📚 Polar Coordinates | 极坐标坐标
Polar coordinates offer a different way to locate a point in a plane by using a distance from a fixed origin and an angle from a fixed direction. This system is especially useful for curves that are naturally circular or radial, such as spirals, roses and cardioids.
极坐标通过一个点到固定极点的距离和与固定极轴所成的角度来确定平面上的点的位置。它可以十分自然地描述圆、螺旋线、玫瑰线和心形线等曲线，在 AQA A-Level 数学中占据重要地位。
1. The Polar Coordinate System | 极坐标系简介
A point P is written as (r, θ), where r is the directed distance from the pole (the origin) to P, and θ is the angle measured anticlockwise from the polar axis (usually the positive x-axis).
点 P 记作 (r, θ)，其中 r 是从极点（原点）到 P 的有向距离，θ 是从极轴（通常取正 x 轴）开始逆时针旋转到 OP 的有向角。
The value of r may be negative. When r < 0, P is located in the opposite direction of the angle θ, that is, the point is drawn at distance |r| along the ray θ + π.
r 可以为负值。当 r < 0 时，P 位于角 θ 的反方向，即沿 θ + π 方向距离 |r| 处。
An ordered pair (r, θ) is not unique: adding 2π to θ gives the same point, and (−r, θ + π) also represents the same point. In AQA you will usually restrict θ to a suitable interval, often 0 ≤ θ < 2π.
有序数对 (r, θ) 并不唯一：θ 增加 2π 得到同一个点，(−r, θ + π) 也表示同一个点。在 AQA 考试中通常会限定 θ 在一个合适区间内，常见的是 0 ≤ θ < 2π。
2. Converting between Polar and Cartesian Coordinates | 极坐标与直角坐标互化
If a point has polar coordinates (r, θ) and Cartesian coordinates (x, y), then the basic conversion formulas are:
若一个点的极坐标为 (r, θ)，直角坐标为 (x, y)，则基本转换公式为：
x = r cos θ, y = r sin θ
To convert from Cartesian to polar, we use r² = x² + y² and tan θ = y / x. The angle must be chosen according to the quadrant in which (x, y) lies.
从直角坐标到极坐标使用 r² = x² + y² 和 tan θ = y / x。θ 必须根据 (x, y) 所在象限仔细判断。
For example, the point (−1, 1) has r = √2 and θ = 3π/4, not θ = −π/4, because it lies in the second quadrant.
例如，点 (−1, 1) 的 r = √2，θ = 3π/4，而不是 θ = −π/4，因为它位于第二象限。
3. Plotting Polar Curves | 绘制极坐标曲线
To sketch a polar curve r = f(θ), you can first make a table of values for θ at convenient angles such as 0, π/6, π/4, π/3, π/2, etc. Then plot the corresponding points and join them smoothly.
绘制极坐标曲线 r = f(θ) 时，可以先取 θ 为 0、π/6、π/4、π/3、π/2 等常用角，列出对应的 r 值表，然后将这些点平滑地连接起来。
It is often helpful to check the range of r. If r is always positive, the curve lies in the same direction as θ; if r can be negative, the curve may have lobes in opposite quadrants.
检查 r 的取值范围往往很有用。若 r 始终为正，曲线总在 θ 方向；若 r 可能为负，曲线可能在相反象限出现叶瓣。
A graphing calculator or a careful mental plot can reveal symmetry, intersections and overall shape before you calculate tangents or areas.
使用图形计算器或仔细地手绘可以发现对称性、交点及整体形状，这有助于后续计算切线和面积。
4. Symmetry Tests | 对称性检验
For a polar curve given by r = f(θ), the following symmetry tests help simplify sketching and integration.
对于极坐标曲线 r = f(θ)，下列对称性检验有助于简化绘图和积分。
If f(−θ) = f(θ), the curve is symmetric about the polar axis (x-axis).
若 f(−θ) = f(θ)，曲线关于极轴（x 轴）对称。
If f(π − θ) = f(θ), the curve is symmetric about the line θ = π/2 (the y-axis).
若 f(π − θ) = f(θ)，曲线关于直线 θ = π/2（即 y 轴）对称。
If f(θ + π) = −f(θ), then replacing r by −r gives the same point, so the curve is symmetric about the pole (origin).
若 f(θ + π) = −f(θ)，则 r 取相反数后对应同一点，因此曲线关于极点（原点）对称。
5. Standard Polar Equations | 常见极坐标方程
The circle r = a has centre at the pole and radius a.
圆 r = a 表示以极点为圆心、半径为 a 的圆。
The ray θ = α is a straight half-line starting from the pole at angle α. If extended to negative r, it becomes the full line through the pole.
射线 θ = α 是从极点出发、方向角为 α 的一条半直线。如果允许 r 为负，则它变成经过极点的一条完整直线。
The circle r = 2a cos θ has centre (a, 0) and radius a, passing through the pole. Similarly r = 2a sin θ has centre (0, a) and radius a.
圆 r = 2a cos θ 的圆心为 (a, 0)，半径为 a，并通过极点。类似地，r = 2a sin θ 的圆心为 (0, a)，半径为 a。
The cardioid r = a(1 + cos θ) or r = a(1 − cos θ) has a heart-like shape, with a cusp at the pole.
心形线 r = a(1 + cos θ) 或 r = a(1 − cos θ) 呈心形，在极点处有一个尖点。
Rose curves r = a cos(nθ) or r = a sin(nθ) have n petals if n is odd, and 2n petals if n is even (for the standard polar sketch over 0 ≤ θ < 2π).
玫瑰线 r = a cos(nθ) 或 r = a sin(nθ)，当 n 为奇数时有 n 片花瓣，当 n 为偶数时有 2n 片花瓣（在 0 ≤ θ < 2π 的标准绘图范围内）。
6. Tangents to Polar Curves | 极坐标曲线切线
To find the slope of the tangent to a polar curve r = f(θ), we use the relation between Cartesian and polar differentials:
为求极坐标曲线 r = f(θ) 的切线斜率，我们利用直角坐标与极坐标微分之间的关系：
dy/dx = ( (dr/dθ) sin θ + r cos θ ) / ( (dr/dθ) cos θ − r sin θ )
This formula comes from differentiating x = r cos θ and y = r sin θ with respect to θ, then dividing dy by dx.
该公式来自 x = r cos θ 和 y = r sin θ 对 θ 求导，再计算 dy 与 dx 的商。
If the denominator is zero but the numerator is not, the tangent is vertical. If both are zero, the point may be a cusp or a self-intersection, so further investigation is needed.
若分母为零而分子不为零，则切线为垂直。若分子分母同时为零，该点可能是尖点或自交点，需要进一步考察。
At the pole (r = 0), the tangent directions are given by the values of θ satisfying f(θ) = 0, provided the curve passes through the pole smoothly.
在极点 (r = 0) 处，切线方向由满足 f(θ) = 0 的 θ 值给出，前提是曲线在该处光滑通过极点。
7. Area Enclosed by a Polar Curve | 极坐标曲线所围面积
The area enclosed by a polar curve r = f(θ) between θ = α and θ = β is given by:
极坐标曲线 r = f(θ) 在 θ = α 与 θ = β 之间围成的面积由下式给出：
A = ½ ∫αβ [f(θ)]² dθ
This formula comes from summing the areas of small circular sectors. It works even if r is negative, because r is squared.
该公式由无数个微小扇形面积求和而来。即使 r 为负值也能成立，因为 r 取平方。
When using symmetry, you may integrate over a smaller interval and multiply by the appropriate factor. Always check that your limits correspond to the region you actually want.
使用对称性时，可以只在较小区间上积分再乘以相应倍数。务必检查积分限是否与实际要求的区域对应。
8. Length of a Polar Curve | 极坐标曲线弧长
For a polar curve r = f(θ) swept from θ = α to θ = β, the arc length is:
极坐标曲线 r = f(θ) 从 θ = α 到 θ = β 的弧长为：
L = ∫αβ √( r² + (dr/dθ)² ) dθ
This is a natural extension of the Cartesian arc-length formula. It is often used with numerical or non-calculator-friendly integrals in exams, so you may be asked to set up the integral rather than evaluate it.
这是直角坐标弧长公式的自然推广。考试中常出现不易甚至无法解析求值的积分，因此题目可能只要求列出定积分而不是算出数值。
9. Converting Equations between Polar and Cartesian Form | 方程的极坐标与直角坐标互化
To convert a Cartesian equation into polar form, substitute x = r cos θ and y = r sin θ.
把直角坐标方程化为极坐标方程时，代入 x = r cos θ 和 y = r sin θ 即可。
For example, x² + y² = ax becomes r² = ar cos θ, so r = a cos θ (assuming r ≠ 0 or including the pole).
例如，x² + y² = ax 化为 r² = ar cos θ，从而 r = a cos θ（假设 r ≠ 0，或把极点也包含在内）。
To convert a polar equation into Cartesian form, replace r by √(x² + y²), cos θ by x/r, and sin θ by y/r, then simplify.
将极坐标方程化为直角坐标方程时，用 r = √(x² + y²)、cos θ = x/r、sin θ = y/r 替换，然后化简。
The equation r = 2 cos θ becomes x² + y² = 2x, equivalent to (x − 1)² + y² = 1.
方程 r = 2 cos θ 化为 x² + y² = 2x，等价于 (x − 1)² + y² = 1。
10. Common Exam Pitfalls | 常见考试陷阱
Many students forget to adjust θ when converting from Cartesian to polar. Always draw the quadrant before choosing the angle.
许多学生在直角坐标转极坐标时忘记调整 θ。选角度之前务必画象限判断。
Another common mistake is using degrees when the formula requires radians. In polar coordinate questions at A-Level, angles are almost always in radians.
另一个常见错误是需要在弧度制中使用公式时误用了角度制。在 A-Level 极坐标题目中，角度几乎总用弧度。
When finding the area between two polar curves, find their intersection points carefully, and remember that the outer and inner curves may switch between limits.
求两条极坐标曲线之间的面积时，要仔细找交点，并注意外曲线和内曲线可能在积分限之间互换。
Do not forget to include the pole when an area is bounded by a curve passing through the origin; omitting it can change the region completely.
当曲线通过原点并围成区域时，不要忽略极点。忽略极点可能会完全改变所求区域。
11. Worked Example: Area of a Cardioid | 例题：心形线的面积
Consider the cardioid r = 2(1 + cos θ). Find the area enclosed by the full curve.
考虑心形线 r = 2(1 + cos θ)，求整条曲线围成的面积。
The curve is symmetric about the polar axis, so we can integrate from 0 to π and double the result.
该曲线关于极轴对称，因此可在 0 到 π 上积分并把结果乘以 2。
A = 2 × ½ ∫0π [2(1 + cos θ)]² dθ = ∫0π 4(1 + cos θ)² dθ
Expand the integrand: 4(1 + 2 cos θ + cos² θ) = 4 + 8 cos θ + 4 cos² θ.
展开被积函数：4(1 + 2 cos θ + cos² θ) = 4 + 8 cos θ + 4 cos² θ。
Use cos² θ = (1 + cos 2θ)/2. Then the integral becomes ∫₀^π (4 + 8 cos θ + 2 + 2 cos 2θ) dθ = ∫₀^π (6 + 8 cos θ + 2 cos 2θ) dθ.
使用 cos² θ = (1 + cos 2θ)/2，则积分变为 ∫₀^π (4 + 8 cos θ + 2 + 2 cos 2θ) dθ = ∫₀^π (6 + 8 cos θ + 2 cos 2θ) dθ。
Now integrate term by term: [6θ + 8 sin θ + sin 2θ]₀^π = 6π + 0 + 0 − 0 = 6π.
现在逐项积分：[6θ + 8 sin θ + sin 2θ]₀^π = 6π + 0 + 0 − 0 = 6π。
Therefore the area enclosed by the cardioid r = 2(1 + cos θ) is 6π square units.
因此心形线 r = 2(1 + cos θ) 所围面积为 6π 平方单位。
12. Summary | 总结
Polar coordinates provide a powerful tool for curves that are awkward in Cartesian form. Master the conversions, symmetry tests, standard shapes, tangent formula, and area formula to solve most exam questions.
极坐标为处理直角坐标中不便描述的曲线提供了有力工具。掌握坐标互化、对称性检验、常见曲线形状、切线公式和面积公式，即可应对绝大多数考试题。
Always remember to work in radians, check quadrants when converting, and use symmetry wisely to simplify integrals.
始终记得使用弧度制，转换时检查象限，并巧妙利用对称性简化积分。
With regular practice, polar coordinate questions become routine and open the door to higher-level topics such as vector fields and complex numbers.
通过经常练习，极坐标题目会变得非常熟练，并为向量场、复数等更高阶主题打下基础。
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