Polar Coordinates Revision for GCSE Edexcel Maths | GCSE Edexcel 数学：极坐标考点精讲

📚 Polar Coordinates Revision for GCSE Edexcel Maths | GCSE Edexcel 数学：极坐标考点精讲

Polar coordinates provide an elegant way to represent points and curves based on a distance from the origin and an angle from the positive x-axis. In the Edexcel GCSE Further Pure Mathematics course, you are expected to master conversions, sketch polar graphs, find intersections, and calculate areas using integration.

极坐标提供了一种优雅的方式，用点到原点的距离和与正 x 轴的夹角来表示点和曲线。在 Edexcel GCSE Further Pure Mathematics 课程中，你需要掌握坐标转换、绘制极坐标图形、求交点，并运用积分计算面积。

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

Instead of (x, y), a point P is given by (r, θ), where r is the radial distance from the origin O, and θ is the anticlockwise angle from the positive x-axis (polar axis). Negative r means the point lies on the opposite ray.

代替 (x, y)，点 P 用 (r, θ) 表示，其中 r 是到原点 O 的径向距离，θ 是从正 x 轴（极轴）逆时针旋转的角度。负的 r 表示点位于相反方向的射线上。

The polar axis is the horizontal line to the right of O; angles are usually measured in radians for calculus work.

极轴是位于原点右侧的水平线；在微积分作业中，角通常以弧度为单位。

2. Converting Between Polar and Cartesian Forms | 极坐标与直角坐标的转换

The key relationships linking polar and Cartesian coordinates are: x = r cos θ, y = r sin θ, and r² = x² + y², tan θ = y/x (for x ≠ 0). These formulas allow you to switch between representations seamlessly.

连接极坐标和直角坐标的关键关系是：x = r cos θ, y = r sin θ，以及 r² = x² + y², tan θ = y/x（当 x ≠ 0）。这些公式让你能够在两种表示之间无缝切换。

Example: Convert (3, π/4) to Cartesian. Here r=3, θ=π/4, so x=3 cos(π/4)=3√2/2, y=3 sin(π/4)=3√2/2, giving (3√2/2, 3√2/2).

示例：将 (3, π/4) 转换为直角坐标。这里 r=3, θ=π/4，所以 x=3 cos(π/4)=3√2/2, y=3 sin(π/4)=3√2/2，得到 (3√2/2, 3√2/2)。

To convert (x,y) to polar, compute r = √(x²+y²) and determine θ using arctan(y/x), considering the quadrant.

将 (x,y) 转换为极坐标时，计算 r = √(x²+y²)，并使用 arctan(y/x) 确定 θ，同时需考虑象限。

3. Basic Polar Equations and Graphs | 基本极坐标方程与图形

Common simple polar equations include r = constant (a circle radius a centred at origin), θ = constant (a line through the origin at angle α). These are building blocks for more complex curves.

常见的简单极坐标方程包括 r = 常数（以原点为中心、半径为 a 的圆），θ = 常数（过原点且角度为 α 的直线）。它们是更复杂曲线的构建基础。

Graphing by hand typically involves compiling a table of θ values and computing corresponding r, then plotting the points.

手工绘图通常需要列出 θ 值表并计算相应的 r，然后描点连线。

4. Circles in Polar Form | 极坐标形式的圆

A circle passing through the pole O with diameter a on the polar axis has equation r = a cos θ. Similarly, r = a sin θ is a circle whose diameter a lies on the line θ = π/2.

过极点 O 且直径 a 位于极轴上的圆方程为 r = a cos θ。同理，r = a sin θ 则是对直径 a 在 θ = π/2 直线上的圆。

If the centre is at the pole, the equation is simply r = a, a circle radius a. For a circle offset from the pole, use r = 2a cos(θ – α) to control the centre’s position.

如果圆心在极点，方程就是 r = a，半径为 a 的圆。对于偏离极点的圆，用 r = 2a cos(θ – α) 来控制圆心的位置。

5. Cardioids and Limaçons | 心形线与蜗牛线

Equations of the form r = a + b cos θ or r = a + b sin θ produce limaçon curves. When a = b, the shape is a cardioid, for example r = a(1 + cos θ) or r = a(1 + sin θ).

形如 r = a + b cos θ 或 r = a + b sin θ 的方程产生蜗牛线。当 a = b 时，形状为心形线，例如 r = a(1 + cos θ) 或 r = a(1 + sin θ)。

If |a| > |b|, the limaçon has no inner loop; if |a| < |b|, there is an inner loop. These curves are symmetric and often appear in exam questions requiring area calculations.

如果 |a| > |b|，蜗牛线没有内环；如果 |a| < |b|，则有一个内环。这些曲线具有对称性，常见于要求面积计算的考题中。

6. The Spiral of Archimedes | 阿基米德螺线

The Archimedean spiral is given by r = aθ, where a is a constant. As θ increases, the radius grows linearly, so the curve winds around the pole with constant separation between successive turns.

阿基米德螺线由 r = aθ 给出，其中 a 为常数。随着 θ 增大，半径线性增长，因此曲线绕极点旋转，相邻圈之间的间隔恒定。

You may be asked to sketch it for θ ≥ 0, noting that negative θ values produce a reflection across the pole.

你可能会被要求绘制 θ ≥ 0 时的图形，注意负的 θ 值会产生关于极点的反射图形。

7. Finding Intersections of Polar Curves | 求极坐标曲线的交点

To find where two polar curves r = f(θ) and r = g(θ) intersect, solve f(θ) = g(θ) for θ in the relevant domain. However, always check if the pole (r=0) is an intersection point, as f(θ)=0 and g(θ)=0 might occur at different θ values.

为求两条极坐标曲线 r = f(θ) 和 r = g(θ) 的交点，在相关定义域内求解 f(θ) = g(θ)。然而，一定要检查极点 (r=0) 是否

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Polar Coordinates Revision for GCSE Edexcel Maths | GCSE Edexcel 数学：极坐标 考点精讲