Polar Coordinates Essentials | 极坐标考点精讲

📚 Polar Coordinates Essentials | 极坐标考点精讲

Polar coordinates offer an alternative way to locate points in the plane, using a distance from the origin and an angle from a fixed direction. They are particularly useful for curves with circular or rotational symmetry. In the IGCSE AQA Mathematics specification, you need to master the conversion between polar and Cartesian forms, sketch common polar curves, find intersections, and determine slopes of tangents.

极坐标为平面上点的定位提供了另一种方式，利用点到原点的距离和与固定方向的夹角来描述位置。对于具有圆形或旋转对称性的曲线，极坐标尤其方便。在 IGCSE AQA 数学考试中，你需要掌握极坐标与直角坐标的互化、绘制常见极坐标曲线、求交点以及确定切线斜率。

---

1. Introduction to Polar Coordinates | 极坐标简介

In the polar coordinate system, a point P is represented by an ordered pair (r, θ). Here r is the radial distance from the origin O (the pole), and θ is the angle measured anticlockwise from the positive x‑axis (the polar axis). The angle θ is usually given in radians. Negative r values are interpreted as moving in the opposite direction along the ray θ + π.

在极坐标系中，点 P 由有序数对 (r, θ) 表示。其中 r 是到原点 O（极点）的径向距离，θ 是从正 x 轴（极轴）逆时针测量的角度。θ 通常以弧度为单位。负的 r 值被解释为沿射线 θ + π 的反方向移动。

The same point can have infinitely many polar representations: (r, θ + 2kπ) and (−r, θ + (2k+1)π) all refer to the same location. This non‑uniqueness is an important feature when solving equations.

同一点可以有无穷多种极坐标表示：(r, θ + 2kπ) 和 (−r, θ + (2k+1)π) 都表示同一位置。这种非唯一性是解方程时的一个重要特征。

---

2. Converting Between Polar and Cartesian | 极坐标与直角坐标互化

The link between polar coordinates (r, θ) and Cartesian coordinates (x, y) is given by:

极坐标 (r, θ) 与直角坐标 (x, y) 之间的联系由以下公式给出：

x = r cos θ,    y = r sin θ

r² = x² + y²,    tan θ = y/x (x ≠ 0)

from Cartesian to polar, determine the quadrant of (x, y) to choose the correct θ. Use r = √x² + y², but be careful with signs if you need a specific polar representation.

从直角坐标转换为极坐标时，需根据 (x, y) 的象限确定正确的 θ。使用 r = √x² + y²，但如果需要特定的极坐标表示，需要注意符号。

Case	θ range	Example (x,y) → (r,θ)
x>0, y any	θ = arctan(y/x)	(3, 4) → (5, 0.927 rad)
x<0	θ = arctan(y/x) ± π	(−3, 4) → (5, 2.214 rad)
x=0, y>0	θ = π/2	(0, 2) → (2, π/2)

---

3. Basic Polar Graphs: Circles and Lines | 基本极坐标图形：圆和直线

The simplest polar equations produce circles and straight lines.

最简单的极坐标方程产生圆和直线。

• r = a (constant): a circle of radius |a| centred at the pole. If a > 0, it is traced once as θ runs from 0 to 2π.
• r = a（常数）：半径为 |a|、圆心在极点的圆。若 a > 0，当 θ 从 0 到 2π 时曲线被描出一次。
• θ = α (constant): a straight line through the pole making an angle α with the polar axis.
• θ = α（常数）：通过极点且与极轴夹角为 α 的直线。
• r cos θ = a: a vertical line x = a. (since x = r cos θ)
• r cos θ = a：垂直线 x = a。（因为 x = r cos θ）
• r sin θ = b: a horizontal line y = b.
• r sin θ = b：水平线 y = b。
• r = 2a cos θ: a circle of radius |a| with centre at (a,0) in Cartesian, passing through the pole.
• r = 2a cos θ：半径为 |a|、圆心在直角坐标 (a,0) 且经过极点的圆。

---

4. Sketching Polar Curves Step‑by‑Step | 逐步绘制极坐标曲线

ly. Pay attention to where r = 0 (the curve passes through the pole) and where r is maximal or minimal.

要绘制极坐标曲线 r = f(θ)，先为关键角度（通常是 π/6 或 π/4 的倍数）建立数值表，描出点 (r, θ)，再平滑连接。注意 r = 0 的位置（曲线经过极点）以及 r 的最大值和最小值。

Example: Sketch r = 1 + cos θ for 0 ≤ θ ≤ 2π. Create a table:

示例：绘制 r = 1 + cos θ，0 ≤ θ ≤ 2π。建立表格：

θ	0	π/3	π/2	2π/3	π	4π/3	3π/2	5π/3	2π
r	2	1.5	1	0.5	0	0.5	1	1.5	2

eveal a cardioid. Mark the pole crossing at θ = π (where r = 0, tangent line is vertical).

描出这些点并连接，呈现出心形线。在 θ = π 处标记极点经过点（r = 0，切线为垂直线）。

---

5. Symmetry in Polar Curves | 极坐标曲线的对称性

Tests for symmetry can save time when sketching. A polar curve is symmetric about:

对称性检验可以在绘图时节省时间。极坐标曲线关于以下对象对称：

• the polar axis (x‑axis) if replacing θ by −θ leaves the equation unchanged, or if f(−θ) = f(θ).
• 极轴（x 轴），若将 θ 替换为 −θ 后方程不变，或 f(−θ) = f(θ)。
• the line θ = π/2 (y‑axis) if replacing θ by π−θ yields the same r, i.e. f(π−θ) = f(θ).
• 直线 θ = π/2（y 轴），若将 θ 替换为 π−θ 得到相同的 r，即 f(π−θ) = f(θ)。
• f replacing r by −r (equivalent to θ → θ+π) gives the same equation, or if f(θ+π) = −f(θ).
• 极点（原点），若将 r 替换为 −r（等价于 θ → θ+π）后方程相同，或 f(θ+π) = −f(θ)。

e because cos 2(θ+π) = cos(2θ+2π) = cos 2θ, and replacing θ by −θ gives cos(−2θ) = cos 2θ.

例如，r = cos 2θ 关于两坐标轴和极点均对称，因为 cos 2(θ+π) = cos(2θ+2π) = cos 2θ，且将 θ 换成 −θ 得 cos(−2θ) = cos 2θ。

---

广告 / Sponsored

6. Special Polar Curves: Cardioid, Limaçon, Rose | 特殊极坐标曲线：心形线、蜗线、玫瑰线

Recognising standard forms helps in quickly sketching and answering exam questions.

识别标准形式有助于快速绘图和解答考题。

• Cardioid: r = a(1 ± cos θ) or r = a(1 ± sin θ). Heart‑shaped with a cusp at the pole.
• 心形线：r = a(1 ± cos θ) 或 r = a(1 ± sin θ)。心形，极点处有一个尖点。
• Limaçon with inner loop: r = a + b cos θ (or sin θ) where b > a > 0. The inner loop occurs when |a/b| < 1.
• 带内环的蜗线：r = a + b cos θ（或 sin θ），其中 b > a > 0。当 |a/b| < 1 时出现内环。
• Rose curves (nθ) or r = a sin(nθ). If n is even, the rose has 2n petals; if n is odd, it has n petals.
• 玫瑰线：r = a cos(nθ) 或 r = a sin(nθ)。若 n 为偶数，有 2n 片花瓣；若 n 为奇数，有 n 片花瓣。

For instance, r = 3 sin 2θ yields a 4‑petal rose, while r = 2 cos 3θ gives a 3‑petal rose. The maximum r is |a| and petals are equally spaced.

例如，r = 3 sin 2θ 产生四叶玫瑰线，而 r = 2 cos 3θ 为三叶玫瑰线。r 的最大值为 |a|，花瓣等角度间距。

---

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

the pole (r = 0) or when the curves meet under different θ representations (e.g. (−r, θ+π)). Always check the pole separately by setting each r = 0.

求两条极坐标曲线 r = f(θ) 和 r = g(θ) 的交点，解方程 f(θ) = g(θ)。但要注意在极点 (r = 0) 或当曲线以不同 θ 表示相遇时（例如 (−r, θ+π)）可能还有额外的交点。务必单独检验极点，令每个 r = 0。

k pole: r = 1 never zero; r = 2 cos θ = 0 at θ = π/2, 3π/2, but these are not on r = 1, so no extra pole intersection.

示例：求 r = 1 和 r = 2 cos θ 的交点。解 1 = 2 cos θ → cos θ = 1/2 → θ = π/3, 5π/3（或 −π/3）。交点：(1, π/3) 和 (1, 5π/3)。检验极点：r = 1 永不为零；r = 2 cos θ 在 θ = π/2, 3π/2 处为零，但不在 r = 1 上，故无多余的极点交点。

---

8. Slope and Tangent Lines in Polar Form | 极坐标下的斜率与切线

can be expressed using parametric derivatives. Since x = r cos θ and y = r sin θ, we have:

极坐标曲线 r = f(θ) 的斜率可用参数导数表示。因为 x = r cos θ，y = r sin θ，得到：

dy/dx = (dy/dθ) / (dx/dθ) = (r' sin θ + r cos θ) / (r' cos θ − r sin θ)

where r' A vertical tangent occurs when dx/dθ = 0 (provided dy/dθ ≠ 0). At the pole, if r = 0 and r' ≠ 0, the curve is tangent to the line θ = α where α satisfies f(α) = 0.

其中 r' = dr/dθ。水平切线出现在 dy/dθ = 0（且 dx/dθ ≠ 0）处。垂直切线出现在 dx/dθ = 0（且 dy/dθ ≠ 0）处。在极点，若 r = 0 且 r' ≠ 0，则曲线与直线 θ = α 相切，其中 α 满足 f(α) = 0。

π (or odd multiples). At θ = π, the tangent line is simply the line θ = π (the negative x‑axis). Alternatively, using derivative, at θ = π, r = 0, r' = −sin π = 0? Wait, careful: r' = −sin θ, at θ = π, r' = 0, so the pole is a cusp, consistent with a cardioid.

示例：对于 r = 1 + cos θ，求极点处的切线。令 r = 0 → cos θ = −1 → θ = π（或奇数倍 π）。在 θ = π 处，切线即为直线 θ = π（负 x 轴）。或者使用导数，在 θ = π 处 r = 0，r' = −sin π = 0，因此极点是一个尖点，与心形线相符。

---

9. Common Mistakes and Exam Tips | 常见错误与考试技巧

Avoid these pitfalls when working with polar coordinates:

处理极坐标时避免以下常见错误：

• Forgetting multiple representations: When solving equations, always consider −r and adding π to θ. Missing these can lose solutions.
• 忽略多重表示：解方程时，总是要考虑 −r 以及 θ + π。遗漏这些可能丢失解。
• Ignoring the pole separately; the pole may be an intersection point even if the equations do not share a θ.
• 忽略极点：单独检验 r = 0；即使方程不共享同一个 θ，极点也可能是交点。
• Angles in degrees vs radians: Polar calculus and formal sketching always use radians. Set calculators correctly.
• 角度单位度与弧度混淆：极坐标的微积分和规范绘图始终使用弧度。请正确设置计算器。
• Drawing unnecessary branches on to : Pay attention to the domain of θ. Many curves are defined for 0 ≤ θ ≤ 2π but some only need a subset.
• 画出多余的分支：注意 θ 的定义域。许多曲线定义在 0 ≤ θ ≤ 2π，但有些只需子区间。

Exam tip: Always label key points on your sketch (intersections, pole, maxima) with their polar coordinates and write the scale or ray directions.

考试技巧：始终在草图上标注关键点（交点、极点、最大值）的极坐标，并写出比例或射线方向。

---

10. Summary and Key Formulas | 总结与关键公式

Here is a quick reference for the essential polar coordinate facts you need to master for IGCSE AQA Mathematics:

以下是你需要掌握的 IGCSE AQA 数学极坐标核心知识速查表：

Concept / 概念	Formula / Result
Polar to Cartesian	x = r cos θ, y = r sin θ
Cartesian to Polar	r² = x² + y², tan θ = y/x
Circle (pole centre)	r = a
Line through pole	θ = α
Vertical/horizontal line	r cos θ = a, r sin θ = b
Slope dy/dx	(r' sin θ + r cos θ) / (r' cos θ − r sin θ)
Tangent at pole (if r=0)	θ = α where f(α)=0, provided r' ≠ 0

Regular practice with plotting, solving intersection equations, and applying symmetry will build confidence. Always connect the geometric picture with the algebraic form.

定期练习绘图、解交点方程以及运用对称性将建立信心。务必把几何图形与代数形式联系起来。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）