 可用流动性, 剩余代币)

    更新 剩余代币
    更新 协议手续费

    if 交换后价格 == 目标价格:
        if tick已初始化：
            价格穿越该tick，更新tick相关字段
            更新 可用流动性

        当前tick := 下一个tick - 1
    else if 交换后价格 != 初始价格:
        当前tick := 根据交换后价格计算tick

首先需要根据当前tick寻找下一个tick，即tickNext，具体逻辑可参考：tickBitmap.nextInitializedTickWithinOneWord。

计算当前step的目标价格：

// get the price for the next tick
step.sqrtPriceNextX96 = TickMath.getSqrtRatioAtTick(step.tickNext);

计算本次（step）交换的输入输出（即执行一次交换）：

// compute values to swap to the target tick, price limit, or point where input/output amount is exhausted
(state.sqrtPriceX96, step.amountIn, step.amountOut, step.feeAmount) = SwapMath.computeSwapStep(
    state.sqrtPriceX96,
    (zeroForOne ? step.sqrtPriceNextX96 &lt; sqrtPriceLimitX96 : step.sqrtPriceNextX96 > sqrtPriceLimitX96)
        ? sqrtPriceLimitX96
        : step.sqrtPriceNextX96,
    state.liquidity,
    state.amountSpecifiedRemaining,
    fee
);

SwapMath.computeSwapStep将根据当前价格、目标价格、可用流动性、可用输入代币等数据，计算本次交换能最多成交的输入代币数量（amountIn），输出代币数量（amountOut），手续费（feeAmount）和成交后价格（sqrtRatioNextX96）。请参考computeSwapStep。

保存本次交易的amountIn和amountOut：

if (exactInput) {
    state.amountSpecifiedRemaining -= (step.amountIn + step.feeAmount).toInt256();
    state.amountCalculated = state.amountCalculated.sub(step.amountOut.toInt256());
} else {
    state.amountSpecifiedRemaining += step.amountOut.toInt256();
    state.amountCalculated = state.amountCalculated.add((step.amountIn + step.feeAmount).toInt256());
}

如果是指定输入代币数量（token0或token1）
- amountSpecifiedRemaining表示（扣除手续费后）剩余可用输入代币数量
- amountCalculated表示已输出代币数量（注意，这里是负值）
如果是指定输出代币数量（token0或token1）
- amountSpecifiedRemaining表示剩余需要输出的代币数量（初始为负值，因此每次交换后需要+= step.amountOut），直到为0
- amountCalculated表示（加入手续费后）已使用的输入代币数量

计算协议手续费：

// if the protocol fee is on, calculate how much is owed, decrement feeAmount, and increment protocolFee
if (cache.feeProtocol > 0) {
    uint256 delta = step.feeAmount / cache.feeProtocol;
    step.feeAmount -= delta;
    state.protocolFee += uint128(delta);
}

如果开启了协议手续费，则从交易手续费中拆出协议手续费。注意，协议手续费的值feeProtocol表示交易手续费的 $\frac{1}{n}$。

// update global fee tracker
if (state.liquidity > 0)
    state.feeGrowthGlobalX128 += FullMath.mulDiv(step.feeAmount, FixedPoint128.Q128, state.liquidity);

计算（每流动性）全局累积手续费。

// shift tick if we reached the next price
if (state.sqrtPriceX96 == step.sqrtPriceNextX96) {
    // if the tick is initialized, run the tick transition
    if (step.initialized) {
        // check for the placeholder value, which we replace with the actual value the first time the swap
        // crosses an initialized tick
        if (!cache.computedLatestObservation) {
            (cache.tickCumulative, cache.secondsPerLiquidityCumulativeX128) = observations.observeSingle(
                cache.blockTimestamp,
                0,
                slot0Start.tick,
                slot0Start.observationIndex,
                cache.liquidityStart,
                slot0Start.observationCardinality
            );
            cache.computedLatestObservation = true;
        }
        int128 liquidityNet =
            ticks.cross(
                step.tickNext,
                (zeroForOne ? state.feeGrowthGlobalX128 : feeGrowthGlobal0X128),
                (zeroForOne ? feeGrowthGlobal1X128 : state.feeGrowthGlobalX128),
                cache.secondsPerLiquidityCumulativeX128,
                cache.tickCumulative,
                cache.blockTimestamp
            );
        // if we're moving leftward, we interpret liquidityNet as the opposite sign
        // safe because liquidityNet cannot be type(int128).min
        if (zeroForOne) liquidityNet = -liquidityNet;

        state.liquidity = LiquidityMath.addDelta(state.liquidity, liquidityNet);
    }

    state.tick = zeroForOne ? step.tickNext - 1 : step.tickNext;
} else if (state.sqrtPriceX96 != step.sqrtPriceStartX96) {
    // recompute unless we're on a lower tick boundary (i.e. already transitioned ticks), and haven't moved
    state.tick = TickMath.getTickAtSqrtRatio(state.sqrtPriceX96);
}

如果本次交换后的价格达到目标价格（即根据寻找的下一个tick计算的价格）：
- 如果该tick已经初始化，则：
  - 通过ticks.cross方法穿越该tick，反向设置相关Outside变量的数据
  - 使用tick净流动性liquidityNet更新可用流动性state.liquidity
    - 由于在mint初始化tick时，tickLower的liquidityNet是正的，即liquidityDelta；tickUpper的liquidityNet是负的，即-liquidityDelta；因此这里需要根据zeroForOne的值来调整liquidityNet的正负
    - 当zeroForOne = true时，随着交易的进行， Pool 中 $x$ 变多， $y$ 变少，价格 $\sqrt{P}$ 逐渐变小，tick朝 lower 方向移动，如果穿越了tickLower，意味着离开区间，因此需要减少流动性；反之，如果穿越了tickUpper，意味着进入区间，因此需要增加流动性；即都使用 -liquidityNet
    - 当zeroForOne = false时，随着交易的进行， Pool 中 $y$ 变多， $x$ 变少，价格 $\sqrt{P}$ 逐渐变大，tick朝 upper 方向移动，如果穿越了tickUpper，意味着离开区间，因此需要减少流动性；反之，如果穿越了tickLower，意味着进入区间，因此需要增加流动性；即都使用 liquidityNet
- 移动当前tick到下一个tick
如果交换后的价格没有达到本次目标价格，但是又不等于初始价格，即表示此时交易结束：
- 使用交换后的价格计算最新的tick值

重复上述step，直到交换完全结束。

完成交换后，更新全局状态：

// update tick and write an oracle entry if the tick change
if (state.tick != slot0Start.tick) {
    (uint16 observationIndex, uint16 observationCardinality) =
        observations.write(
            slot0Start.observationIndex,
            cache.blockTimestamp,
            slot0Start.tick,
            cache.liquidityStart,
            slot0Start.observationCardinality,
            slot0Start.observationCardinalityNext
        );
    (slot0.sqrtPriceX96, slot0.tick, slot0.observationIndex, slot0.observationCardinality) = (
        state.sqrtPriceX96,
        state.tick,
        observationIndex,
        observationCardinality
    );
} else {
    // otherwise just update the price
    slot0.sqrtPriceX96 = state.sqrtPriceX96;
}

如果交换后的tick与交换前的tick不同：
- 记录一次（预言机）观测点数据，因为tickCumulative发生了改变
- 更新slot0.sqrtPriceX96, slot0.tick等值，注意此时sqrtPriceX96与tick并不一定对应，sqrtPriceX96才能准确反映当前价格
如果交换前后tick值相同，则只需要修改价格：
- 更新slot0.sqrtPriceX96

同样，如果全局流动性发生改变，则更新liquidity：

// update liquidity if it changed
if (cache.liquidityStart != state.liquidity) liquidity = state.liquidity;

更新累积手续费和协议手续费：

// update fee growth global and, if necessary, protocol fees
// overflow is acceptable, protocol has to withdraw before it hits type(uint128).max fees
if (zeroForOne) {
    feeGrowthGlobal0X128 = state.feeGrowthGlobalX128;
    if (state.protocolFee > 0) protocolFees.token0 += state.protocolFee;
} else {
    feeGrowthGlobal1X128 = state.feeGrowthGlobalX128;
    if (state.protocolFee > 0) protocolFees.token1 += state.protocolFee;
}

注意，如果是从token0交换token1，则只能收取token0作为手续费；反之，只能收取token1作为手续费。

(amount0, amount1) = zeroForOne == exactInput
    ? (amountSpecified - state.amountSpecifiedRemaining, state.amountCalculated)
    : (state.amountCalculated, amountSpecified - state.amountSpecifiedRemaining);

计算本次交换需要的具体amount0和amount1。

// do the transfers and collect payment
if (zeroForOne) {
    if (amount1 &lt; 0) TransferHelper.safeTransfer(token1, recipient, uint256(-amount1));

    uint256 balance0Before = balance0();
    IUniswapV3SwapCallback(msg.sender).uniswapV3SwapCallback(amount0, amount1, data);
    require(balance0Before.add(uint256(amount0)) &lt;= balance0(), 'IIA');
} else {
    if (amount0 &lt; 0) TransferHelper.safeTransfer(token0, recipient, uint256(-amount0));

    uint256 balance1Before = balance1();
    IUniswapV3SwapCallback(msg.sender).uniswapV3SwapCallback(amount0, amount1, data);
    require(balance1Before.add(uint256(amount1)) &lt;= balance1(), 'IIA');
}

emit Swap(msg.sender, recipient, amount0, amount1, state.sqrtPriceX96, state.liquidity, state.tick);
slot0.unlocked = true;

合约将输出代币转账给recipient，同时，调用方需要在uniswapV3SwapCallback方法将输入代币转给交易对合约：

至此，整个swap流程就结束了。

flash

本方法实现Uniswap v3闪电贷功能。

方法参数：

recipient：闪电贷接收者
amount0：借出token0的数量
amount1：借出token1的数量
data：回调方法参数

/// @inheritdoc IUniswapV3PoolActions
function flash(
    address recipient,
    uint256 amount0,
    uint256 amount1,
    bytes calldata data
) external override lock noDelegateCall {
    uint128 _liquidity = liquidity;
    require(_liquidity > 0, 'L');

    uint256 fee0 = FullMath.mulDivRoundingUp(amount0, fee, 1e6);
    uint256 fee1 = FullMath.mulDivRoundingUp(amount1, fee, 1e6);

闪电贷手续费与swap手续费相同，都是 $\frac{fee}{10^6}$。

    uint256 balance0Before = balance0();
    uint256 balance1Before = balance1();

    if (amount0 > 0) TransferHelper.safeTransfer(token0, recipient, amount0);
    if (amount1 > 0) TransferHelper.safeTransfer(token1, recipient, amount1);

    IUniswapV3FlashCallback(msg.sender).uniswapV3FlashCallback(fee0, fee1, data);

    uint256 balance0After = balance0();
    uint256 balance1After = balance1();

    require(balance0Before.add(fee0) &lt;= balance0After, 'F0');
    require(balance1Before.add(fee1) &lt;= balance1After, 'F1');

向收款人转入贷出的代币数量，flash方法的调用方需实现IUniswapV3FlashCallback.uniswapV3FlashCallback接口方法，并在该方法中归还代币，包含手续费。

    // sub is safe because we know balanceAfter is gt balanceBefore by at least fee
    uint256 paid0 = balance0After - balance0Before;
    uint256 paid1 = balance1After - balance1Before;

    if (paid0 > 0) {
        uint8 feeProtocol0 = slot0.feeProtocol % 16;
        uint256 fees0 = feeProtocol0 == 0 ? 0 : paid0 / feeProtocol0;
        if (uint128(fees0) > 0) protocolFees.token0 += uint128(fees0);
        feeGrowthGlobal0X128 += FullMath.mulDiv(paid0 - fees0, FixedPoint128.Q128, _liquidity);
    }
    if (paid1 > 0) {
        uint8 feeProtocol1 = slot0.feeProtocol >> 4;
        uint256 fees1 = feeProtocol1 == 0 ? 0 : paid1 / feeProtocol1;
        if (uint128(fees1) > 0) protocolFees.token1 += uint128(fees1);
        feeGrowthGlobal1X128 += FullMath.mulDiv(paid1 - fees1, FixedPoint128.Q128, _liquidity);
    }

    emit Flash(msg.sender, recipient, amount0, amount1, paid0, paid1);
}

根据收取的手续费，计算协议手续费（注意，token0和token1的协议手续费是单独设置的，参考：setFeeProtocol），最后更新protocolFees和feeGrowthGlobal1X128。

collect

该方法实现取回代币功能，包括销毁流动性记录的代币和手续费代币。

参数如下：

recipient：代币接收者
tickLower：头寸低点
tickUpper：头寸高点
amount0Requested：请求取回的token0数量
amount1Requested：请求取回的token1数量

/// @inheritdoc IUniswapV3PoolActions
function collect(
    address recipient,
    int24 tickLower,
    int24 tickUpper,
    uint128 amount0Requested,
    uint128 amount1Requested
) external override lock returns (uint128 amount0, uint128 amount1) {
    // we don't need to checkTicks here, because invalid positions will never have non-zero tokensOwed{0,1}
    Position.Info storage position = positions.get(msg.sender, tickLower, tickUpper);

    amount0 = amount0Requested > position.tokensOwed0 ? position.tokensOwed0 : amount0Requested;
    amount1 = amount1Requested > position.tokensOwed1 ? position.tokensOwed1 : amount1Requested;

    if (amount0 > 0) {
        position.tokensOwed0 -= amount0;
        TransferHelper.safeTransfer(token0, recipient, amount0);
    }
    if (amount1 > 0) {
        position.tokensOwed1 -= amount1;
        TransferHelper.safeTransfer(token1, recipient, amount1);
    }

    emit Collect(msg.sender, recipient, tickLower, tickUpper, amount0, amount1);
}

上述代码比较简单，这里不再展开。需要注意，如果希望取回所有代币，则需要指定比tokensOwned更大的数，比如可以使用type(uint128).max。

increaseObservationCardinalityNext

扩容预言机观测点的可写入空间，该方法调用Oracle.sol的grow方法实现扩容。

/// @inheritdoc IUniswapV3PoolActions
function increaseObservationCardinalityNext(uint16 observationCardinalityNext)
    external
    override
    lock
    noDelegateCall
{
    uint16 observationCardinalityNextOld = slot0.observationCardinalityNext; // for the event
    uint16 observationCardinalityNextNew =
        observations.grow(observationCardinalityNextOld, observationCardinalityNext);
    slot0.observationCardinalityNext = observationCardinalityNextNew;
    if (observationCardinalityNextOld != observationCardinalityNextNew)
        emit IncreaseObservationCardinalityNext(observationCardinalityNextOld, observationCardinalityNextNew);
}

observe

批量获取指定时间的观测点数据，该方法调用Oracle.sol的observe实现。

/// @inheritdoc IUniswapV3PoolDerivedState
function observe(uint32[] calldata secondsAgos)
    external
    view
    override
    noDelegateCall
    returns (int56[] memory tickCumulatives, uint160[] memory secondsPerLiquidityCumulativeX128s)
{
    return
        observations.observe(
            _blockTimestamp(),
            secondsAgos,
            slot0.tick,
            slot0.observationIndex,
            liquidity,
            slot0.observationCardinality
        );
}

setFeeProtocol

设置协议手续费的比例，该方法仅允许工厂合约的owner执行。

注意，需要分别设置token0和token1的协议手续费比例，该比例是交易手续费的占比，合法值为0（不开启协议手续费）或者 $4 \leq n \leq 10$，也就是可以设置协议手续费为交易手续费的 $\frac{1}{n}$。

/// @inheritdoc IUniswapV3PoolOwnerActions
function setFeeProtocol(uint8 feeProtocol0, uint8 feeProtocol1) external override lock onlyFactoryOwner {
    require(
        (feeProtocol0 == 0 || (feeProtocol0 >= 4 && feeProtocol0 &lt;= 10)) &&
            (feeProtocol1 == 0 || (feeProtocol1 >= 4 && feeProtocol1 &lt;= 10))
    );
    uint8 feeProtocolOld = slot0.feeProtocol;
    slot0.feeProtocol = feeProtocol0 + (feeProtocol1 &lt;&lt; 4);
    emit SetFeeProtocol(feeProtocolOld % 16, feeProtocolOld >> 4, feeProtocol0, feeProtocol1);
}

slot0.feeProtocol类型为uint8，保存两种代币的协议手续费比例，高4位为token1，低4位为token0：

$$ slot0.feeProtocol = \overbrace{0000}^{fee1}\overbrace{0000}^{fee0} $$

因此，feeProtocolOld % 16表示token0的协议手续费比例fee0，feeProtocolOld >> 4表示token1的协议手续费比例fee1。

collectProtocol

取回协议手续费，该方法仅允许工厂合约的owner执行。

协议手续费有两个来源：

swap产生的交易手续费
flash产生的闪电贷手续费

/// @inheritdoc IUniswapV3PoolOwnerActions
function collectProtocol(
    address recipient,
    uint128 amount0Requested,
    uint128 amount1Requested
) external override lock onlyFactoryOwner returns (uint128 amount0, uint128 amount1) {
    amount0 = amount0Requested > protocolFees.token0 ? protocolFees.token0 : amount0Requested;
    amount1 = amount1Requested > protocolFees.token1 ? protocolFees.token1 : amount1Requested;

    if (amount0 > 0) {
        if (amount0 == protocolFees.token0) amount0--; // ensure that the slot is not cleared, for gas savings
        protocolFees.token0 -= amount0;
        TransferHelper.safeTransfer(token0, recipient, amount0);
    }
    if (amount1 > 0) {
        if (amount1 == protocolFees.token1) amount1--; // ensure that the slot is not cleared, for gas savings
        protocolFees.token1 -= amount1;
        TransferHelper.safeTransfer(token1, recipient, amount1);
    }

    emit CollectProtocol(msg.sender, recipient, amount0, amount1);
}

上述代码比较简单，此处不再展开。

Tick.sol

Tick.sol管理Tick内部状态。

tickSpacingToMaxLiquidityPerTick

根据tickSpacing计算每个tick最大流动性，只有能够被tickSpacing整除的tick才能够存放流动性：

/// @notice Derives max liquidity per tick from given tick spacing
/// @dev Executed within the pool constructor
/// @param tickSpacing The amount of required tick separation, realized in multiples of `tickSpacing`
///     e.g., a tickSpacing of 3 requires ticks to be initialized every 3rd tick i.e., ..., -6, -3, 0, 3, 6, ...
/// @return The max liquidity per tick
function tickSpacingToMaxLiquidityPerTick(int24 tickSpacing) internal pure returns (uint128) {
    int24 minTick = (TickMath.MIN_TICK / tickSpacing) * tickSpacing;
    int24 maxTick = (TickMath.MAX_TICK / tickSpacing) * tickSpacing;
    uint24 numTicks = uint24((maxTick - minTick) / tickSpacing) + 1;
    return type(uint128).max / numTicks;
}

getFeeGrowthInside

计算两个tick区间内部的每流动性累积手续费，该方法实现白皮书公式6.17-6.19:

$$ f_a(i) = \begin{cases} f_g - f_o(i) & \text{$i_c \geq i$}\ f_o(i) & \text{$i_c < i$} \end{cases} \quad \text{(6.17)} $$

$$ f_b(i) = \begin{cases} f_o(i) & \text{$i_c \geq i$}\ f_g - f_o(i) & \text{$i_c < i$}\end{cases} \quad \text{(6.18)} $$

$$ f_r = f_g - f_b(i_l) - f_a(i_u) \quad \text{(6.19)} $$

代码如下：

/// @notice Retrieves fee growth data
/// @param self The mapping containing all tick information for initialized ticks
/// @param tickLower The lower tick boundary of the position
/// @param tickUpper The upper tick boundary of the position
/// @param tickCurrent The current tick
/// @param feeGrowthGlobal0X128 The all-time global fee growth, per unit of liquidity, in token0
/// @param feeGrowthGlobal1X128 The all-time global fee growth, per unit of liquidity, in token1
/// @return feeGrowthInside0X128 The all-time fee growth in token0, per unit of liquidity, inside the position's tick boundaries
/// @return feeGrowthInside1X128 The all-time fee growth in token1, per unit of liquidity, inside the position's tick boundaries
function getFeeGrowthInside(
    mapping(int24 => Tick.Info) storage self,
    int24 tickLower,
    int24 tickUpper,
    int24 tickCurrent,
    uint256 feeGrowthGlobal0X128,
    uint256 feeGrowthGlobal1X128
) internal view returns (uint256 feeGrowthInside0X128, uint256 feeGrowthInside1X128) {
    Info storage lower = self[tickLower];
    Info storage upper = self[tickUpper];

    // calculate fee growth below
    uint256 feeGrowthBelow0X128;
    uint256 feeGrowthBelow1X128;
    if (tickCurrent >= tickLower) {
        feeGrowthBelow0X128 = lower.feeGrowthOutside0X128;
        feeGrowthBelow1X128 = lower.feeGrowthOutside1X128;
    } else {
        feeGrowthBelow0X128 = feeGrowthGlobal0X128 - lower.feeGrowthOutside0X128;
        feeGrowthBelow1X128 = feeGrowthGlobal1X128 - lower.feeGrowthOutside1X128;
    }

    // calculate fee growth above
    uint256 feeGrowthAbove0X128;
    uint256 feeGrowthAbove1X128;
    if (tickCurrent &lt; tickUpper) {
        feeGrowthAbove0X128 = upper.feeGrowthOutside0X128;
        feeGrowthAbove1X128 = upper.feeGrowthOutside1X128;
    } else {
        feeGrowthAbove0X128 = feeGrowthGlobal0X128 - upper.feeGrowthOutside0X128;
        feeGrowthAbove1X128 = feeGrowthGlobal1X128 - upper.feeGrowthOutside1X128;
    }

    feeGrowthInside0X128 = feeGrowthGlobal0X128 - feeGrowthBelow0X128 - feeGrowthAbove0X128;
    feeGrowthInside1X128 = feeGrowthGlobal1X128 - feeGrowthBelow1X128 - feeGrowthAbove1X128;
}

首先根据当前tickCurrent，分别计算tickLower和tickUpper的 $f_a$ , $f_b$ ，最后计算出区间内手续费 $f_r$：

$$ \underbrace{\overbrace{..., i_l - 1}^{f_b(i_l)}, \overbrace{i_l, i_l + 1, ..., i_u - 1, i_u}^{f_r}, \overbrace{i_u + 1, ...}^{f_a(iu)}}{f_g} $$

为什么需要计算区间内每流动性累计手续费呢？因为每个头寸（Position）会在mint/burn时根据该值计算自己的应收手续费：

$$ liquidityDelta \cdot (feeGrowthInside - feeGrowthInsideLast) $$

update

更新tick状态，并返回该tick是否翻转flipped：

/// @notice Updates a tick and returns true if the tick was flipped from initialized to uninitialized, or vice versa
/// @param self The mapping containing all tick information for initialized ticks
/// @param tick The tick that will be updated
/// @param tickCurrent The current tick
/// @param liquidityDelta A new amount of liquidity to be added (subtracted) when tick is crossed from left to right (right to left)
/// @param feeGrowthGlobal0X128 The all-time global fee growth, per unit of liquidity, in token0
/// @param feeGrowthGlobal1X128 The all-time global fee growth, per unit of liquidity, in token1
/// @param secondsPerLiquidityCumulativeX128 The all-time seconds per max(1, liquidity) of the pool
/// @param tickCumulative The tick * time elapsed since the pool was first initialized
/// @param time The current block timestamp cast to a uint32
/// @param upper true for updating a position's upper tick, or false for updating a position's lower tick
/// @param maxLiquidity The maximum liquidity allocation for a single tick
/// @return flipped Whether the tick was flipped from initialized to uninitialized, or vice versa
function update(
    mapping(int24 => Tick.Info) storage self,
    int24 tick,
    int24 tickCurrent,
    int128 liquidityDelta,
    uint256 feeGrowthGlobal0X128,
    uint256 feeGrowthGlobal1X128,
    uint160 secondsPerLiquidityCumulativeX128,
    int56 tickCumulative,
    uint32 time,
    bool upper,
    uint128 maxLiquidity
) internal returns (bool flipped) {
    Tick.Info storage info = self[tick];

    uint128 liquidityGrossBefore = info.liquidityGross;
    uint128 liquidityGrossAfter = LiquidityMath.addDelta(liquidityGrossBefore, liquidityDelta);

    require(liquidityGrossAfter &lt;= maxLiquidity, 'LO');

    flipped = (liquidityGrossAfter == 0) != (liquidityGrossBefore == 0);

如果tick从无流动性到有流动性，或者从有流动性变成无流动性，则表示tick需要翻转flipped。

    if (liquidityGrossBefore == 0) {
        // by convention, we assume that all growth before a tick was initialized happened _below_ the tick
        if (tick &lt;= tickCurrent) {
            info.feeGrowthOutside0X128 = feeGrowthGlobal0X128;
            info.feeGrowthOutside1X128 = feeGrowthGlobal1X128;
            info.secondsPerLiquidityOutsideX128 = secondsPerLiquidityCumulativeX128;
            info.tickCumulativeOutside = tickCumulative;
            info.secondsOutside = time;
        }
        info.initialized = true;
    }

如果tick之前没有流动性，则进行初始化；对于小于当前tickCurrent的tick，设置Outside等变量。

    info.liquidityGross = liquidityGrossAfter;

liquidityGross表示总流动性，用于判断tick是否需要初始化：

如果mint，则增加流动性；如果burn，则减少流动性
该变量与tick在不同头寸中是否作为边界低点或高点无关，只与mint或burn操作有关
如果一个tick同时被用作tickLower和tickUpper，则其liquidityNet可能是0，但liquidityGross仍然会大于0，因此不需要再次初始化

    // when the lower (upper) tick is crossed left to right (right to left), liquidity must be added (removed)
    info.liquidityNet = upper
        ? int256(info.liquidityNet).sub(liquidityDelta).toInt128()
        : int256(info.liquidityNet).add(liquidityDelta).toInt128();
}

liquidityNet表示净流动性，当swap穿越tick时，用于更新全局可用流动性liquidity：

如果作为tickLower，即边界低点（左边界点），则增加liquidityDelta（mint时为正，burn时为负）
如果作为tickUpper，即边界高点（右边界点），则减少liquidityDelta（mint时为正，burn时为负）

clear

当tick翻转后，如果没有流动性关联该tick，即liquidityGross = 0，则清空tick状态：

/// @notice Clears tick data
/// @param self The mapping containing all initialized tick information for initialized ticks
/// @param tick The tick that will be cleared
function clear(mapping(int24 => Tick.Info) storage self, int24 tick) internal {
    delete self[tick];
}

cross

当tick被穿越时，需要翻转Outside等变量的方向，如白皮书公式6.20:

$$ f_o(i) := f_g - f_o(i) \quad \text{(6.20)} $$

这些变量在getFeeGrowthInside等方法被用到。

/// @notice Transitions to next tick as needed by price movement
/// @param self The mapping containing all tick information for initialized ticks
/// @param tick The destination tick of the transition
/// @param feeGrowthGlobal0X128 The all-time global fee growth, per unit of liquidity, in token0
/// @param feeGrowthGlobal1X128 The all-time global fee growth, per unit of liquidity, in token1
/// @param secondsPerLiquidityCumulativeX128 The current seconds per liquidity
/// @param tickCumulative The tick * time elapsed since the pool was first initialized
/// @param time The current block.timestamp
/// @return liquidityNet The amount of liquidity added (subtracted) when tick is crossed from left to right (right to left)
function cross(
    mapping(int24 => Tick.Info) storage self,
    int24 tick,
    uint256 feeGrowthGlobal0X128,
    uint256 feeGrowthGlobal1X128,
    uint160 secondsPerLiquidityCumulativeX128,
    int56 tickCumulative,
    uint32 time
) internal returns (int128 liquidityNet) {
    Tick.Info storage info = self[tick];
    info.feeGrowthOutside0X128 = feeGrowthGlobal0X128 - info.feeGrowthOutside0X128;
    info.feeGrowthOutside1X128 = feeGrowthGlobal1X128 - info.feeGrowthOutside1X128;
    info.secondsPerLiquidityOutsideX128 = secondsPerLiquidityCumulativeX128 - info.secondsPerLiquidityOutsideX128;
    info.tickCumulativeOutside = tickCumulative - info.tickCumulativeOutside;
    info.secondsOutside = time - info.secondsOutside;
    liquidityNet = info.liquidityNet;
}

TickMath.sol

TickMath主要包含两个方法：

getSqrtRatioAtTick：根据tick计算开根号价格 $\sqrt{P}$
getTickAtSqrtRatio：根据开根号价格 $\sqrt{P}$ 计算tick

getSqrtRatioAtTick

该方法对应白皮书公式6.2:

$$ \sqrt{p}(i) = \sqrt{1.0001}^i = 1.0001^{\frac{i}{2}} $$

其中， $i$ 即为tick。

因为Uniswap v3支持的价格（ $\frac{token1}{token0}$ ）区间为 $[2^{-128}, 2^{128}]$ ，根据白皮书公式6.1:

$$ p(i) = 1.0001^i $$

因此，对应的最大tick（MAX_TICK）为：

$$ i = \lfloor log{1.0001}{p(i)} \rfloor = \lfloor log{1.0001}{2^{128}} \rfloor = \lfloor 887272.7517970635 \rfloor = 887272 $$

最小tick（MIN_TICK）为：

$$ i = \lceil log_{1.0001}{2^{-128}} \rceil = \lceil -887272.7517970635 \rceil = -887272 $$

假设 $i$ $\geq 0$，对于一个给定的tick $i$，它总可以表示为二进制，因此以下式子总是成立：

$$ \begin{cases} i = \sum_{n=0}^{19}{(x_n \cdot 2^n)} = x_0 \cdot 1 + x_1 \cdot 2 + x2 \cdot 4 + ... + x{19}\cdot 524288 \ \forall x_n \in {0, 1} \end{cases} \quad \text{(1.1)} $$

其中， $x_n$ 为 $i$ 的二进制位。如 $i=6$ ，其对应的二进制为：000000000000000000000110，则 $x_1 = 1, x_2 = 1$ ，其余 $x_n$ 均为0。

同样可以推出 $i < 0$ 也可以用类似的公式表示。

我们先看 $i < 0$ 的情况：

如果 $i < 0$，则：

$$ \sqrt{p}(i) = 1.0001^{\frac{i}{2}} = 1.0001^{-\frac{|i|}{2}} = \frac{1}{1.0001^{\frac{|i|}{2}}} = \frac{1}{1.0001^{\frac{1}{2}(\sum_{n=0}^{19}{(x_n \cdot 2^n)})}} \ = \frac{1}{1.0001^{\frac{1}{2} \cdot x_0}} \cdot \frac{1}{1.0001^{\frac{2}{2} \cdot x_1}} \cdot \frac{1}{1.0001^{\frac{4}{2} \cdot x2}} \cdot ... \cdot \frac{1}{1.0001^{\frac{524288}{2} \cdot x{19}}} $$

根据二进制位 $x_n$ 的值，可以总结如下：

$$ \frac{1}{1.0001^{\frac{x_n \cdot 2^n}{2}}} \begin{cases} = 1 & \text{$x_n = 0, n \geq 0, i < 0$}\ < 1 & \text{$x_n = 1, n \geq 0, i < 0$} \end{cases} $$

为了最小化精度误差，在计算过程中，使用Q128.128（128位定点数）表示中间价格，对于每一个价格 $p$ ，均需要左移128位。由于 $i < 0, x_n = 1$ 时， $\frac{1}{1.0001^{\frac{x_n \cdot 2^n}{2}}} < 1$ ，因此在连续乘积过程中不会有溢出问题。

可以总结计算 $\sqrt{p}(i)$ 的方法：

初始值为1，从第0位开始，从低位到高位（从右往左）循环遍历 $i$ 的二进制比特位
如果该位不为0，则乘以对应的 $\frac{2^{128}}{1.0001^{\frac{2^n}{2}}}$ ，其中 $2^{128}$ 表示左移128位
如果该位为0，则乘以1，可以省略

/// @notice Calculates sqrt(1.0001^tick) * 2^96
/// @dev Throws if |tick| > max tick
/// @param tick The input tick for the above formula
/// @return sqrtPriceX96 A Fixed point Q64.96 number representing the sqrt of the ratio of the two assets (token1/token0)
/// at the given tick
function getSqrtRatioAtTick(int24 tick) internal pure returns (uint160 sqrtPriceX96) {
    uint256 absTick = tick &lt; 0 ? uint256(-int256(tick)) : uint256(int256(tick));
    require(absTick &lt;= uint256(MAX_TICK), 'T');

    // 如果第0位非0，则ratio = 0xfffcb933bd6fad37aa2d162d1a594001 ，即：2^128 / 1.0001^0.5
    uint256 ratio = absTick & 0x1 != 0 ? 0xfffcb933bd6fad37aa2d162d1a594001 : 0x100000000000000000000000000000000;
    // 如果第1位非0，则乘以 0xfff97272373d413259a46990580e213a ，即：2^128 / 1.0001^1，因为两个乘数均为Q128.128，最终结果多乘了2^128，因此需要右移128
    if (absTick & 0x2 != 0) ratio = (ratio * 0xfff97272373d413259a46990580e213a) >> 128;
    // 如果第2位非0，则乘以 0xfff2e50f5f656932ef12357cf3c7fdcc ，即：2^128 / 1.0001^2，
    if (absTick & 0x4 != 0) ratio = (ratio * 0xfff2e50f5f656932ef12357cf3c7fdcc) >> 128;
    // 以此类推
    if (absTick & 0x8 != 0) ratio = (ratio * 0xffe5caca7e10e4e61c3624eaa0941cd0) >> 128;
    if (absTick & 0x10 != 0) ratio = (ratio * 0xffcb9843d60f6159c9db58835c926644) >> 128;
    if (absTick & 0x20 != 0) ratio = (ratio * 0xff973b41fa98c081472e6896dfb254c0) >> 128;
    if (absTick & 0x40 != 0) ratio = (ratio * 0xff2ea16466c96a3843ec78b326b52861) >> 128;
    if (absTick & 0x80 != 0) ratio = (ratio * 0xfe5dee046a99a2a811c461f1969c3053) >> 128;
    if (absTick & 0x100 != 0) ratio = (ratio * 0xfcbe86c7900a88aedcffc83b479aa3a4) >> 128;
    if (absTick & 0x200 != 0) ratio = (ratio * 0xf987a7253ac413176f2b074cf7815e54) >> 128;
    if (absTick & 0x400 != 0) ratio = (ratio * 0xf3392b0822b70005940c7a398e4b70f3) >> 128;
    if (absTick & 0x800 != 0) ratio = (ratio * 0xe7159475a2c29b7443b29c7fa6e889d9) >> 128;
    if (absTick & 0x1000 != 0) ratio = (ratio * 0xd097f3bdfd2022b8845ad8f792aa5825) >> 128;
    if (absTick & 0x2000 != 0) ratio = (ratio * 0xa9f746462d870fdf8a65dc1f90e061e5) >> 128;
    if (absTick & 0x4000 != 0) ratio = (ratio * 0x70d869a156d2a1b890bb3df62baf32f7) >> 128;
    if (absTick & 0x8000 != 0) ratio = (ratio * 0x31be135f97d08fd981231505542fcfa6) >> 128;
    if (absTick & 0x10000 != 0) ratio = (ratio * 0x9aa508b5b7a84e1c677de54f3e99bc9) >> 128;
    if (absTick & 0x20000 != 0) ratio = (ratio * 0x5d6af8dedb81196699c329225ee604) >> 128;
    if (absTick & 0x40000 != 0) ratio = (ratio * 0x2216e584f5fa1ea926041bedfe98) >> 128;
    // 如果第19位非0，因为（2^19 = 0x80000=524288），则乘以 0x2216e584f5fa1ea926041bedfe98，即：2^128 / 1.0001^(524288/2)
    // tick的最大值为887272，因此其二进制最多只需要20位表示，从0开始计数，最后一位为第19位。
    if (absTick & 0x80000 != 0) ratio = (ratio * 0x48a170391f7dc42444e8fa2) >> 128;

    if (tick > 0) ratio = type(uint256).max / ratio;

    // this divides by 1&lt;&lt;32 rounding up to go from a Q128.128 to a Q128.96.
    // we then downcast because we know the result always fits within 160 bits due to our tick input constraint
    // we round up in the division so getTickAtSqrtRatio of the output price is always consistent
    sqrtPriceX96 = uint160((ratio >> 32) + (ratio % (1 &lt;&lt; 32) == 0 ? 0 : 1));
}

假设 $i > 0$ 时：

$$ \sqrt{p{Q128128}(i)} = 2^{128} \cdot \sqrt{p(i)} = 2^{128} \cdot 1.0001^{\frac{i}{2}} \ = \frac{2^{128}}{1.0001^{-\frac{i}{2}}} = \frac{2^{256}}{2^{128} \cdot \sqrt{p(-i)}} = \frac{2^{256}}{\sqrt{p{Q128128}(-i)}} $$

因此，只需要算出 $i < 0$ 时的 ratio 值，使用 $2^{256}$ 除以ratio即可得出 $i > 0$ 时，使用Q128.128表示的ratio值：

if (tick > 0) ratio = type(uint256).max / ratio;

代码最后一行将ratio右移32位，转化为Q128.96格式的定点数：

sqrtPriceX96 = uint160((ratio >> 32) + (ratio % (1 &lt;&lt; 32) == 0 ? 0 : 1));

这里算的是开根号价格 $\sqrt{p}$ ，由于价格 $p$ 最大为 $2^{128}$ ，因此 $\sqrt{p}$ 最大为 $2^{64}$ ，也就是整数部分最大只需要64位表示，因此最终的sqrtPriceX96一定可以用160位（64+96，即Q64.96格式的定点数）表示。

getTickAtSqrtRatio

该方法对应白皮书中的公式6.8：

$$ ic = \lfloor \log{\sqrt{1.0001}} \sqrt{P} \rfloor $$

本方法涉及在Solidity中计算对数，根据对数公式，可以推出：

$$ \log_{\sqrt{1.0001}} \sqrt{P} = \frac{log_2{\sqrt{P}}}{log_2{\sqrt{1.0001}}} = log2{\sqrt{P}} \cdot log{\sqrt{1.0001}}{2} $$

由于 $log_{\sqrt{1.0001}}{2}$ 是一个常数，因此我们只需要计算 $log_2{\sqrt{P}}$ 即可。

将输入的参数为 $\sqrt{P}$ 看作 $x$ ，问题转化为求 $log_2{x}$。

把结果分为整数部分 $n$ 和小数部分 $m$ ，则：

$$ n \leq log_2{x} = n + m < n + 1 $$

整数部分

对于 $n$ ，因为：

$$ 2^n \leq x < 2^{n+1} $$

可以通过二分查找找到 $n$ 值：

对于256位的数， $0 \leq n < 256$ ，可以用8位比特表示 $n$
从二进制表示的第8位（k=7）到第1位（k=0）（从最高位到最低位），依次比较 $x$ 是否大于 $2^{2^{k}} - 1$ ，如果大于则标记该位为1，并右移 $2^k$ 位；否则标记0
最终标记后的8位二进制即为 $n$ 值

使用Python代码描述如下：

def find_msb(x):
    msb = 0
    for k in reversed(range(8)): // k = 7, 6, 5. 4, 3, 2, 1, 0
        if x > 2 ** (2 ** k) - 1:
            msb += 2 ** k // 标记该位为1，即加上 2 ** k
            x /= 2 ** (2 ** k) // 右移 2 ** k 位
    return msb

Uniswap v3中的Solidity代码如下（请参考代码中注释）：

/// @notice Calculates the greatest tick value such that getRatioAtTick(tick) &lt;= ratio
/// @dev Throws in case sqrtPriceX96 &lt; MIN_SQRT_RATIO, as MIN_SQRT_RATIO is the lowest value getRatioAtTick may
/// ever return.
/// @param sqrtPriceX96 The sqrt ratio for which to compute the tick as a Q64.96
/// @return tick The greatest tick for which the ratio is less than or equal to the input ratio
function getTickAtSqrtRatio(uint160 sqrtPriceX96) internal pure returns (int24 tick) {
    // second inequality must be &lt; because the price can never reach the price at the max tick
    require(sqrtPriceX96 >= MIN_SQRT_RATIO && sqrtPriceX96 &lt; MAX_SQRT_RATIO, 'R');
    uint256 ratio = uint256(sqrtPriceX96) &lt;&lt; 32; // 左移32位，转化为Q128.128格式

    uint256 r = ratio;
    uint256 msb = 0;

    assembly {
        // 如果大于2 ** (2 ** 7) - 1，则保存临时变量：2 ** 7
        let f := shl(7, gt(r, 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF))
        // msb += 2 ** 7
        msb := or(msb, f)
        // r /= (2 ** (2 ** 7))，即右移 2 ** 7
        r := shr(f, r)
    }
    assembly {
        let f := shl(6, gt(r, 0xFFFFFFFFFFFFFFFF))
        msb := or(msb, f)
        r := shr(f, r)
    }
    assembly {
        let f := shl(5, gt(r, 0xFFFFFFFF))
        msb := or(msb, f)
        r := shr(f, r)
    }
    assembly {
        let f := shl(4, gt(r, 0xFFFF))
        msb := or(msb, f)
        r := shr(f, r)
    }
    assembly {
        let f := shl(3, gt(r, 0xFF))
        msb := or(msb, f)
        r := shr(f, r)
    }
    assembly {
        let f := shl(2, gt(r, 0xF))
        msb := or(msb, f)
        r := shr(f, r)
    }
    assembly {
        let f := shl(1, gt(r, 0x3))
        msb := or(msb, f)
        r := shr(f, r)
    }
    assembly {
        let f := gt(r, 0x1)
        msb := or(msb, f)
    }

小数部分

对于小数部分 $m$：

$$ 0 \leq m = log_2{x} - n = log_2{\frac{x}{2^n}} < 1 \quad \text{(1.2)} $$

其中， $n$ 为上文算出的msb，即整数部分。

我们先将 $\frac{x}{2^n}$ 看做一个整体 $r$，则：

$$ 0 \leq log_2{r} < 1 $$

$$ 1 \leq r = \frac{x}{2^n} < 2 $$

这里我们希望求出 $log_2{r}$ ，如果能够将 $log_2{r}$ 表示成一个不断收敛的数列，当小数位足够多时，就可以近似求出 $log_2{r}$ 的值。

根据对数公式，我们可以推导以下两个等式：

$$ log_2{r} = \frac{2 \cdot log_2{r}}{2} = \frac{log_2{r^2}}{2} \quad \text{(1.3)} $$

$$ log_2{r} = log_2{2 \cdot \frac{r}{2}} = 1 + log_2{\frac{r}{2}} \quad \text{(1.4)} $$

我们循环套用上述两个公式，可以整理以下方法：

因为初始时 $log_2{r} < 1$，因此先应用公式1.3，将问题转化为求 $log_2{r^2}$ ，注意此时基数为 $\frac{1}{2}$ ；
- 事实上，每一次进入步骤1，新的基数都是上一次基数的 $\frac{1}{2}$ ，比如第二次进入步骤1的基数为 $\frac{1}{4}$ ，以此类推。
如果 $r^2$ >= 2 ，则应用公式1.4，分离出1，并将问题转化为求 $log_2{\frac{r^2}{2}}$ ；
- 因为公式1.4是在公式1.3之后判断，因此这里的1需要乘以上一次步骤1的基数，如果是第一次则记录 $\frac{1}{2}$ ，第二次则记录 $\frac{1}{4}$ ，以此类推；
- 因为 $1 \leq r < 2$ ，且 $2 \leq r^2 < 4$ ，因此 $1 \leq \frac{r^2}{2} < 2$ ，将 $\frac{r^2}{2}$ 看做一个整体 $r$ ，又回到步骤1求解 $log_2{r}$ ，并且 $1 \leq r < 2$ 。
如果 $r^2 < 2$ ，则回到步骤1继续。

可以将上述步骤总结为以下公式：

$$ log_2{r} = m_1 \cdot \frac{1}{2} + m_2 \cdot \frac{1}{4} + ... + mn \cdot \frac{1}{2^n} = \sum^{\infty}{i=1}(m_i \cdot \frac{1}{2^i}) \quad \text{(1.5)} $$

其中， $\forall m_i \in {0, 1}$。

我们继续看 Uniswap v3 中计算小数部分的代码：

        if (msb >= 128) r = ratio >> (msb - 127);
        else r = ratio &lt;&lt; (127 - msb);

这里msb即为整数部分 $n$ 。因为ratio是Q128.128，如果msb >= 128则表示ratio >= 1，因此需要右移整数位数得到小数部分ratio >> msb；-127表示左移127位，使用Q129.127表示小数部分；同样，如果msb < 128，则表示ratio < 1，其本身就只有小数部分，因此通过左移127 - msb位，将小数部分凑齐127位，也用Q129.127表示小数部分。

实际上，ratio >> msb即为公式1.2中的 $\frac{x}{2^n}$ ，也就是步骤1中的 $r$ ，在后续迭代算法（步骤1-3）中需要用到。

    int256 log_2 = (int256(msb) - 128) &lt;&lt; 64;

因为msb是基于Q128.128的ratio计算的，int256(msb) - 128表示 $n$ 的真正值。<< 64使用Q192.64表示 $n$ 。这一行代码实际上是使用Q192.64保存整数部分的值。

下面代码循环计算二进制表示的小数部分的前14位小数：

    assembly {
        // 根据步骤1，计算r^2，右移127位是因为两个r都是Q129.127
        r := shr(127, mul(r, r))
        // 因为1 &lt;= r^2 &lt; 4，仅需2位表示r^2的整数，
        // 因此从右往左数第129和128位表示r^2的整数部分，
        // 右移128位，仅剩129位，
        // 该值为1，则表示r >= 2；该值为0，则表示r &lt; 2
        let f := shr(128, r)
        // 如果f == 1，则log_2 += Q192.64的1/2
        log_2 := or(log_2, shl(63, f))
        // 根据步骤2（即公式1.4），如果r >= 2（即f == 1），则r /= 2；否则不操作，即步骤3
        r := shr(f, r)
    }
    // 重复进行上述过程
    assembly {
        r := shr(127, mul(r, r))
        let f := shr(128, r)
        log_2 := or(log_2, shl(62, f))
        r := shr(f, r)
    }
    assembly {
        r := shr(127, mul(r, r))
        let f := shr(128, r)
        log_2 := or(log_2, shl(61, f))
        r := shr(f, r)
    }
    assembly {
        r := shr(127, mul(r, r))
        let f := shr(128, r)
        log_2 := or(log_2, shl(60, f))
        r := shr(f, r)
    }
    assembly {
        r := shr(127, mul(r, r))
        let f := shr(128, r)
        log_2 := or(log_2, shl(59, f))
        r := shr(f, r)
    }
    assembly {
        r := shr(127, mul(r, r))
        let f := shr(128, r)
        log_2 := or(log_2, shl(58, f))
        r := shr(f, r)
    }
    assembly {
        r := shr(127, mul(r, r))
        let f := shr(128, r)
        log_2 := or(log_2, shl(57, f))
        r := shr(f, r)
    }
    assembly {
        r := shr(127, mul(r, r))
        let f := shr(128, r)
        log_2 := or(log_2, shl(56, f))
        r := shr(f, r)
    }
    assembly {
        r := shr(127, mul(r, r))
        let f := shr(128, r)
        log_2 := or(log_2, shl(55, f))
        r := shr(f, r)
    }
    assembly {
        r := shr(127, mul(r, r))
        let f := shr(128, r)
        log_2 := or(log_2, shl(54, f))
        r := shr(f, r)
    }
    assembly {
        r := shr(127, mul(r, r))
        let f := shr(128, r)
        log_2 := or(log_2, shl(53, f))
        r := shr(f, r)
    }
    assembly {
        r := shr(127, mul(r, r))
        let f := shr(128, r)
        log_2 := or(log_2, shl(52, f))
        r := shr(f, r)
    }
    assembly {
        r := shr(127, mul(r, r))
        let f := shr(128, r)
        log_2 := or(log_2, shl(51, f))
        r := shr(f, r)
    }
    assembly {
        r := shr(127, mul(r, r))
        let f := shr(128, r)
        log_2 := or(log_2, shl(50, f))
    }

上述计算的log_2即为Q192.64表示的 $log_2{\sqrt{P}}$ ，精度为 $2^{-14}$ 。

    int256 log_sqrt10001 = log_2 * 255738958999603826347141; // 128.128 number

因为：

$$ \log_{\sqrt{1.0001}} \sqrt{P} = \frac{log_2{\sqrt{P}}}{log_2{\sqrt{1.0001}}} = log2{\sqrt{P}} \cdot log{\sqrt{1.0001}}{2} $$

这里255738958999603826347141即为 $log_{\sqrt{1.0001}}{2} \cdot 2^{64}$ ，两个Q192.64乘以的结果为Q128.64（不会发生溢出）。

由于这里算出的 $log_2{\sqrt{P}}$ 精度为 $2^{-14}$ ，乘以255738958999603826347141后误差进一步放大，因此需要修正并确保结果是最接近给定价格的tick。

    int24 tickLow = int24((log_sqrt10001 - 3402992956809132418596140100660247210) >> 128);
    int24 tickHi = int24((log_sqrt10001 + 291339464771989622907027621153398088495) >> 128);

    tick = tickLow == tickHi ? tickLow : getSqrtRatioAtTick(tickHi) &lt;= sqrtPriceX96 ? tickHi : tickLow;

其中，3402992956809132418596140100660247210表示0.01000049749154292 << 128，291339464771989622907027621153398088495表示0.8561697375276566 << 128。

参考abdk的这篇文章，当精度为 $2^{-14}$ 时，tick的最小误差为 $− 0.85617$ ，最大误差为 $0.0100005$ 。同时，这篇文章也从理论上证明了，只有当精度等于（或高于） $2^{-14}$ 时，只需要一次计算即可得出所需的tick值。

以下是本节参考文章，有兴趣的朋友请扩展阅读：

TickBitmap.sol

TickBitmap使用Bitmap（位图）保存Tick的初始化状态，提供以下几个方法：

position：根据tick返回位图索引数据
flipTick：翻转tick状态
nextInitializedTickWithinOneWord：寻找同组内下一个初始化的tick

position

/// @notice Computes the position in the mapping where the initialized bit for a tick lives
/// @param tick The tick for which to compute the position
/// @return wordPos The key in the mapping containing the word in which the bit is stored
/// @return bitPos The bit position in the word where the flag is stored
function position(int24 tick) private pure returns (int16 wordPos, uint8 bitPos) {
    wordPos = int16(tick >> 8);
    bitPos = uint8(tick % 256);
}

只有能被tickSpacing整除的tick才能记录在位图中，因此此处的参数： $tick = \frac{tick}{tickSpacing}$。

tick类型为int24，其二进制从右到左，从低位到高位，前8位表示bitPos，后16位表示wordPos，如下图所示：

$$ \overbrace{23,...,8}^{wordPos},\overbrace{7,...,0}^{bitPos} $$

tick的bitmap表示为：self[wordPos] ^= 1 << bitPos。

flipTick

当tick翻转初始化状态时，如果其位图的值为0，则需要修改为1；否则，修改为0；即对该位“取反”。

/// @notice Flips the initialized state for a given tick from false to true, or vice versa
/// @param self The mapping in which to flip the tick
/// @param tick The tick to flip
/// @param tickSpacing The spacing between usable ticks
function flipTick(
    mapping(int16 => uint256) storage self,
    int24 tick,
    int24 tickSpacing
) internal {
    require(tick % tickSpacing == 0); // ensure that the tick is spaced
    (int16 wordPos, uint8 bitPos) = position(tick / tickSpacing);
    uint256 mask = 1 &lt;&lt; bitPos;
    self[wordPos] ^= mask;
}

首先获取tick对应的wordPos和bitPos，由于最后针对bitPos执行按位“异或”操作：

因为1与任何值b（0或1）异或等于~b；0与任何值b（0或1）异或等于b。

对于tick对应的位（bit），mask为1
- 如果旧值为1，1^1=0，因此tick状态由“初始化”变成“未初始化”；
- 如果旧值为0，0^1=1，因此tick状态由“未初始化”变成“初始化”
对于非tick对应的位，mask为0
- 如果旧值为1，1^0=1，因此状态不变
- 如果旧值为0，0^0=0，因此状态不变

所以，上述代码实现了tick位取反的效果。

nextInitializedTickWithinOneWord

根据参数tick，寻找位图上最近一个已初始化的tick，如未找到，返回本组最后一个未初始化的tick。

/// @notice Returns the next initialized tick contained in the same word (or adjacent word) as the tick that is either
/// to the left (less than or equal to) or right (greater than) of the given tick
/// @param self The mapping in which to compute the next initialized tick
/// @param tick The starting tick
/// @param tickSpacing The spacing between usable ticks
/// @param lte Whether to search for the next initialized tick to the left (less than or equal to the starting tick)
/// @return next The next initialized or uninitialized tick up to 256 ticks away from the current tick
/// @return initialized Whether the next tick is initialized, as the function only searches within up to 256 ticks
function nextInitializedTickWithinOneWord(
    mapping(int16 => uint256) storage self,
    int24 tick,
    int24 tickSpacing,
    bool lte
) internal view returns (int24 next, bool initialized) {
    int24 compressed = tick / tickSpacing;
    if (tick &lt; 0 && tick % tickSpacing != 0) compressed--; // round towards negative infinity

    if (lte) {
        (int16 wordPos, uint8 bitPos) = position(compressed);
        // all the 1s at or to the right of the current bitPos
        uint256 mask = (1 &lt;&lt; bitPos) - 1 + (1 &lt;&lt; bitPos);
        uint256 masked = self[wordPos] & mask;

        // if there are no initialized ticks to the right of or at the current tick, return rightmost in the word
        initialized = masked != 0;
        // overflow/underflow is possible, but prevented externally by limiting both tickSpacing and tick
        next = initialized
            ? (compressed - int24(bitPos - BitMath.mostSignificantBit(masked))) * tickSpacing
            : (compressed - int24(bitPos)) * tickSpacing;
    } else {
        // start from the word of the next tick, since the current tick state doesn't matter
        (int16 wordPos, uint8 bitPos) = position(compressed + 1);
        // all the 1s at or to the left of the bitPos
        uint256 mask = ~((1 &lt;&lt; bitPos) - 1);
        uint256 masked = self[wordPos] & mask;

        // if there are no initialized ticks to the left of the current tick, return leftmost in the word
        initialized = masked != 0;
        // overflow/underflow is possible, but prevented externally by limiting both tickSpacing and tick
        next = initialized
            ? (compressed + 1 + int24(BitMath.leastSignificantBit(masked) - bitPos)) * tickSpacing
            : (compressed + 1 + int24(type(uint8).max - bitPos)) * tickSpacing;
    }
}

如果lte == true，即寻找小于等于当前tick的值，因此问题转化为：寻找低比特位上是否有1。

if (lte) {
    (int16 wordPos, uint8 bitPos) = position(compressed);
    // all the 1s at or to the right of the current bitPos
    uint256 mask = (1 &lt;&lt; bitPos) - 1 + (1 &lt;&lt; bitPos);
    uint256 masked = self[wordPos] & mask;
}

其中，mask的值为所有小于等于bitPos的位全部置1，比如bitPos = 7，则mask二进制 = 1111111；masked保留位图中小于等于bitPos的位值，比如self[wordPos] = 110101011，则masked = 110101011 & 1111111 = 000101011。

如果masked != 0，则表示当前方向（lte）该wordPos上有初始化的tick；否则，则表示该方向都是未初始化的tick。

// if there are no initialized ticks to the right of or at the current tick, return rightmost in the word
initialized = masked != 0;

如果存在已初始化的tick，则需要定位到masked中最高位的1；如果不存在，则返回当前方向最后一个tick。

// overflow/underflow is possible, but prevented externally by limiting both tickSpacing and tick
next = initialized
    ? (compressed - int24(bitPos - BitMath.mostSignificantBit(masked))) * tickSpacing
    : (compressed - int24(bitPos)) * tickSpacing;

BitMath.mostSignificantBit(masked)通过二分查找法找到masked最高位的1，关于该算法的具体说明，可参考本文TickMath对数计算部分。简单而言，mostSignificantBit(masked)会返回一个数n，使得：

$$ 2^n \leq masked < 2^{n+1} $$

比如，如果masked = 000101011，mostSignificantBit将返回5，因为最高位的1在（从0开始）第5位：000101011。

compressed - int24(bitPos)表示当前wordPos第一个tick，compressed - int24(bitPos - BitMath.mostSignificantBit(masked))表示该最高位bitPos对应的tick；* tickSpacing将恢复到原始tick值，因为保存位图时，tick需要先除以tickSpacing。

同样，如果向大于当前tick方向寻找第一个已初始化的tick时，方法和上述类似，只是mostSignificantBit需要换成leastSignificantBit，即从tick比特位开始（不含），往高位寻找第一个bitPos位为1的tick。

else {
    // start from the word of the next tick, since the current tick state doesn't matter
    (int16 wordPos, uint8 bitPos) = position(compressed + 1);
    // all the 1s at or to the left of the bitPos
    uint256 mask = ~((1 &lt;&lt; bitPos) - 1);
    uint256 masked = self[wordPos] & mask;

    // if there are no initialized ticks to the left of the current tick, return leftmost in the word
    initialized = masked != 0;
    // overflow/underflow is possible, but prevented externally by limiting both tickSpacing and tick
    next = initialized
        ? (compressed + 1 + int24(BitMath.leastSignificantBit(masked) - bitPos)) * tickSpacing
        : (compressed + 1 + int24(type(uint8).max - bitPos)) * tickSpacing;
}

Position.sol

Position.sol管理头寸相关信息，包括以下方法：

get：获取头寸对象
update：更新头寸

一个头寸（Position）可由“所有者”owner、“区间低点”tickLower和“区间高点”tickUpper唯一确定，对于同一个交易对池子，每个用户可以创建多个不同价格区间的头寸，但只能创建一个相同价格区间的头寸；不同用户可以创建相同价格区间的头寸。

get

根据owner、tickLower和tickUpper返回一个头寸对象：

/// @notice Returns the Info struct of a position, given an owner and position boundaries
/// @param self The mapping containing all user positions
/// @param owner The address of the position owner
/// @param tickLower The lower tick boundary of the position
/// @param tickUpper The upper tick boundary of the position
/// @return position The position info struct of the given owners' position
function get(
    mapping(bytes32 => Info) storage self,
    address owner,
    int24 tickLower,
    int24 tickUpper
) internal view returns (Position.Info storage position) {
    position = self[keccak256(abi.encodePacked(owner, tickLower, tickUpper))];
}

update

更新头寸的流动性和可取回代币，注意，该方法只会在mint和burn时被触发，swap并不会更新头寸信息。

/// @notice Credits accumulated fees to a user's position
/// @param self The individual position to update
/// @param liquidityDelta The change in pool liquidity as a result of the position update
/// @param feeGrowthInside0X128 The all-time fee growth in token0, per unit of liquidity, inside the position's tick boundaries
/// @param feeGrowthInside1X128 The all-time fee growth in token1, per unit of liquidity, inside the position's tick boundaries
function update(
    Info storage self,
    int128 liquidityDelta,
    uint256 feeGrowthInside0X128,
    uint256 feeGrowthInside1X128
) internal {
    Info memory _self = self;

    uint128 liquidityNext;
    if (liquidityDelta == 0) {
        require(_self.liquidity > 0, 'NP'); // disallow pokes for 0 liquidity positions
        liquidityNext = _self.liquidity;
    } else {
        liquidityNext = LiquidityMath.addDelta(_self.liquidity, liquidityDelta);
    }

更新流动性：

如果是mint，则liquidityDelta > 0
如果是burn，则liquidityDelta < 0

    // calculate accumulated fees
    uint128 tokensOwed0 =
        uint128(
            FullMath.mulDiv(
                feeGrowthInside0X128 - _self.feeGrowthInside0LastX128,
                _self.liquidity,
                FixedPoint128.Q128
            )
        );
    uint128 tokensOwed1 =
        uint128(
            FullMath.mulDiv(
                feeGrowthInside1X128 - _self.feeGrowthInside1LastX128,
                _self.liquidity,
                FixedPoint128.Q128
            )
        );

根据头寸区间自上一次更新后的每流动性手续费增长值，分别计算token0和token1的应收手续费。

    // update the position
    if (liquidityDelta != 0) self.liquidity = liquidityNext;
    self.feeGrowthInside0LastX128 = feeGrowthInside0X128;
    self.feeGrowthInside1LastX128 = feeGrowthInside1X128;
    if (tokensOwed0 > 0 || tokensOwed1 > 0) {
        // overflow is acceptable, have to withdraw before you hit type(uint128).max fees
        self.tokensOwed0 += tokensOwed0;
        self.tokensOwed1 += tokensOwed1;
    }
}

更新头寸流动性、本次每流动性手续费和可取回代币数。

SwapMath.sol

computeSwapStep

SwapMath只有一个方法，即computeSwapStep，计算单步交换的输入输出。

该方法实际上实现的是白皮书6.2.3小节：在一个Tick内交易。在该场景下，流动性 $L$ 不变，仅价格 $\sqrt{P}$ 发生变化。

参数如下：

sqrtRatioCurrentX96：当前价格
sqrtRatioTargetX96：目标价格
liquidity：可用流动性
amountRemaining：剩余输入代币
feePips：手续费

返回值：

sqrtRatioNextX96：交换后价格
amountIn：本次交换消耗的输入代币
amountOut：本次交换消耗的输出代币
feeAmount：交易手续费（含协议手续费）

function computeSwapStep(
    uint160 sqrtRatioCurrentX96,
    uint160 sqrtRatioTargetX96,
    uint128 liquidity,
    int256 amountRemaining,
    uint24 feePips
)
    internal
    pure
    returns (
        uint160 sqrtRatioNextX96,
        uint256 amountIn,
        uint256 amountOut,
        uint256 feeAmount
    )
{
    bool zeroForOne = sqrtRatioCurrentX96 >= sqrtRatioTargetX96;
    bool exactIn = amountRemaining >= 0;

我们在swap章节提到，对于一个交换操作，根据zeroForOne和exactIn的值，有四种组合：

zeroForOne	exactInput	swap
true	true	输入固定数量`token0`，输出最大数量`token1`
true	false	输入最小数量`token0`，输出固定数量`token1`
false	true	输入固定数量`token1`，输出最大数量`token0`
false	false	输入最小数量`token1`，输出固定数量`token0`

if (exactIn) {
    uint256 amountRemainingLessFee = FullMath.mulDiv(uint256(amountRemaining), 1e6 - feePips, 1e6);
    amountIn = zeroForOne
        ? SqrtPriceMath.getAmount0Delta(sqrtRatioTargetX96, sqrtRatioCurrentX96, liquidity, true)
        : SqrtPriceMath.getAmount1Delta(sqrtRatioCurrentX96, sqrtRatioTargetX96, liquidity, true);
    if (amountRemainingLessFee >= amountIn) sqrtRatioNextX96 = sqrtRatioTargetX96;
    else
        sqrtRatioNextX96 = SqrtPriceMath.getNextSqrtPriceFromInput(
            sqrtRatioCurrentX96,
            liquidity,
            amountRemainingLessFee,
            zeroForOne
        );
}

如果是输入固定数量，则交易手续费需要从输入代币中扣除。

因为feePips的单位是百分之一基点，即 $\frac{1}{10^6}$ ，因此，按照如下公式扣除手续费：

$$ amountRemaining \cdot (1 - \frac{feePips}{10^6}) $$

使用getAmount0Delta或getAmount1Delta，根据当前价格、目标价格和可用流动性计算所需的输入代币数量amountIn：

如果从token0交换token1，则输入代币为token0，因此计算amount0
如果从token1交换token0，则输入代币为token1，因此计算amount1

注意，方法的最后一个参数roundUp，当计算amountIn时需要设置roundUp = true，也就是合约只能多收钱，而不能少收，否则合约资金将出现损失；同样，当计算amountOut时设置roundUp = false，也就是合约可以少付钱，而不能多付。

如果可用代币数量大于amountIn，则表示该步交易可以完成，因此交换后的价格等于目标价格
否则，根据当前价格，可用流动性和可用代币amountRemainingLessFee，计算交换后价格，我们在SqrtPriceMath.getNextSqrtPriceFromInput会具体介绍计算方法

else {
    amountOut = zeroForOne
        ? SqrtPriceMath.getAmount1Delta(sqrtRatioTargetX96, sqrtRatioCurrentX96, liquidity, false)
        : SqrtPriceMath.getAmount0Delta(sqrtRatioCurrentX96, sqrtRatioTargetX96, liquidity, false);
    if (uint256(-amountRemaining) >= amountOut) sqrtRatioNextX96 = sqrtRatioTargetX96;
    else
        sqrtRatioNextX96 = SqrtPriceMath.getNextSqrtPriceFromOutput(
            sqrtRatioCurrentX96,
            liquidity,
            uint256(-amountRemaining),
            zeroForOne
        );
}

如果不是输入固定数量，也就是输出固定数量，需要将amountRemaining作为输出代币数量使用：

如果从token0交换token1，则输出代币为token1，因此计算amount1
如果从token1交换token0，则输出代币为token0，因此计算amount0

首先根据当前价格、目标价格和可用流动性，计算可产生的输出代币数量amountOut，注意，这里与上面相反，如果是token0交换token1，需要计算token1的数量，需使用SqrtPriceMath.getAmount1Delta方法。

当表示固定输出代币时，传入的参数amountRemaining是负值：

如果amountOut的绝对值小于应输出代币数量，则表示可完全交换，交换后价格等于目标价格
否则，根据可用输出amountRemaining，计算交换后价格

    bool max = sqrtRatioTargetX96 == sqrtRatioNextX96;

如果交换后价格等于目标价格，则表示完全交换。

// get the input/output amounts
if (zeroForOne) {
    amountIn = max && exactIn
        ? amountIn
        : SqrtPriceMath.getAmount0Delta(sqrtRatioNextX96, sqrtRatioCurrentX96, liquidity, true);
    amountOut = max && !exactIn
        ? amountOut
        : SqrtPriceMath.getAmount1Delta(sqrtRatioNextX96, sqrtRatioCurrentX96, liquidity, false);
} else {
    amountIn = max && exactIn
        ? amountIn
        : SqrtPriceMath.getAmount1Delta(sqrtRatioCurrentX96, sqrtRatioNextX96, liquidity, true);
    amountOut = max && !exactIn
        ? amountOut
        : SqrtPriceMath.getAmount0Delta(sqrtRatioCurrentX96, sqrtRatioNextX96, liquidity, false);
}

计算本次交换所需输入amountIn和所得输出amountOut：

如果从token0交换token1
- 所需输入amountIn
  - 如果完全交换，并且固定输入，则所需输入即为amountIn
  - 否则（非完全交换，或者固定输出），则使用getAmount0Delta，根据价格区间和流动性计算所需输入
- 所得输出amountOut
  - 如果完全交换，并且固定输出，则所得输出即为amountOut
  - 否则（非完全交换，或者固定输入），则使用getAmount1Delta，根据价格区间和流动性计算所得输出
如果从token1交换token1
- 所需输入amountIn
  - 如果完全交换，并且固定输入，则所需输入即为amountIn
  - 否则（非完全交换，或者固定输出），则使用getAmount1Delta，根据价格区间和流动性计算所需输入
- 所得输出amountOut
  - 如果完全交换，并且固定输出，则所得输出即为amountOut
  - 否则（非完全交换，或者固定输入），则使用getAmount0Delta，根据价格区间和流动性计算所得输出

// cap the output amount to not exceed the remaining output amount
if (!exactIn && amountOut > uint256(-amountRemaining)) {
    amountOut = uint256(-amountRemaining);
}

确认所得输出没有超过指定输出。

if (exactIn && sqrtRatioNextX96 != sqrtRatioTargetX96) {
    // we didn't reach the target, so take the remainder of the maximum input as fee
    feeAmount = uint256(amountRemaining) - amountIn;
} else {
    feeAmount = FullMath.mulDivRoundingUp(amountIn, feePips, 1e6 - feePips);
}

计算交易手续费（包括协议手续费）：

如果固定输入，并且没有达到目标价格，相当于是最后一次交换，则原始输入代币amountRemaining扣除本次交换所需输入amountIn，即为手续费；
否则，需要根据amountIn计算手续费；注意，除了上述最后一次交换外，这里的amountIn和amountOut都是不包括手续费的，因此计算手续费需要除以1e6 - feePips，而非1e6。

SqrtPriceMath.sol

getNextSqrtPriceFromAmount0RoundingUp

根据当前价格、liquidity和 $\Delta{x}$ ，计算目标价格。

根据白皮书公式6.16:

$$ \Delta{x} = \Delta{\frac{1}{\sqrt{P}}} \cdot L $$

假设 $\sqrt{P_a} > \sqrt{P_b}$ ，则 $x_a < x_b$ ：

$$ \Delta{x} = x_b - x_a $$

如果已知 $\sqrt{P_a}$ 计算 $\sqrt{P_b}$ ，则：

$$ \frac{1}{\sqrt{P_b}} = \frac{\Delta{x}}{L} + \frac{1}{\sqrt{P_a}} = \frac{1}{L} \cdot (\Delta{x} + \frac{L}{\sqrt{P_a}}) \quad \text{(1.1)} $$

$$ \sqrt{P_b} = \frac{L}{\Delta{x} + \frac{L}{\sqrt{P_a}}} \quad \text{(1.2)} $$

$$ {\sqrt{P_b}} = \frac{L \cdot \sqrt{P_a}}{L + \Delta{x} \cdot \sqrt{P_a}} \quad \text{(1.3)} $$

如果已知 $\sqrt{P_b}$ 计算 $\sqrt{P_a}$ ，则：

$$ \frac{1}{\sqrt{P_a}} = \frac{1}{\sqrt{P_b}} - \frac{\Delta{x}}{L} \quad \text{(1.4)} $$

$$ {\sqrt{P_a}} = \frac{L \cdot \sqrt{P_b}}{L - \Delta{x} \cdot \sqrt{P_b}} \quad \text{(1.5)} $$

/// @notice Gets the next sqrt price given a delta of token0
/// @dev Always rounds up, because in the exact output case (increasing price) we need to move the price at least
/// far enough to get the desired output amount, and in the exact input case (decreasing price) we need to move the
/// price less in order to not send too much output.
/// The most precise formula for this is liquidity * sqrtPX96 / (liquidity +- amount * sqrtPX96),
/// if this is impossible because of overflow, we calculate liquidity / (liquidity / sqrtPX96 +- amount).
/// @param sqrtPX96 The starting price, i.e. before accounting for the token0 delta
/// @param liquidity The amount of usable liquidity
/// @param amount How much of token0 to add or remove from virtual reserves
/// @param add Whether to add or remove the amount of token0
/// @return The price after adding or removing amount, depending on add
function getNextSqrtPriceFromAmount0RoundingUp(
    uint160 sqrtPX96,
    uint128 liquidity,
    uint256 amount,
    bool add
) internal pure returns (uint160) {
    // we short circuit amount == 0 because the result is otherwise not guaranteed to equal the input price
    if (amount == 0) return sqrtPX96;
    uint256 numerator1 = uint256(liquidity) &lt;&lt; FixedPoint96.RESOLUTION;

    if (add) {
        uint256 product;
        if ((product = amount * sqrtPX96) / amount == sqrtPX96) {
            uint256 denominator = numerator1 + product;
            if (denominator >= numerator1)
                // always fits in 160 bits
                return uint160(FullMath.mulDivRoundingUp(numerator1, sqrtPX96, denominator));
        }

        return uint160(UnsafeMath.divRoundingUp(numerator1, (numerator1 / sqrtPX96).add(amount)));
    } else {
        uint256 product;
        // if the product overflows, we know the denominator underflows
        // in addition, we must check that the denominator does not underflow
        require((product = amount * sqrtPX96) / amount == sqrtPX96 && numerator1 > product);
        uint256 denominator = numerator1 - product;
        return FullMath.mulDivRoundingUp(numerator1, sqrtPX96, denominator).toUint160();
    }
}

当add = true时，即已知 $\sqrt{P_a}$ 计算 $\sqrt{P_b}$
- 如果 $\Delta{x} \cdot \sqrt{P_a}$ 没有发生溢出，则使用公式1.3计算 $\sqrt{P_b}$ ，因为这种方式精度最高
- 否则，使用公式1.2计算 $\sqrt{P_b}$
当add = false时，即已知 $\sqrt{P_b}$ ，根据上述公式1.5计算 $\sqrt{P_a}$

getNextSqrtPriceFromAmount1RoundingDown

根据当前价格、liquidity和 $\Delta{y}$ ，计算目标价格。

根据白皮书公式6.13:

$$ \Delta{\sqrt{P}} = \frac{\Delta{y}}{L} \quad \text{(6.13)} $$

假设 $\sqrt{P_a} > \sqrt{P_b}$ ，则 $y_a > y_b$：

$$ \Delta{y} = y_a - y_b $$

如果已知 $\sqrt{P_b}$ 计算 $\sqrt{P_a}$ ，则：

$$ \sqrt{P_a} = \sqrt{P_b} + \frac{\Delta{y}}{L} \quad \text{(1.6)} $$

如果已知 $\sqrt{P_a}$ 计算 $\sqrt{P_b}$ ，则：

$$ \sqrt{P_b} = \sqrt{P_a} - \frac{\Delta{y}}{L} \quad \text{(1.7)} $$

/// @notice Gets the next sqrt price given a delta of token1
/// @dev Always rounds down, because in the exact output case (decreasing price) we need to move the price at least
/// far enough to get the desired output amount, and in the exact input case (increasing price) we need to move the
/// price less in order to not send too much output.
/// The formula we compute is within &lt;1 wei of the lossless version: sqrtPX96 +- amount / liquidity
/// @param sqrtPX96 The starting price, i.e., before accounting for the token1 delta
/// @param liquidity The amount of usable liquidity
/// @param amount How much of token1 to add, or remove, from virtual reserves
/// @param add Whether to add, or remove, the amount of token1
/// @return The price after adding or removing `amount`
function getNextSqrtPriceFromAmount1RoundingDown(
    uint160 sqrtPX96,
    uint128 liquidity,
    uint256 amount,
    bool add
) internal pure returns (uint160) {
    // if we're adding (subtracting), rounding down requires rounding the quotient down (up)
    // in both cases, avoid a mulDiv for most inputs
    if (add) {
        uint256 quotient =
            (
                amount &lt;= type(uint160).max
                    ? (amount &lt;&lt; FixedPoint96.RESOLUTION) / liquidity
                    : FullMath.mulDiv(amount, FixedPoint96.Q96, liquidity)
            );

        return uint256(sqrtPX96).add(quotient).toUint160();
    } else {
        uint256 quotient =
            (
                amount &lt;= type(uint160).max
                    ? UnsafeMath.divRoundingUp(amount &lt;&lt; FixedPoint96.RESOLUTION, liquidity)
                    : FullMath.mulDivRoundingUp(amount, FixedPoint96.Q96, liquidity)
            );

        require(sqrtPX96 > quotient);
        // always fits 160 bits
        return uint160(sqrtPX96 - quotient);
    }
}

当add = true时，即按照公式1.6，根据 $\sqrt{P_b}$ 计算 $\sqrt{P_a}$
当add = false时，按照公式1.7，根据 $\sqrt{P_a}$ 计算 $\sqrt{P_b}$

getNextSqrtPriceFromInput

根据输入代币计算下一个价格，即添加amountIn数量的代币后的价格：

/// @notice Gets the next sqrt price given an input amount of token0 or token1
/// @dev Throws if price or liquidity are 0, or if the next price is out of bounds
/// @param sqrtPX96 The starting price, i.e., before accounting for the input amount
/// @param liquidity The amount of usable liquidity
/// @param amountIn How much of token0, or token1, is being swapped in
/// @param zeroForOne Whether the amount in is token0 or token1
/// @return sqrtQX96 The price after adding the input amount to token0 or token1
function getNextSqrtPriceFromInput(
    uint160 sqrtPX96,
    uint128 liquidity,
    uint256 amountIn,
    bool zeroForOne
) internal pure returns (uint160 sqrtQX96) {
    require(sqrtPX96 > 0);
    require(liquidity > 0);

    // round to make sure that we don't pass the target price
    return
        zeroForOne
            ? getNextSqrtPriceFromAmount0RoundingUp(sqrtPX96, liquidity, amountIn, true)
            : getNextSqrtPriceFromAmount1RoundingDown(sqrtPX96, liquidity, amountIn, true);
}

getNextSqrtPriceFromOutput

根据输出代币计算下一个价格，即移除amountOut数量的代币后的价格：

/// @notice Gets the next sqrt price given an output amount of token0 or token1
/// @dev Throws if price or liquidity are 0 or the next price is out of bounds
/// @param sqrtPX96 The starting price before accounting for the output amount
/// @param liquidity The amount of usable liquidity
/// @param amountOut How much of token0, or token1, is being swapped out
/// @param zeroForOne Whether the amount out is token0 or token1
/// @return sqrtQX96 The price after removing the output amount of token0 or token1
function getNextSqrtPriceFromOutput(
    uint160 sqrtPX96,
    uint128 liquidity,
    uint256 amountOut,
    bool zeroForOne
) internal pure returns (uint160 sqrtQX96) {
    require(sqrtPX96 > 0);
    require(liquidity > 0);

    // round to make sure that we pass the target price
    return
        zeroForOne
            ? getNextSqrtPriceFromAmount1RoundingDown(sqrtPX96, liquidity, amountOut, false)
            : getNextSqrtPriceFromAmount0RoundingUp(sqrtPX96, liquidity, amountOut, false);
}

getAmount0Delta

该方法计算白皮书中的公式6.16：

$$ \Delta{x} = \Delta{\frac{1}{\sqrt{P}}} \cdot L $$

展开公式为：

$$ amount0 = x_b - x_a = L \cdot (\frac{1}{\sqrt{P_b}} - \frac{1}{\sqrt{P_a}}) = L \cdot (\frac{\sqrt{P_a} - \sqrt{P_b}}{\sqrt{P_a} \cdot \sqrt{P_b}}) $$

/// @notice Gets the amount0 delta between two prices
/// @dev Calculates liquidity / sqrt(lower) - liquidity / sqrt(upper),
/// i.e. liquidity * (sqrt(upper) - sqrt(lower)) / (sqrt(upper) * sqrt(lower))
/// @param sqrtRatioAX96 A sqrt price
/// @param sqrtRatioBX96 Another sqrt price
/// @param liquidity The amount of usable liquidity
/// @param roundUp Whether to round the amount up or down
/// @return amount0 Amount of token0 required to cover a position of size liquidity between the two passed prices
function getAmount0Delta(
    uint160 sqrtRatioAX96,
    uint160 sqrtRatioBX96,
    uint128 liquidity,
    bool roundUp
) internal pure returns (uint256 amount0) {
    if (sqrtRatioAX96 > sqrtRatioBX96) (sqrtRatioAX96, sqrtRatioBX96) = (sqrtRatioBX96, sqrtRatioAX96);

    uint256 numerator1 = uint256(liquidity) &lt;&lt; FixedPoint96.RESOLUTION;
    uint256 numerator2 = sqrtRatioBX96 - sqrtRatioAX96;

    require(sqrtRatioAX96 > 0);

    return
        roundUp
            ? UnsafeMath.divRoundingUp(
                FullMath.mulDivRoundingUp(numerator1, numerator2, sqrtRatioBX96),
                sqrtRatioAX96
            )
            : FullMath.mulDiv(numerator1, numerator2, sqrtRatioBX96) / sqrtRatioAX96;
}

getAmount1Delta

同样，该方法计算白皮书公式6.7：

$$ L = \frac{\Delta{Y}}{\Delta{\sqrt{P}}} $$

展开公式为：

$$ amount1 = y_b - y_a = L \cdot \Delta{\sqrt{P}} = L \cdot (\sqrt{P_b} - \sqrt{P_a}) $$

/// @notice Gets the amount1 delta between two prices
/// @dev Calculates liquidity * (sqrt(upper) - sqrt(lower))
/// @param sqrtRatioAX96 A sqrt price
/// @param sqrtRatioBX96 Another sqrt price
/// @param liquidity The amount of usable liquidity
/// @param roundUp Whether to round the amount up, or down
/// @return amount1 Amount of token1 required to cover a position of size liquidity between the two passed prices
function getAmount1Delta(
    uint160 sqrtRatioAX96,
    uint160 sqrtRatioBX96,
    uint128 liquidity,
    bool roundUp
) internal pure returns (uint256 amount1) {
    if (sqrtRatioAX96 > sqrtRatioBX96) (sqrtRatioAX96, sqrtRatioBX96) = (sqrtRatioBX96, sqrtRatioAX96);

    return
        roundUp
            ? FullMath.mulDivRoundingUp(liquidity, sqrtRatioBX96 - sqrtRatioAX96, FixedPoint96.Q96)
            : FullMath.mulDiv(liquidity, sqrtRatioBX96 - sqrtRatioAX96, FixedPoint96.Q96);

Oracle.sol

本合约定义预言机相关方法。

在交易对合约创建时，默认初始化长度为1的数组用于存储观测点数据；任何用户都可以调用UniswapV3Pool.sol的increaseObservationCardinalityNext方法扩展预言机的观测点空间，但需要承担扩展空间带来的手续费。

一个预言机观测点包括以下内容：

blockTimestamp：该观测点写入时的时间
tickCumulative：截止该观测点时间的累计tick
secondsPerLiquidityCumulativeX128：截止该观测点的累计每流动性持续时间
initialized：观测点是否初始化

观测点结构定义如下：

struct Observation {
    // the block timestamp of the observation
    uint32 blockTimestamp;
    // the tick accumulator, i.e. tick * time elapsed since the pool was first initialized
    int56 tickCumulative;
    // the seconds per liquidity, i.e. seconds elapsed / max(1, liquidity) since the pool was first initialized
    uint160 secondsPerLiquidityCumulativeX128;
    // whether or not the observation is initialized
    bool initialized;
}

本合约包含以下方法：

transform：基于上一个观测点，返回新观测点对象（但是不写入观测点）
initialize：初始化观测点数组
write：写入一个观测点
grow：扩展预言机观测点空间
lte：判断两个timestamp大小
binarySearch：二分查找指定时间的观测点边界
getSurroundingObservations：获取指定时间的观测点边界
observeSingle：获取指定时间的观测点数据
observe：批量获取指定时间的观测点数据

transform

基于上一个观测点，返回临时观测点对象，但是不写入观测点。

function transform(
    Observation memory last,
    uint32 blockTimestamp,
    int24 tick,
    uint128 liquidity
) private pure returns (Observation memory) {
    uint32 delta = blockTimestamp - last.blockTimestamp;
    return
        Observation({
            blockTimestamp: blockTimestamp,
            tickCumulative: last.tickCumulative + int56(tick) * delta,
            secondsPerLiquidityCumulativeX128: last.secondsPerLiquidityCumulativeX128 +
                ((uint160(delta) &lt;&lt; 128) / (liquidity > 0 ? liquidity : 1)),
            initialized: true
        });
}

首先，计算当前时间与上一个观测点的时间差：delta = blockTimestamp - last.blockTimestamp。

接着主要计算tickCumulative和secondsPerLiquidityCumulativeX128,

其中：tickCumulative: last.tickCumulative + int56(tick) * delta。

根据白皮书公式5.3-5.5：

$$ \log{1.0001}(P{t_1,t2}) = \frac{\sum{i=t_1}^{t2} \log{1.0001}(P_i)}{t_2 - t_1} \quad \text{(5.3)} $$

$$ \log{1.0001}(P{t_1,t2}) = \frac{a{t2} - a{t_1}}{t_2 - t_1} \quad \text{(5.4)} $$

$$ P_{t_1,t2} = 1.0001^{\frac{a{t2} - a{t_1}}{t_2 - t_1}} \quad \text{(5.5)} $$

这里保存的tickCumulative即为 $a_{t_n}$ ，其对应的公式为：

$$ tickCumulative = \sum_{i=0}^{tn} \log{1.0001}(P_i) $$

同样，每流动性持续时间secondsPerLiquidityCumulative为：

$$ secondsPerLiquidityCumulative = \sum_{i=0}^{n} \frac{t_i}{L_i} $$

initialize

初始化预言机存储空间，设置第一个slot。

/// @notice Initialize the oracle array by writing the first slot. Called once for the lifecycle of the observations array
/// @param self The stored oracle array
/// @param time The time of the oracle initialization, via block.timestamp truncated to uint32
/// @return cardinality The number of populated elements in the oracle array
/// @return cardinalityNext The new length of the oracle array, independent of population
function initialize(Observation[65535] storage self, uint32 time)
    internal
    returns (uint16 cardinality, uint16 cardinalityNext)
{
    self[0] = Observation({
        blockTimestamp: time,
        tickCumulative: 0,
        secondsPerLiquidityCumulativeX128: 0,
        initialized: true
    });
    return (1, 1);
}

默认返回的cardinality和cardinalityNext都为1，即仅能存放一个观测点数据。

该方法只能被调用一次，实际上是在UniswapV3Pool.sol的initialize方法中调用：

(uint16 cardinality, uint16 cardinalityNext) = observations.initialize(_blockTimestamp());

write

写入一次观测点数据。根据之前的描述，只有在UniswapV3Pool发生mint、burn和swap操作时，才可能触发写入操作。

可以将预言机观测点数组看作一个循环列表，可写入空间由cardinality和cardinalityNext决定；当数组空间写满以后，会继续从0位置开始覆盖写。

本方法的参数如下：

self：预言机观测点数组
index：最后一次写入的观测点索引，从0开始
blockTimestamp：待添加观测点的时间
tick：待添加观测点的tick
liquidity：待添加观测点时间的全局可用流动性
cardinality：观测点数组当前长度（可写入空间）
cardinalityNext：观测点数组（扩展后的）最新长度

function write(
    Observation[65535] storage self,
    uint16 index,
    uint32 blockTimestamp,
    int24 tick,
    uint128 liquidity,
    uint16 cardinality,
    uint16 cardinalityNext
) internal returns (uint16 indexUpdated, uint16 cardinalityUpdated) {
    Observation memory last = self[index];

    // early return if we've already written an observation this block
    if (last.blockTimestamp == blockTimestamp) return (index, cardinality);

如果新观测点时间与最后一次观测点时间相同，则直接返回，确保每个区块最多只能写入一次观测点。

    // if the conditions are right, we can bump the cardinality
    if (cardinalityNext > cardinality && index == (cardinality - 1)) {
        cardinalityUpdated = cardinalityNext;
    } else {
        cardinalityUpdated = cardinality;
    }

如果cardinalityNext > cardinality，则表示预言机数组被扩容过；如果index == (cardinality - 1)即上一次写入的位置是最后一个观测点，则本次需要继续写入扩容后的空间，因此cardinalityUpdated使用扩容后的数组长度cardinalityNext；
否则继续使用旧的数组长度cardinality，因为还未写满。

    indexUpdated = (index + 1) % cardinalityUpdated;

更新本次写入观测点数组的索引indexUpdated，% cardinalityUpdated是为了计算循环写的索引。

    self[indexUpdated] = transform(last, blockTimestamp, tick, liquidity);

调用transform方法计算最新的观测点数据，并写入观测点数组的indexUpdated位置。

grow

注意，grow方法使用internal修饰，用户实际上是通过UniswapV3Pool.sol的increaseObservationCardinalityNext方法进行扩容。

扩容方法会触发SSTORE操作，因此调用该方法的用户需要支付由此带来的gas开销。

/// @notice Prepares the oracle array to store up to `next` observations
/// @param self The stored oracle array
/// @param current The current next cardinality of the oracle array
/// @param next The proposed next cardinality which will be populated in the oracle array
/// @return next The next cardinality which will be populated in the oracle array
function grow(
    Observation[65535] storage self,
    uint16 current,
    uint16 next
) internal returns (uint16) {
    require(current > 0, 'I');
    // no-op if the passed next value isn't greater than the current next value
    if (next &lt;= current) return current;
    // store in each slot to prevent fresh SSTOREs in swaps
    // this data will not be used because the initialized boolean is still false
    for (uint16 i = current; i &lt; next; i++) self[i].blockTimestamp = 1;
    return next;
}

lte

比较两个时间戳的大小。

由于合约使用uint32类型表示时间戳，其最大值 $2^{32}-1 = 4294967295$，对应的时间为February 7, 2106 6:28:15 AM GMT+00:00，如果两个时间a和b（ $a < b$ ）正好位于uint32最大值的两边，b由于溢出，因此 $a > b$，直接比较数值会导致错误的结果，因此需要统一时间。

注：实际上每136年左右就会有溢出问题。

方法接受3个参数：time、a和b；其中，time为基准时间，a和b在逻辑（时间）上小于等于time；方法返回一个bool，表示a是否在逻辑时间点上小于等于b。

/// @notice comparator for 32-bit timestamps
/// @dev safe for 0 or 1 overflows, a and b _must_ be chronologically before or equal to time
/// @param time A timestamp truncated to 32 bits
/// @param a A comparison timestamp from which to determine the relative position of `time`
/// @param b From which to determine the relative position of `time`
/// @return bool Whether `a` is chronologically &lt;= `b`
function lte(
    uint32 time,
    uint32 a,
    uint32 b
) private pure returns (bool) {
    // if there hasn't been overflow, no need to adjust
    if (a &lt;= time && b &lt;= time) return a &lt;= b;

    uint256 aAdjusted = a > time ? a : a + 2**32;
    uint256 bAdjusted = b > time ? b : b + 2**32;

    return aAdjusted &lt;= bAdjusted;
}

如果a和b都小于等于time，则表示没有发生溢出，因此直接返回a <= b
由于a和b在时间线上均小于等于time
- 如果 $a > time$ ，则表示发生溢出，aAdjusted为补齐2**32后的时间a
- 同理，bAdjusted为补齐后的时间b
- 最后比较两个补齐后的时间aAdjusted和bAdjusted

因此，该方法在a、b与time存在0到1个 $2^{32}$ 的时间差时是溢出安全的。

不能跨越两个 $2^{32} - 1$ 。

binarySearch

二分查找指定目标时间的观测点。

参数如下：

self：观测点数组
time：当前区块时间
target：目标时间
index：最后一次写入的观测点索引
cardinality：观测点数组当前长度（可写入空间）

/// @notice Fetches the observations beforeOrAt and atOrAfter a target, i.e. where [beforeOrAt, atOrAfter] is satisfied.
/// The result may be the same observation, or adjacent observations.
/// @dev The answer must be contained in the array, used when the target is located within the stored observation
/// boundaries: older than the most recent observation and younger, or the same age as, the oldest observation
/// @param self The stored oracle array
/// @param time The current block.timestamp
/// @param target The timestamp at which the reserved observation should be for
/// @param index The index of the observation that was most recently written to the observations array
/// @param cardinality The number of populated elements in the oracle array
/// @return beforeOrAt The observation recorded before, or at, the target
/// @return atOrAfter The observation recorded at, or after, the target
function binarySearch(
    Observation[65535] storage self,
    uint32 time,
    uint32 target,
    uint16 index,
    uint16 cardinality
) private view returns (Observation memory beforeOrAt, Observation memory atOrAfter) {

对于二分查找算法，首先需要确认左右边界点。

    uint256 l = (index + 1) % cardinality; // oldest observation
    uint256 r = l + cardinality - 1; // newest observation

由于最后一次索引为index，因此循环继续右移一位（并取模）即为左边界点l（最老的索引，如果已写满一轮的话）；右边界点为l + cardinality - 1，注意，右边界点r没有取模，因为在二分查找中右边节点一定不能小于左边节点。

    uint256 i;
    while (true) {
        i = (l + r) / 2;

        beforeOrAt = self[i % cardinality];

        // we've landed on an uninitialized tick, keep searching higher (more recently)
        if (!beforeOrAt.initialized) {
            l = i + 1;
            continue;
        }

        atOrAfter = self[(i + 1) % cardinality];

如果计算的中点未初始化（即此时数组空间未写满），则使用右半部分区间（往时间点更近的方向）继续进行二分查找。

atOrAfter为右侧紧邻的观测点。

        bool targetAtOrAfter = lte(time, beforeOrAt.blockTimestamp, target);

        // check if we've found the answer!
        if (targetAtOrAfter && lte(time, target, atOrAfter.blockTimestamp)) break;

        if (!targetAtOrAfter) r = i - 1;
        else l = i + 1;
    }
}

如果目标时间target位于beforeOrAt与atOrAfter时间之间，则退出二分查找，返回beforeOrAt与atOrAfter两个观测点
否则：
- 如果beforeOrAt时间大于目标时间target，则继续在左半部分（往更小的时间）进行二分查找
- 如果目标时间target大于atOrAfter时间，则继续往右半部分（往更大的时间）进行二分查找

假设cardinality为10，index为5，我们可以画出几个变量的逻辑关系如下：

getSurroundingObservations

获取目标时间target的观测点数据beforeOrAt和atOrAfter，满足target位于[beforeOrAt, atOrAfter]之间。

/// @notice Fetches the observations beforeOrAt and atOrAfter a given target, i.e. where [beforeOrAt, atOrAfter] is satisfied
/// @dev Assumes there is at least 1 initialized observation.
/// Used by observeSingle() to compute the counterfactual accumulator values as of a given block timestamp.
/// @param self The stored oracle array
/// @param time The current block.timestamp
/// @param target The timestamp at which the reserved observation should be for
/// @param tick The active tick at the time of the returned or simulated observation
/// @param index The index of the observation that was most recently written to the observations array
/// @param liquidity The total pool liquidity at the time of the call
/// @param cardinality The number of populated elements in the oracle array
/// @return beforeOrAt The observation which occurred at, or before, the given timestamp
/// @return atOrAfter The observation which occurred at, or after, the given timestamp
function getSurroundingObservations(
    Observation[65535] storage self,
    uint32 time,
    uint32 target,
    int24 tick,
    uint16 index,
    uint128 liquidity,
    uint16 cardinality
) private view returns (Observation memory beforeOrAt, Observation memory atOrAfter) {
    // optimistically set before to the newest observation
    beforeOrAt = self[index];

    // if the target is chronologically at or after the newest observation, we can early return
    if (lte(time, beforeOrAt.blockTimestamp, target)) {
        if (beforeOrAt.blockTimestamp == target) {
            // if newest observation equals target, we're in the same block, so we can ignore atOrAfter
            return (beforeOrAt, atOrAfter);
        } else {
            // otherwise, we need to transform
            return (beforeOrAt, transform(beforeOrAt, target, tick, liquidity));
        }
    }

首先将beforeOrAt设置为最近一次的观测点：

如果beforeOrAt时间小于等于目标时间target：

如果等于目标时间，则直接返回beforeOrAt，忽略atOrAfter
如果小于目标时间，则使用transform基于beforeOrAt和target临时生成一个观测点作为atOrAfter

    // now, set before to the oldest observation
    beforeOrAt = self[(index + 1) % cardinality];
    if (!beforeOrAt.initialized) beforeOrAt = self[0];

否则，可以确认最后（最晚）一个观测点的时间大于target。

设置beforeOrAt为循环右移一个位置的观测点，即最老的观测点；如果该观测点未初始化，则表示数组没有写满，因此最早的一定是索引为0的观测点。

    // ensure that the target is chronologically at or after the oldest observation
    require(lte(time, beforeOrAt.blockTimestamp, target), 'OLD');

确认target时间大于等于最早的观测点时间，因此此时target一定位于最早的和最晚观测点的时间区间内，可以使用binarySearch进行二分查找。

    // if we've reached this point, we have to binary search
    return binarySearch(self, time, target, index, cardinality);
}

observeSingle

获取指定时间的观测点数据。

方法的参数如下：

self：预言机观测点数组
time：当前区块时间
secondsAgo：距离当前时间多少秒以前的指定时间，根据该时间寻找观测点
tick：目标时间的tick（价格）
index：最后一次写入的观测点索引
liquidity：当前全局可用流动性
cardinality：预言机数组当前长度（可写入空间）

tickCumulative：从交易对池子创建以来，截止到secondsAgo的累计tick
secondsPerLiquidityCumulativeX128：从交易对池子创建以来，截止到secondsAgo的累计每流动性持续时间

/// @dev Reverts if an observation at or before the desired observation timestamp does not exist.
/// 0 may be passed as `secondsAgo' to return the current cumulative values.
/// If called with a timestamp falling between two observations, returns the counterfactual accumulator values
/// at exactly the timestamp between the two observations.
/// @param self The stored oracle array
/// @param time The current block timestamp
/// @param secondsAgo The amount of time to look back, in seconds, at which point to return an observation
/// @param tick The current tick
/// @param index The index of the observation that was most recently written to the observations array
/// @param liquidity The current in-range pool liquidity
/// @param cardinality The number of populated elements in the oracle array
/// @return tickCumulative The tick * time elapsed since the pool was first initialized, as of `secondsAgo`
/// @return secondsPerLiquidityCumulativeX128 The time elapsed / max(1, liquidity) since the pool was first initialized, as of `secondsAgo`
function observeSingle(
    Observation[65535] storage self,
    uint32 time,
    uint32 secondsAgo,
    int24 tick,
    uint16 index,
    uint128 liquidity,
    uint16 cardinality
) internal view returns (int56 tickCumulative, uint160 secondsPerLiquidityCumulativeX128) {
    if (secondsAgo == 0) {
        Observation memory last = self[index];
        if (last.blockTimestamp != time) last = transform(last, time, tick, liquidity);
        return (last.tickCumulative, last.secondsPerLiquidityCumulativeX128);
    }

如果secondsAgo == 0，表示取当前时间的观测点。如果当前时间不等于最后一次写入的观测点时间，则使用transform生成一个临时观测点（注意，这里没有写入该观测点），并返回相关数据。

    uint32 target = time - secondsAgo;

    (Observation memory beforeOrAt, Observation memory atOrAfter) =
        getSurroundingObservations(self, time, target, tick, index, liquidity, cardinality);

根据secondsAgo计算目标时间target，使用getSurroundingObservations方法寻找距离目标时间最近的观测点边界beforeOrAt和atOrAfter。

    if (target == beforeOrAt.blockTimestamp) {
        // we're at the left boundary
        return (beforeOrAt.tickCumulative, beforeOrAt.secondsPerLiquidityCumulativeX128);
    } else if (target == atOrAfter.blockTimestamp) {
        // we're at the right boundary
        return (atOrAfter.tickCumulative, atOrAfter.secondsPerLiquidityCumulativeX128);
    } else {
        // we're in the middle
        uint32 observationTimeDelta = atOrAfter.blockTimestamp - beforeOrAt.blockTimestamp;
        uint32 targetDelta = target - beforeOrAt.blockTimestamp;
        return (
            beforeOrAt.tickCumulative +
                ((atOrAfter.tickCumulative - beforeOrAt.tickCumulative) / observationTimeDelta) *
                targetDelta,
            beforeOrAt.secondsPerLiquidityCumulativeX128 +
                uint160(
                    (uint256(
                        atOrAfter.secondsPerLiquidityCumulativeX128 - beforeOrAt.secondsPerLiquidityCumulativeX128
                    ) * targetDelta) / observationTimeDelta
                )
        );
    }

根据getSurroundingObservations方法，需要优先使用左边界点beforeOrAt：

如果目标时间等于beforeOrAt时间，则直接返回该观测点的相关数据
如果目标时间等于atOrAfter时间，则也返回相关数据
如果目标时间介于beforeOrAt和atOrAfter之间，则需要根据时间比例计算相关值：
- observationTimeDelta为beforeOrAt和atOrAfter的时间差值（下面用 $\Delta{t}$ 表示），targetDelta为beforeOrAt和target的时间差值
- 因为 $\Delta{tickCumulative} = tick \cdot \Delta{t}$，则截止到target的值应为： $\frac{\Delta{tickCumulative}}{\Delta{t}} \cdot targetDelta$
- 同样， $\Delta{secondsPerLiquidityCumulativeX128} = \frac{\Delta{t}}{liquidity}$，则截止到target的值应为： $\frac{\Delta{secondsPerLiquidityCumulativeX128}}{\Delta{t}} \cdot targetDelta$

observe

批量获取指定时间的观测点数据。

该方法主要调用observeSingle获取单个指定时间的观测点数据，然后批量返回。

/// @notice Returns the accumulator values as of each time seconds ago from the given time in the array of `secondsAgos`
/// @dev Reverts if `secondsAgos` > oldest observation
/// @param self The stored oracle array
/// @param time The current block.timestamp
/// @param secondsAgos Each amount of time to look back, in seconds, at which point to return an observation
/// @param tick The current tick
/// @param index The index of the observation that was most recently written to the observations array
/// @param liquidity The current in-range pool liquidity
/// @param cardinality The number of populated elements in the oracle array
/// @return tickCumulatives The tick * time elapsed since the pool was first initialized, as of each `secondsAgo`
/// @return secondsPerLiquidityCumulativeX128s The cumulative seconds / max(1, liquidity) since the pool was first initialized, as of each `secondsAgo`
function observe(
    Observation[65535] storage self,
    uint32 time,
    uint32[] memory secondsAgos,
    int24 tick,
    uint16 index,
    uint128 liquidity,
    uint16 cardinality
) internal view returns (int56[] memory tickCumulatives, uint160[] memory secondsPerLiquidityCumulativeX128s) {
    require(cardinality > 0, 'I');

    tickCumulatives = new int56[](secondsAgos.length);
    secondsPerLiquidityCumulativeX128s = new uint160[](secondsAgos.length);
    for (uint256 i = 0; i &lt; secondsAgos.length; i++) {
        (tickCumulatives[i], secondsPerLiquidityCumulativeX128s[i]) = observeSingle(
            self,
            time,
            secondsAgos[i],
            tick,
            index,
            liquidity,
            cardinality
        );
    }
}

本文转载自： github.com/adshao/public...

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深入理解 Uniswap v3 智能合约 - Core（一）

深入理解 Uniswap v3 智能合约 （一）

概述

Uniswap-v3-core

UniswapV3Factory.sol

createPool

setOwner

enableFeeAmount

UniswapV3Pool.sol

initialize

mint

_modifyPosition

_updatePosition

burn

swap

flash

collect

increaseObservationCardinalityNext

observe

setFeeProtocol

collectProtocol

Tick.sol

tickSpacingToMaxLiquidityPerTick

getFeeGrowthInside

update

clear

cross

TickMath.sol

getSqrtRatioAtTick

getTickAtSqrtRatio

整数部分

小数部分

TickBitmap.sol

position

flipTick

nextInitializedTickWithinOneWord

Position.sol

get

update

SwapMath.sol

computeSwapStep

SqrtPriceMath.sol

getNextSqrtPriceFromAmount0RoundingUp

getNextSqrtPriceFromAmount1RoundingDown

getNextSqrtPriceFromInput

getNextSqrtPriceFromOutput

getAmount0Delta

getAmount1Delta

Oracle.sol

transform

initialize

write

grow

lte

binarySearch

getSurroundingObservations

observeSingle

observe

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深入理解 Uniswap v3 智能合约（一）