The Relationship Between Cartesian and Polar Coordinates | 笛卡尔坐标与极坐标的关系

📚 The Relationship Between Cartesian and Polar Coordinates | 笛卡尔坐标与极坐标的关系

In AQA A-level Mathematics, polar coordinates offer an alternative way to describe points in the plane using a distance from the origin and an angle from the positive x-axis. Understanding how Cartesian and polar systems are linked is essential for converting equations, sketching curves such as cardioids and limacons, and tackling calculus problems involving polar curves.

在 AQA A-level 数学中，极坐标提供了另一种描述平面上点的方式，使用到原点的距离和从正 x 轴起算的角度。理解直角坐标与极坐标之间的联系，对于转换方程、绘制心形线和蜗线等曲线、以及处理涉及极坐标曲线的微积分问题都至关重要。

1. Why Polar Coordinates Matter | 为什么需要极坐标

Some plane curves, especially those involving circles, spirals, and rotations, have much simpler equations in polar coordinates than in Cartesian coordinates. For example, a circle centred at the origin is just r = a, whereas in Cartesian form it is x²+y²=a². This simplicity makes polar coordinates powerful for both sketching and integrating rotational curves.

有些平面曲线，特别是涉及圆、螺线和旋转的曲线，在极坐标下的方程比直角坐标下简单得多。例如，圆心在原点的圆只是 r = a，而在直角坐标下为 x²+y²=a²。这种简洁性使极坐标在绘制旋转曲线和积分时非常有用。

2. Defining the Polar Coordinate System | 极坐标系的定义

A point P is specified by (r, θ), where r is the distance OP from the pole O and θ is the angle from the initial line to OP, measured anticlockwise as positive. The same Cartesian point can have infinitely many polar representations, such as (r, θ+2πn) and (−r, θ+π), where n is an integer.

点 P 由 (r, θ) 确定，其中 r 是从极点 O 到 P 的距离 OP，θ 是从极轴到 OP 的角度，逆时针为正。同一个直角坐标点可以有无穷多种极坐标表示，例如 (r, θ+2πn) 和 (−r, θ+π)，其中 n 为整数。

Polar graph paper uses concentric circles for constant r and radial lines for constant θ, so a point (r, θ) is plotted by rotating θ from the initial line and moving r units along that direction.

极坐标图纸用同心圆表示常值 r，用射线表示常值 θ，因此绘制点 (r, θ) 时从极轴旋转 θ，再沿该方向移动 r 个单位。

3. Core Conversion Formulas | 核心转换公式

From the definitions of cosine and sine in a right-angled triangle, the Cartesian coordinates are x = r cos θ and y = r sin θ. Squaring and adding gives r² = x² + y², hence r = √(x²+y²). Dividing gives tan θ = y/x for x ≠ 0.

根据直角三角形中余弦和正弦的定义，直角坐标为 x = r cos θ、y = r sin θ。将两式平方相加得 r² = x² + y²，因此 r = √(x²+y²)。两式相除得 tan θ = y/x（x ≠ 0）。

x = r cos θ, y = r sin θ

r² = x² + y², tan θ = y/x

These formulas assume the pole is at the origin and the initial line is the positive x-axis. When using negative r, the point is plotted in the opposite direction from the angle θ.

这些公式假设极点位于原点，极轴为正 x 轴。当 r 为负时，点沿角 θ 的反方向绘制。

4. Finding r and θ from Cartesian Coordinates | 由直角坐标求 r 和 θ

Given (x,y), first compute r = √(x²+y²), which is always non-negative if we use the principal polar form. To find θ, start with arctan(y/x) and adjust according to the quadrant.

给定 (x,y)，先计算 r = √(x²+y²)，若使用主值极坐标形式，r 始终非负。求 θ 时，先取 arctan(y/x)，再根据象限进行调整。

For example, the point (−1,1) has r = √2, and arctan(−1) = −π/4. Because the point lies in the second quadrant, θ = −π/4 + π = 3π/4. For (1,−1), if we require 0 ≤ θ < 2π, then θ = 7π/4.

例如，点 (−1,1) 的 r = √2，而 arctan(−1) = −π/4，因点在第二象限，所以 θ = −π/4 + π = 3π/4。点 (1,−1) 若要求 0 ≤ θ < 2π，则 θ = 7π/4。

5. Converting Polar Equations to Cartesian Form | 将极坐标方程化为直角坐标方程

To convert r = f(θ), use r² = x²+y², cos θ = x/r, sin θ = y/r, and sometimes tan θ = y/x. Multiplying both sides of an equation by r often helps eliminate the denominator.

要将 r = f(θ) 转换，使用 r² = x²+y²、cos θ = x/r、sin θ = y/r，有时也使用 tan θ = y/x。将方程两边乘以 r 通常有助于消去分母

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