Polar Coordinates | 极坐标

📚 Polar Coordinates | 极坐标

Polar coordinates offer a useful alternative to Cartesian coordinates when a curve is naturally described by a distance from a fixed point and an angle from a fixed direction. In AQA A-Level Mathematics, you will be expected to convert between polar and Cartesian forms, sketch standard polar curves, find tangents, and calculate areas enclosed by polar curves.

极坐标是直角坐标之外的另一种坐标系统，当曲线需要由到固定点的距离和从固定方向量起的角度来描述时尤其有用。在 AQA A-Level 数学中，你需要掌握极坐标与直角坐标的互化、常见极坐标曲线的作图、切线斜率求解以及极坐标曲线围成面积的计算。

1. What Are Polar Coordinates? | 什么是极坐标

In polar coordinates, a point P is located by a distance r from a fixed point O, called the pole, and an angle θ measured from a fixed ray, called the initial line or polar axis.

在极坐标中，一个点 P 的位置由到固定点 O（称为极点）的距离 r 以及从固定射线（称为初始线或极轴）量起的角 θ 确定。

The angle θ is usually measured in radians, and anticlockwise is taken as positive. The point with polar coordinates (r, θ) is not unique: adding any integer multiple of 2π to θ gives the same point.

角 θ 通常以弧度为单位，逆时针方向为正。极坐标 (r, θ) 表示的点并不唯一，给 θ 加上任意 2π 的整数倍仍然表示同一点。

On AQA papers, r is normally taken to be non-negative, but you should know that (-r, θ) represents the same ray as (r, θ + π). This can help when θ is outside the usual 0 to 2π interval.

在 AQA 试题中，r 通常取非负值，但你应知道 (-r, θ) 与 (r, θ + π) 表示同一条射线。这在 θ 超出常规 0 到 2π 区间时会很有用。

2. Converting Between Polar and Cartesian Coordinates | 极坐标与直角坐标的互化

The conversion equations come directly from right-angled trigonometry. If the initial line is the positive x-axis and the pole is the origin, then:

转换公式直接来自直角三角形中的三角关系。若以极点为原点、初始线为正 x 轴，则有：

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