Representation of Curves in Polar Form | 极坐标形式下曲线的表示

📚 Representation of Curves in Polar Form | 极坐标形式下曲线的表示

In AQA A-Level Mathematics, polar coordinates offer an alternative way to describe plane curves by specifying the distance r from a fixed pole and the angle θ from the initial line. This article reviews how to represent curves in polar form, sketch them, and apply the key calculus results for area and tangents.

在 AQA A-Level 数学中，极坐标提供了一种描述平面曲线的替代方式：用点到固定极点的距离 r 和从极轴起算的角度 θ 来表示。本文复习如何用极坐标形式表示曲线、绘制曲线，并应用面积与切线的关键微积分结论。

1. From Cartesian to Polar Coordinates | 从直角坐标到极坐标

The polar coordinate system uses a pole O and an initial line. A point P is located by r, the directed distance from O, and θ, the angle measured from the initial line, usually in radians.

极坐标系使用极点 O 和极轴。点 P 由 r（从 O 出发的有向距离）和 θ（从极轴开始测量的角，通常以弧度表示）确定。

The conversion equations are x = r cos θ, y = r sin θ, r² = x² + y², and tan θ = y/x. When finding θ from tan θ, you must check the quadrant of the point.

转换公式为 x = r cos θ，y = r sin θ，r² = x² + y²，tan θ = y/x。在从 tan θ 求 θ 时，必须检查点所在的象限。

Because angles repeat every 2π, the same point can be written as (r, θ + 2πk) for integer k, and also as (−r, θ + π). This matters when solving polar equations.

由于角度每 2π 重复一次，同一点可写作 (r, θ + 2πk)（k 为整数），也可写作 (−r, θ + π)。这在解极坐标方程时很重要。

x = r cos θ, y = r sin θ, r² = x² + y²

2. Polar Curves and Simple Equations | 极坐标曲线与简单方程

A polar curve is usually written as r = f(θ) or as an equation involving r and θ. For example, r = 4 is a circle of radius 4 centred at the pole, while θ = π/3 is a straight line through the pole.

极坐标曲线通常写作 r = f(θ) 或关于 r 和 θ 的方程。例如，r = 4 是以极点为中心、半径为 4 的圆，而 θ = π/3 是过极点的直线。

Since θ is an angle, polar equations often repeat. A full sketch usually needs only 0 ≤ θ < 2π, but symmetry can reduce the work even further.

由于 θ 是角，极坐标方程往往会重复。完整图形通常只需要 0 ≤ θ < 2π 的区间，但对称性可以进一步减少工作量。

The curve r = 2 + 2 cos θ is a cardioid, but r = 2 cos θ is a circle. Pay attention to whether a constant is added to the trigonometric term, because it changes both the shape and the position.

曲线 r = 2 + 2 cos θ 是心形线，但 r = 2 cos θ 是圆。注意三角函数项前是否加了常数，因为这会同时改变图形的形状和位置。

3. Symmetry Tests | 对称性检验

To test symmetry about the initial line, replace θ by −θ. If the equation is unchanged or equivalent, the curve is symmetric about the initial line.

要检验关于极轴的对称性，可将 θ 替换为 −θ。如果方程不变或等价，则曲线关于极轴对称。

To test symmetry about the vertical line θ = π/2, replace θ by π − θ. Symmetry about the pole can be tested by replacing r by −r or by replacing θ with θ + π.

要检验关于竖直线 θ = π/2 的对称性，可将 θ 替换为 π − θ。关于极点的对称性可通过将 r 替换为 −r 或将 θ 替换为 θ + π 来检验。

These tests are especially useful when sketching curves such as r = cos 2θ, which has symmetry about the initial line, the vertical line θ = π/2, and the pole.

这些检验在绘制 r = cos 2θ 等曲线时特别有用，该曲线关于极轴、竖直线 θ = π/2 以及极点都对称。

4. Sketching Polar Curves | 极坐标曲线绘图

A reliable sketch begins by finding the zeros of r, the maximum value of |r|, and any symmetries. Then evaluate r at key angles such as 0, π/6, π/4, π/3, π/2 and their multiples.

可靠的草图从求 r 的零点、|r| 的最大值以及任何对称性开始。然后计算关键角度（如 0、π/6、π/4、π/3、π/2 及其倍数）处的 r 值。

Plot the points in polar form and join them smoothly, respecting the direction in which θ increases. Mark the pole wherever r = 0, because the curve passes through the origin there.

在极坐标中描点并平滑连接，注意 θ 增大的方向。在 r = 0 处标出极点，因为曲线在那里经过原点。

If r becomes negative, the point is plotted on the opposite side of the pole at the angle θ. This often creates inner loops or parts of the curve reflected through the pole.

如果 r 变为负值，点会画在极点的另一侧、角度 θ 的相反方向上。这通常会产生内环或经过极点反射的曲线部分。

5. Circles in Polar Form | 圆的极坐标形式

The equation r = a represents a circle of radius a centred at the pole. The equation r = 2a cos θ is a circle with centre (a, 0) and radius a, while r = 2a sin θ has centre (0, a) and radius a, assuming a > 0.

方程 r = a 表示以极点为中心、半径为 a 的圆。方程 r = 2a cos θ

Published by TutorHao | A-Level Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）