📚 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 are expected to convert between polar and Cartesian forms, sketch standard polar curves, find tangents, and calculate areas enclosed by polar curves, and every one of those skills is assessed through the same small set of standard techniques.
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，而这些技能都依赖同一套标准方法。
This revision guide keeps the whole topic in one place: definitions, conversions, standard curves, symmetry, tangents, maxima, intersections, areas, arc length, exam technique and a final checklist.
本篇复习指南把整个专题集中在一处：定义、互化、常见曲线、对称性、切线、极值、交点、面积、弧长、考试技巧以及最后的复习清单。
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. The standard convention is to place the pole at the Cartesian origin and the initial line along the positive x-axis, so that the polar system sits on top of the familiar x-y plane.
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。标准约定是把极点放在直角坐标原点，把初始线放在正 x 轴上，这样极坐标系就叠加在我们熟悉的 x-y 平面上。
The angle θ is usually measured in radians, and anticlockwise is taken as positive, so a clockwise angle appears as a negative value. The point with polar coordinates (r, θ) is not unique: adding any integer multiple of 2π to θ gives the same point, which is why a single polar curve may be traced several times as θ runs from 0 to 2π.
角 θ 通常以弧度为单位，逆时针方向为正，因此顺时针方向的角表现为负值。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，这正是同一条极坐标曲线在 θ 从 0 变到 2π 时可能被重复描绘的原因。
de the usual 0 to 2π interval, and it is also the key to understanding curves such as r² = a²cos 2θ, where negative values of r² are simply not used.
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，同时也是理解 r² = a²cos 2θ 这类曲线的关键，因为 r² 为负的区间直接舍弃即可。
and a polar curve is the set of all points whose coordinates satisfy an equation r = f(θ).
极点本身是所有满足 r = 0 的点的集合，与 θ 的取值无关；而一条极坐标曲线就是所有坐标满足方程 r = f(θ) 的点的集合。
- Pole (O) | 极点：the fixed origin of the system. 坐标系统的固定原点。
- Initial line / polar axis | 初始线/极轴：the fixed ray from which θ is measured, normally the positive x-axis. 量角所依据的固定射线，通常取正 x 轴。
- Polar coordinates (r, θ) | 极坐标 (r, θ)：distance then angle, always in that order. 先距离后角度，顺序不可颠倒。
- Polar equation r = f(θ) | 极坐标方程 r = f(θ)：r expressed explicitly as a function of θ. 把 r 显式表示为 θ 的函数。
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 the horizontal and vertical components of the point give the two standard results below.
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，则该点的水平与竖直分量直接给出下面两个标准公式。
x = r cos θ y = r sin θ
r² = x² + y² tan θ = y ÷ x
he weakest of the four because the tangent function repeats every π, so it cannot distinguish between a point in the first quadrant and a point in the third quadrant.
第一对公式用于从极坐标化为直角坐标，第二对公式用于反向转换。四个公式中 tan θ = y ÷ x 最容易出错，因为正切函数以 π 为周期，无法区分第一象限和第三象限的点。
1) gives tan θ = 1, but the required angle is 5π/4, not π/4, and the correct radius is r = √2.
从直角坐标化为极坐标时，一定要画出点的位置，或者直接使用 cos θ 与 sin θ。第三象限的点如 (-1, -1) 满足 tan θ = 1，但所需的角是 5π/4 而不是 π/4，正确的极径是 r = √2。
| Conversion | 转换方向 | Use | 使用公式 |
| 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 | r = √x² + y², then find θ from a sketch |
| Check | 检验 | r² = x² + y² must hold for every point |
Worked example. 3) in Cartesian form. Here r = 4 and θ = π/3, so x = 4 cos(π/3) = 2 and y = 4 sin(π/3) = 2√3, giving the point (2, 2√3).
例题。把 (4, π/3) 化为直角坐标。此处 r = 4，θ = π/3，故 x = 4 cos(π/3) = 2，y = 4 sin(π/3) = 2√3，得到点 (2, 2√3)。
Worked example. Express (-1, √3) in polar form. Then r = √1 + 3 and the point lies in the second quadrant, so θ = π − π/3 = 2π/3. The answer is (2, 2π/3).
例题。把 (-1, √3) 化为极坐标。此时 r = √1 + 3 = 2，点在第二象限，故 θ = π − π/3 = 2π/3，答案为 (2, 2π/3)。
3. Sketching Standard Polar Curves | 常见极坐标曲线的作图
Standard polar curves must be recognised instantly, because a sketch question usually carries marks for shape, intercepts and symmetry rather than for a large table of values. Learn the family names and the number of petals or loops, and most sketches become routine.
常见极坐标曲线必须一眼认出，因为作图题的分值通常落在形状、截距和对称性上，而不是落在大量描点上。记住这些曲线的名称以及花瓣数或环数，大多数作图题就变成常规操作。
| Equation | 方程 | Name | 名称 | Key features | 关键特征 |
| r = a | Circle | 圆 | Centre at the pole, radius a | 圆心在极点，半径为 a |
| θ = α | Half-line | 射线 | A straight line through the pole at angle α | 过极点、倾角为 α 的直线 |
| r = a sec θ | Vertical line | 竖直线 | The line x = a | 直线 x = a |
| r = a cosec θ | Horizontal line | 水平线 | The line y = a | 直线 y = a |
| r = 2a cos θ | Circle through the pole | 过极点的圆 | Centre (a, 0), radius a | 圆心 (a, 0)，半径 a |
| r = 2a sin θ | Circle through the pole | 过极点的圆 | Centre (0, a), radius a | 圆心 (0, a)，半径 a |
| r = a(1 + cos θ) | Cardioid | 心形线 | Max r = 2a at θ = 0, r = 0 at θ = π | θ = 0 时 r = 2a，θ = π 时 r = 0 |
| r = a cos 2θ | Rose, 4 petals | 四瓣玫瑰线 | Even n gives 2n petals | n 为偶数时花瓣数为 2n |
| r = a cos 3θ | Rose, 3 petals | 三瓣玫瑰线 | Odd n gives n petals | n 为奇数时花瓣数为 n |
| r² = a² cos 2θ | Lemniscate | 双纽线 | Only where cos 2θ ≥ 0 | 仅在 cos 2θ ≥ 0 处有定义 |
| r = aθ | Archimedean spiral | 阿基米德螺线 | r grows steadily with θ | r 随 θ 均匀增大 |
and where r is greatest, then join the points smoothly using any symmetry you have identified. Always indicate the pole, the initial line and the maximum value of r on your diagram.
高效作图的方法是：以 π/6 或 π/4 为间隔列出简短的数值表，标出 r = 0 以及 r 取最大值的点，再利用已识别的对称性把点光滑连接。图上一定要标出极点、初始线以及 r 的最大值。
Curves of the form r = a + b cos θ with b less than a are limaç has an inner loop. Knowing this lets you predict the shape before you plot a single point.
形如 r = a + b cos θ 的曲线，
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