Representation of Curves in Polar Form | 极坐标形式下曲线的表示

📚 Representation of Curves in Polar Form | 极坐标形式下曲线的表示

In Cartesian coordinates, a point is located by horizontal and vertical distances, x and y. Polar coordinates instead locate a point by its direct distance r from a fixed pole and an angle θ measured from an initial line. This representation often makes circles, cardioids, roses and spirals far simpler to describe, and it is a key topic in AQA A-level Mathematics.

在直角坐标中，一个点由水平距离 x 和垂直距离 y 确定；而极坐标则用点到极点 O 的直接距离 r 以及从极轴量起的角度 θ 来确定位置。这种表示方法常常使圆、心形线、玫瑰线和螺线的方程更加简洁，是 AQA A-level 数学的重要考点。

1. Polar Coordinate System | 极坐标系

The pole is the fixed origin O, and the initial line is usually the positive x-axis. A point P is specified by an ordered pair (r, θ), where r is the distance OP and θ is the angle from the initial line to OP, measured anticlockwise.

极点是固定原点 O，极轴通常取正 x 轴。点 P 用有序数对 (r, θ) 表示，其中 r 是 OP 的距离，θ 是从极轴到 OP 的逆时针夹角。

In AQA questions, θ is usually in radians unless otherwise stated. The principal range is often 0 ≤ θ < 2π, but negative values and angles outside this range can be used for convenience.

AQA 试题中，除另有说明外 θ 通常以弧度为单位。主值范围常取 0 ≤ θ < 2π，但为了方便也可以使用负角或超出该范围的角。

If r is negative, the point (−r, θ) is plotted in the opposite direction along the line θ, so it is the same as (r, θ + π).

如果 r 为负，点 (−r, θ)

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