📚 Polar Coordinates and Their Applications | 极坐标方程及其应用
Cartesian coordinates describe points using horizontal and vertical distances from an origin, but many curves are described more naturally by a distance from a fixed point and an angle from a fixed direction. Polar coordinates provide a compact and powerful alternative.
直角坐标系用水平距离和垂直距离描述点的位置，但许多曲线更适合用“到一个固定点的距离”和“与固定方向的夹角”来描述。极坐标系为此提供了一种简洁而强大的工具。
This revision guide covers the core definitions, coordinate conversions, standard curve shapes, tangent slopes, area and arc length formulas, and practical applications of polar equations.
本复习指南涵盖极坐标的基本定义、坐标转换、常见曲线图像、切线斜率、面积与弧长公式，以及极坐标方程的实际应用。
1. The Polar Coordinate System | 极坐标系
In the polar coordinate system, a fixed point O is called the pole, and a fixed ray from O is called the polar axis. A point P is represented by an ordered pair (r, θ), where r is the directed distance from the pole to P, and θ is the directed angle measured counterclockwise from the polar axis.
在极坐标系中，固定点 O 称为极点，从 O 出发的一条固定射线称为极轴。点 P 用有序数对 (r, θ) 表示，其中 r 是极点 O 到 P 的有向距离，θ 是从极轴出发逆时针旋转到 OP 方向的有向角。
The coordinate θ is usually measured in radians. If r is positive, P lies in the direction of angle θ; if r is negative, P lies in the opposite direction.
角 θ 通常以弧度为单位。若 r 为正，点 P 位于角 θ 所指的方向；若 r 为负，点 P 位于角 θ 所指方向的相反方向上。
Because angles are periodic, a polar point has infinitely many equivalent representations: (r, θ + 2πn) and (-r, θ + (2n+1)π), where n is any integer.
由于角度具有周期性，一个极坐标点拥有无限多种等价表示：(r, θ + 2πn) 和 (-r, θ + (2n+1)π)，其中 n 为任意整数。
2. Converting Between Polar and Cartesian Coordinates | 极坐标与直角坐标的互化
To convert polar coordinates to Cartesian coordinates, use the two relations below.
将极坐标转换为直角坐标时，使用下面两个关系式。
x = r cos θ, y = r sin θ
To convert Cartesian coordinates to polar coordinates, use the distance formula and the tangent ratio.
将直角坐标转换为极坐标时，使用距离公式和正切比值。
r² = x² + y², tan θ = y / x (x ≠ 0)
When finding θ, you must select the quadrant that matches the sign of x and y. Simply using tan⁻¹(y/x) alone can give the wrong angle.
求 θ 时必须根据 x 和 y 的符号选择正确的象限。仅使用 tan⁻¹(y/x) 可能会得到错误的角度。
| Conversion | 转换 | Formula | 公式 |
| Polar → Cartesian 极坐标 → 直角坐标 |
x = r cos θ, y = r sin θ |
| Cartesian → Polar 直角坐标 → 极坐标 |
r² = x² + y², tan θ = y/x |
For example, the point with polar coordinates (2, π/3) has Cartesian coordinates x = 2 cos(π/3) = 1 and y = 2 sin(π/3) = √3.
例如，极坐标 (2, π/3) 对应的直角坐标为 x = 2 cos(π/3) = 1，y = 2 sin(π/3) = √3。
3. Plotting Polar Curves | 绘制极坐标曲线
To sketch a polar curve, make a table of r-values for selected θ-values, plot the resulting points, and then connect them smoothly.
绘制极坐标曲线时，先列出若干 θ 值对应的 r 值表，然后描点并用平滑曲线连接。
Always pay attention to negative r-values. When r is negative, plot the point in the direction opposite to the given angle.
始终注意负的 r 值。当 r 为负数时，应在给定角度的相反方向上描点。
Symmetry can reduce your work. For a curve r = f(θ):
对称性可以简化作图。对于曲线 r = f(θ)：
- If f(-θ) = f(θ), the curve is symmetric about the polar axis. 若 f(-θ) = f(θ)，则曲线关于极轴对称。
- If f(π - θ) = f(θ), the curve is symmetric about the line θ = π/2. 若 f(π - θ) = f(θ)，则曲线关于直线 θ = π/2 对称。
- If f(θ + π) = f(θ), the curve is symmetric about the pole. 若 f(θ + π) = f(θ)，则曲线关于极点对称。
These tests are convenient, but they are sufficient checks rather than necessary conditions for symmetry.
这些检验方法使用方便，但它们是判断对称性的充分条件，而非必要条件。
4. Standard Polar Curves | 常见极坐标曲线
The following standard forms appear frequently in examinations and applications.
下列标准形式在考试和实际应用中经常出现。
| Curve | 曲线 | Equation | 方程 | Description | 说明 |
| Circle | 圆 | r = a | Circle centred at the pole, radius a. 以极点为中心、半径为 a 的圆。 |
| Circle | 圆 | r = 2a cos θ | Circle passing through the pole, centre on the polar axis. 过极点且圆心在极轴上的圆。 |
| Circle | 圆 | r = 2a sin θ | Circle passing through the pole, centre on the line θ = π/2. 过极点且圆心在 θ = π/2 直线上的圆。 |
| Line through pole | 过极点直线 | θ = α | A line through the pole making angle α with the polar axis. 过极点且与极轴成 α 角的直线。 |
| Line | 直线 | r = p / cos(θ - α)Published by TutorHao | Mathematics Revision Series | aleveler.com
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