📚 Representation of Curves in Polar Form | 极坐标形式下曲线的表示
In Cartesian coordinates, a curve is often described by an equation relating x and y, such as y = f(x) or x² + y² = a². However, many curves — circles, spirals, and rose patterns — are far more elegantly expressed using polar coordinates, where a point is located by its distance from a fixed origin and the angle from a fixed direction. This article explores how curves are represented in polar form, a key topic in AQA A-Level Mathematics.
在直角坐标系中，曲线通常用 x 与 y 的关系式表示，例如 y = f(x) 或 x² + y² = a²。然而，许多曲线——如圆、螺线和玫瑰线——用极坐标表示会更为简洁。极坐标通过一个点到固定原点的距离以及与固定方向所成的角度来确定点的位置。本文将深入探讨如何用极坐标形式表示曲线，这是 AQA A-Level 数学中的一个重要考点。
1. The Polar Coordinate System | 极坐标系
A polar coordinate system is defined by a fixed point O called the pole (usually taken as the origin) and a fixed half-line from O known as the initial line or polar axis, typically drawn along the positive x-axis. A point P is given by an ordered pair (r, θ), where r is the radial distance from O to P, and θ is the angular coordinate measured anticlockwise from the initial line to the line OP.
极坐标系由一个固定点 O（称为极点，通常取原点）和从 O 出发的固定半直线（称为极轴或始线，通常沿 x 轴正方向）确定。点 P 用有序数对 (r, θ) 表示，其中 r 是从 O 到 P 的径向距离，θ 是从极轴逆时针旋转到直线 OP 所成的角。
Angles are measured in radians in A-Level work. The polar angle θ is not unique; adding any multiple of 2π gives the same point. Also, r may be negative: the point (−r, θ) is the same as (r, θ + π), as the negative radius means the point lies in the opposite direction along the same straight line.
在 A-Level 课程中，角度以弧度为单位。极角 θ 并不唯一；加上任意 2π 的整数倍后仍表示同一点。此外，r 可以是负数：点 (−r, θ) 与点 (r, θ + π) 表示同一点，因为负半径意味着该点位于同一直线的相反方向上。
(−r, θ) = (r, θ + π)
This convention allows curves such as spirals to be represented without forcing r to remain positive.
这一约定使得一些曲线（如螺线）可以在无需强制 r 保持正数的前提下得到表示。
2. Converting Between Polar and Cartesian Coordinates | 极坐标与直角坐标的互化
The connection between polar coordinates (r, θ) and Cartesian coordinates (x, y) is fundamental. From the right triangle formed by the point, the pole and the projection onto the x-axis, we obtain:
极坐标 (r, θ) 与直角坐标 (x, y) 之间的联系是基础性的。由点、极点和向 x 轴作投影所形成的直角三角形，可得以下关系：
x = r cos θ, y = r sin θ
Conversely, to convert from Cartesian to polar, we use:
反过来，由直角坐标化为极坐标时，使用：
r² = x² + y², tan θ = y ⁄ x (x ≠ 0)
When determining θ, you must consider the quadrant in which the point lies, rather than simply taking the inverse tangent on a calculator. For example, the point (−1, 1) has θ = 3π ⁄ 4, not −π ⁄ 4.
在确定 θ 时，必须考虑点所在的象限，而不是简单地用计算器取反正切。例如，点 (−1, 1) 的 θ = 3π ⁄ 4，而不是 −π ⁄ 4。
To convert an entire curve from polar form to Cartesian form, substitute x = r cos θ and y = r sin θ, and use r² = x² + y² whenever possible. For instance, the polar equation r = 2a cos θ becomes, after multiplying by r, a Cartesian circle equation.
若要将整条曲线从极坐标形式转换为直角坐标形式，可代入 x = r cos θ 和 y = r sin θ，并尽可能使用 r² = x² + y²。例如，极坐标方程 r = 2a cos θ，两边同乘 r 后即可化为直角坐标下的圆方程。
3. Plotting Polar Curves | 绘制极坐标曲线
A curve in polar form is given by an equation r = f(θ), where r is a function of θ. To sketch such a curve, one useful method is to construct a table of values for θ at key angles — such as 0, π ⁄ 6, π ⁄ 4, π ⁄ 3, π ⁄ 2, 2π ⁄ 3, π — and compute the corresponding r values.
极坐标形式下的曲线由方程 r = f(θ) 给出，其中 r 是 θ 的函数。要绘制这样的曲线，一种有效的方法是在关键角处（例如 0、π ⁄ 6、π ⁄ 4、π ⁄ 3、π ⁄ 2、2π ⁄ 3、π）建立 θ 与 r 的数值表并计算对应的 r 值。
Each point (r, θ) is plotted by measuring an angle from the polar axis and then marking a point at radial distance r along that direction. If r is negative, plot the point in the opposite direction. Joining the plotted points smoothly gives the overall shape of the curve.
绘制每个点 (r, θ) 时，先从极轴量出角度 θ，然后沿该方向在径向距离 r 处标记点。如果 r 为负，则在相反方向标记。将这些点平滑连接即可得到曲线的整体形状。
For example, to sketch r = cos θ, we note that as θ goes from 0 to π, r decreases from 1 to −1. The curve is actually a circle with diameter 1 on the polar axis. Plotting sufficient points reveals the circle clearly.
例如，要绘制 r = cos θ，可注意到当 θ 从 0 变到 π 时，r 从 1 减小到 −1。该曲线实际上是以极轴为直径、直径为 1 的圆。绘制足够多的点后，圆便清晰可见。
When sketching, always consider the domain of θ required to generate the whole curve. For trigonometric functions, often 0 ≤ θ < 2π is enough, but for some curves a smaller interval may suffice due to symmetry.
绘图时，务必考虑生成整条曲线所需的 θ 取值范围。对于三角函数，通常 0 ≤ θ < 2π 足够，但由于对称性，某些曲线可以在更小的区间内完全绘制出来。
4. Symmetry in Polar Curves | 极坐标曲线的对称性
Symmetry is a powerful tool for sketching polar curves because it reduces the amount of plotting required. There are three common types of symmetry in polar graphs:
对称性是绘制极坐标曲线的重要工具，因为它可以减少所需的描点数量。极坐标图通常具有以下三种对称性：
- Symmetry about the initial line (polar axis): If replacing θ by −θ leaves the equation unchanged, then the curve is symmetric about the polar axis.
- Symmetry about the line θ = π ⁄ 2 (the y-axis): If replacing θ by π − θ leaves the equation unchanged, the curve is symmetric about the vertical line through the pole.
- Symmetry about the pole (origin): If replacing r by −r leaves the equation unchanged — or equivalently replacing θ by θ + π — then the curve has rotational symmetry of order 2 about the pole.
- 关于极轴（始线）对称：若用 −θ 代换 θ 后方程不变，则曲线关于极轴对称。
- 关于直线 θ = π ⁄ 2（即 y 轴）对称：若用 π − θ 代换 θ 后方程不变，则曲线关于过极点的竖直直线对称。
- 关于极点（原点）对称：若用 −r 代换 r 后方程不变（或等价地用 θ + π 代换 θ），则曲线关于极点具有二阶旋转对称性。
For example, r = a(1 + cos θ) is symmetric about the initial line, because cos(−θ) = cos θ. Therefore we only need to sketch the curve for 0 ≤ θ ≤ π and reflect it across the polar axis.
例如，r = a(1 + cos θ) 关于极轴对称，因为 cos(−θ) = cos θ。因此只需在 0 ≤ θ ≤ π 上绘制曲线，再关于极轴反射即可得到完整图像。
Recognising symmetry early can also help avoid mistakes when finding intersections or calculating areas.
尽早识别对称性还有助于避免在求交点或计算面积时出现错误。
5. Lines and Circles | 直线与圆
Several standard curves have very simple polar equations. A line through the pole making an angle α with the initial line has equation θ = α, because r can take any real value along that line.
有几类标准曲线的极坐标方程非常简单。经过极点且与极轴成角 α 的直线的方程为 θ = α，因为该直线上 r 可以取任意实数值。
A circle centred at the pole with radius a has equation r = a, since the distance from the pole is constant.
以极点为圆心、半径为 a 的圆的方程为 r = a，因为到极点的距离恒定。
More interestingly, a circle of diameter a that passes through the pole and whose centre lies on the polar axis is given by:
更有趣的是，直径为 a、经过极点且圆心在极轴上的圆，其方程为：
r = a cos θ
To verify this, multiply both sides by r to obtain r² = a r cos θ, which becomes x² + y² = a x, i.e. (x − a ⁄ 2)² + y² = (a ⁄ 2)². This is a circle with centre (a ⁄ 2, 0) and radius a ⁄ 2.
要验证这一点，可将方程两边同乘 r，得 r² = a r cos θ，即 x² + y² = a x，也就是 (x − a ⁄ 2)² + y² = (a ⁄ 2)²。这是一个圆心在 (a ⁄ 2, 0)、半径为 a ⁄ 2 的圆。
Similarly, a circle of diameter a passing through the pole with its centre on the vertical line θ = π ⁄ 2 is represented by r = a sin θ.
类似地，直径为 a、经过极点且圆心在直线 θ = π ⁄ 2 上的圆可表示为 r = a sin θ。
6. Limaçons and Cardioids | 蜗线与心形线
A family of curves known as limaçons (from the Latin for “slug”) are described by polar equations of the form:
一类被称为蜗线（limaçon，源自拉丁语“蛞蝓”）的曲线，其极坐标方程形式为：
r = a + b cos θ or r = a + b sin θ
The shape of the curve depends on the ratio |a ⁄ b|. The most famous special case occurs when a = b; the curve is then called a cardioid because it resembles a heart.
曲线的形状取决于 |a ⁄ b| 的值。最著名的特殊情况是 a = b，此时曲线称为心形线（cardioid），因为其形状像一颗心。
For r = a + a cos θ, the cardioid has a cusp at the pole and a maximum radius of 2a. It is symmetric about the initial line. Sketching on 0 ≤ θ ≤ π and reflecting gives the complete heart shape.
对于 r = a + a cos θ，心形线在极点处有一个尖点，最大半径为 2a。它关于极轴对称。在 0 ≤ θ ≤ π 上绘制并反射，即可得到完整的心形。
| Condition |a ⁄ b| | Shape | Example |
| a / b = 1 | Cardioid (one cusp) | r = 1 + cos θ |
| a / b > 1 | Limaçon with no inner loop and a dimple | r = 2 + cos θ |
| 0 < a / b < 1 | Limaçon with an inner loop | r = 1 + 2 cos θ |
The inner loop occurs because r becomes negative for some interval of θ. For example, when r = 1 + 2 cos θ, r is negative when cos θ < −1 ⁄ 2, producing a small loop on the left side of the pole.
内环的出现是因为在 θ 的某些区间内 r 变为负数。例如，当 r = 1 + 2 cos θ 时，若 cos θ < −1 ⁄ 2，则 r < 0，从而在极点左侧产生一个小环。
7. Rose Curves | 玫瑰线
Rose curves are polar curves with equations of the form:
玫瑰线是指具有如下方程形式的极坐标曲线：
r = a cos(n θ) or r = a sin(n θ)
where n is a positive integer. These curves produce flower-like patterns with a number of petals that depends on n.
其中 n 为正整数。这类曲线会形成类似花瓣的图案，其花瓣数取决于 n。
If n is odd, the rose has exactly n petals. If n is even, it has 2n petals. For example, r = a cos(3θ) has 3 petals, while r = a cos(2θ) has 4 petals.
若 n 为奇数，玫瑰线有 n 个花瓣；若 n 为偶数，则有 2n 个花瓣。例如，r = a cos(3θ) 有 3 个花瓣，而 r = a cos(2θ) 有 4 个花瓣。
The petals are evenly spaced around the pole. For r = a cos(n θ), one petal is centred on the initial line. For r = a sin(n θ), the whole curve is rotated by π ⁄ (2n) relative to the cosine version.
花瓣均匀分布在极点周围。对于 r = a cos(n θ)，其中一个花瓣以极轴为中心。对于 r = a sin(n θ)，整条曲线相对于余弦版本旋转 π ⁄ (2n)。
To sketch a rose curve, first find the angles at which r = 0: these are the roots of cos(n θ) = 0 or sin(n θ) = 0. Between consecutive zeros, r attains a maximum magnitude of |a|, which gives the tip of each petal.
绘制玫瑰线时，首先求出 r = 0 的角度：即 cos(n θ) = 0 或 sin(n θ) = 0 的根。在两个相邻零点之间，r 达到最大绝对值 |a|，对应每个花瓣的尖端。
8. Spirals | 螺线
Spirals are curves in which r continually increases or decreases as θ changes. One of the simplest is the Archimedean spiral, given by:
螺线是指 r 随 θ 的变化而持续增大或减小的曲线。其中最简单的是阿基米德螺线，其方程为：
r = a θ
where a is a positive constant. As θ increases through positive values, r increases linearly, producing a spiral that winds infinitely outward. Taking negative θ gives a second branch winding in the opposite direction.
其中 a 为正数。当 θ 取正值并增大时，r 线性增大，产生向外无限盘旋的螺线。若 θ 取负值，则得到向相反方向旋转的另一支。
Another common spiral is the logarithmic spiral, r = a e^{bθ}, which appears in nature in shells and hurricanes. In this curve, the distances between successive turns grow geometrically.
另一种常见的螺线是对数螺线，r = a e^{bθ}，出现在贝壳和飓风等自然现象中。这种螺线相邻圈之间的距离按几何级数增长。
When sketching spirals, remember that r is not periodic in θ; you must continue plotting for as many turns as required by the problem. The polar angle θ is conventionally given in radians, and the spiral is usually drawn for θ ≥ 0 unless otherwise stated.
绘制螺线时，要注意 r 关于 θ 并不具有周期性；必须根据题目要求绘制足够多的圈数。极角 θ 通常以弧度为单位，除非特别说明，一般只绘制 θ ≥ 0 的部分。
9. Tangents to Polar Curves | 极坐标曲线的切线
To find the gradient of a polar curve r = f(θ) at a point, we can use the Cartesian parametric representation. Since x = r cos θ and y = r sin θ, we differentiate both with respect to θ:
要求极坐标曲线 r = f(θ) 在某点的斜率，我们可以利用直角坐标下的参数表示。由于 x = r cos θ，y = r sin θ，分别对 θ 求导：
dx ⁄ dθ = dr ⁄ dθ cos θ − r sin θ, dy ⁄ dθ = dr ⁄ dθ sin θ + r cos θ
Then the tangent gradient is dy ⁄ dx = (dy ⁄ dθ) ⁄ (dx ⁄ dθ), provided dx ⁄ dθ ≠ 0.
于是切线斜率为 dy ⁄ dx = (dy ⁄ dθ) ⁄ (dx ⁄ dθ)，前提是 dx ⁄ dθ ≠ 0。
At the pole itself, where r = 0, the tangent direction is given by the angle θ that satisfies f(θ) = 0. For a cardioid r = a(1 + cos θ), the polar equation gives r = 0 at θ = π, so the tangent at the pole is the horizontal line through the pole.
在极点本身，即 r = 0 处，切线方向由满足 f(θ) = 0 的 θ 给出。对于心形线 r = a(1 + cos θ)，r = 0 出现在 θ = π，因此极点处的切线是过极点的水平直线。
To locate horizontal or vertical tangents, set dy ⁄ dθ = 0 or dx ⁄ dθ = 0 respectively. Always check whether dx ⁄ dθ and dy ⁄ dθ vanish simultaneously; if they do, the point may be a cusp rather than a smooth tangent.
要找水平或竖直切线，可分别令 dy ⁄ dθ = 0 或 dx ⁄ dθ = 0。同时，需要检查 dx ⁄ dθ 与 dy ⁄ dθ 是否同时为零；若同时为零，该点可能是尖点而不是光滑切线。
10. Summary and Exam Tips | 总结与考试技巧
Polar coordinates provide a powerful and elegant way to represent many curves. The essential tools are the conversion formulas, the ability to sketch r = f(θ), and the recognition of standard curves such as lines, circles, limaçons, cardioids, rose curves, and spirals.
极坐标为表示许多曲线提供了强大而优雅的方式。核心工具包括坐标转换公式、绘制 r = f(θ) 的能力，以及对直线、圆、蜗线、心形线、玫瑰线和螺线等标准曲线的识别。
In AQA A-Level exams, you should be comfortable with the following:
在 AQA A-Level 考试中，你应该熟练掌握以下内容：
- Converting between (r, θ) and (x, y) accurately, paying attention to quadrants.
- Sketching simple polar curves after constructing a table of values.
- Using symmetry to reduce the interval of θ needed for a full sketch.
- Recognising cardioids and rose curves quickly from their standard equations.
- Finding tangents by using the parametric derivative formulas.
- 准确地在 (r, θ) 与 (x, y) 之间进行转换，注意象限。
- 构造数值表后绘制简单极坐标曲线。
- 利用对称性缩短绘制完整图形所需的 θ 区间。
- 快速从标准方程中识别心形线。
- 利用参数导数公式求切线。
One common mistake is forgetting that r can be negative. Another is incorrectly identifying the number of petals for rose curves — remember the odd/even rule. Also, when converting equations, always multiply by r rather than dividing by r unless r is guaranteed non-zero.
一个常见错误是忘记 r 可以为负；另一个错误是错误判断玫瑰线的花瓣数——请牢记奇偶规则。此外，在转换方程时，应尽量乘 r 而不是除以 r，除非保证 r 不为零。
Practice drawing the standard curves by hand; having a mental image of r = a + b cos θ, r = a cos(n θ), and r = a θ will save valuable time in the exam. With these skills, polar curves will become a straightforward topic.
请多加练习手绘标准曲线；脑中若能对 r = a + b cos θ、r = a cos(n θ) 和 r = a θ 有清晰的图像，就能在考试中节省宝贵时间。掌握这些技能后，极坐标曲线将成为一个轻松得分的主题。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导Cancel reply