0 / 4 页已完成

弧度 — 另一种角度单位Radians — Another Unit for Angles

4.1 为什么要学第二种单位？4.1 Why Learn a Second Unit?

我们平时用"度"来表示角度，这是给人看的 -- 0 度、90 度、180 度，很直觉。We normally use "degrees" for angles — it's intuitive for humans: 0°, 90°, 180°.

但 C++ 代码里的 sin() 和 cos() 函数不接受度，它们只接受另一种单位：弧度。But in C++ code, the sin() and cos() functions don't accept degrees — they only accept another unit: radians.

常见 BugCommon Bug

写 sin(90) 不会得到 1。因为 C++ 的 sin() 会把 90 当作 90 弧度（大约是 5156 度），结果完全不对！Writing sin(90) won't give you 1. C++'s sin() treats 90 as 90 radians (about 5156 degrees) — a completely wrong result!

正确写法：sin(90 * M_PI / 180) 或者 sin(1.5708)Correct way: sin(90 * M_PI / 180) or sin(1.5708)

而且弧度还有一个超好用的公式（下面会讲）。所以弧度不只是"另一种写法"，它在数学上有独特的优势。Plus, radians have a super useful formula (explained below). So radians aren't just "another way of writing" — they have unique mathematical advantages.

等一下，π 是什么？Wait, what is π?

π（读作 "派"）≈ 3.14159，是一个数学常数。π (pronounced "pie") ≈ 3.14159, is a mathematical constant.

它的含义：任何圆的周长 ÷ 直径 = π。不管圆有多大多小，这个比值永远是 π。Its meaning: circumference of any circle ÷ diameter = π. No matter how big or small the circle, this ratio is always π.

记住 π ≈ 3.14 就够用了。代码里写 M_PI 就是 π。Just remember π ≈ 3.14. In code, M_PI represents π.

4.2 弧度的定义4.2 Definition of Radians

换算关系很简单：The conversion is simple:

度和弧度的换算Converting Between Degrees and Radians

一圈 = 360 度 = 2π 弧度（约 6.28）Full circle = 360° = 2π radians (about 6.28)
半圈 = 180 度 = π 弧度（约 3.14）Half circle = 180° = π radians (about 3.14)
四分之一圈 = 90 度 = π/2 弧度（约 1.57）Quarter circle = 90° = π/2 radians (about 1.57)

换算公式：Conversion formula:

弧度 = 角度 × π / 180radians = degrees × π / 180

在代码里，通常定义一个宏或常量来做转换：In code, you typically define a macro or constant for the conversion:

C++

// 度转弧度
double toRad(double deg) {
    return deg * M_PI / 180.0;
}

// 弧度转度
double toDeg(double rad) {
    return rad * 180.0 / M_PI;
}

4.3 弧度的杀手级应用4.3 The Killer App of Radians

弧度有一个非常简洁的公式：Radians have a beautifully simple formula:

弧长公式（只对弧度有效！）Arc Length Formula (Only works with radians!)

弧长 = 半径 × 弧度Arc length = radius × radians

只有用弧度，这个公式才成立。用度不行。This formula only works with radians, not degrees.

举个例子：半径 10cm 的圆弧，转了 90 度（= π/2 弧度），弧长是多少？Example: an arc with radius 10cm, turned 90 degrees (= π/2 radians). What's the arc length?

弧长 = 10 × π/2 = 10 × 1.5708 = 15.7cmArc length = 10 × π/2 = 10 × 1.5708 = 15.7cm

这个公式在定位轮里非常重要 -- 定位轮的轮子转过的弧长，就等于半径乘以转过的弧度。这是定位轮测距的数学基础。This formula is crucial for tracking wheels — the arc length traveled by the wheel equals the radius times the radians turned. This is the mathematical foundation of tracking wheel distance measurement.

检查点Checkpoint

代码里写 sin(90) 会得到正确结果吗？Will writing sin(90) in code give the correct result?

准备好了！You're Ready!

你已经掌握了理解定位轮需要的所有数学工具。回顾一下：You've mastered all the math tools needed to understand tracking wheels. Let's review:

坐标 = 用两个数字描述位置（x 是左右，y 是前后）Coordinates = describe position with two numbers (x is left-right, y is front-back)
角度 = 描述机器人朝哪个方向Angles = describe which direction the robot is facing
sin / cos = 把"斜着走"拆成"往右多少 + 往前多少"sin / cos = split "diagonal movement" into "how far right + how far forward"
弧度 = 代码里使用的角度单位，弧长 = 半径 × 弧度Radians = the angle unit used in code; arc length = radius × radians

这四个工具组合在一起，就能理解定位轮的全部数学原理了。These four tools combined let you understand all the math behind tracking wheels.