Representation of Curves in Polar Form | 极坐标形式的曲线表示

📚 Representation of Curves in Polar Form | 极坐标形式的曲线表示

In polar coordinates, a point is located by a radial distance r from a fixed pole (origin) and an angle θ measured from a fixed polar axis (usually the positive x-axis). A curve in polar form is expressed as an equation relating r and θ, typically written as r = f(θ). This representation often simplifies curves that are otherwise complicated in Cartesian coordinates, such as circles centred on an axis, cardioids, and rose curves.

在极坐标系中，一个点由到固定极点（原点）的径向距离 r 和相对固定极轴（通常为 x 轴正方向）的角度 θ 确定。极坐标形式的曲线用 r 与 θ 的关系方程表示，通常写作 r = f(θ)。这种表示方式常能简化在直角坐标中很复杂的曲线，例如圆心在轴上的圆、心形线和玫瑰线。

1. The Polar Coordinate System | 极坐标系

A point P in the plane can be written as (r, θ), where r is the distance from the origin O, and θ is the anticlockwise angle from the positive x-axis to the line OP. The same physical point can be represented by multiple polar pairs: adding 2π to θ gives the same direction, and using a negative r means moving in the opposite direction along the line θ + π.

平面上的点 P 可以写成 (r, θ)，其中 r 是到原点 O 的距离，θ 是从 x 轴正方向到直线 OP 的逆时针夹角。同一个物理点可由多个极坐标对表示：给 θ 加上 2π 得到相同方向；若 r 为负，则表示沿 θ + π 方向移动。

2. Conversion between Polar and Cartesian Coordinates | 极坐标与直角坐标的转换

The relationships between polar coordinates (r, θ) and Cartesian coordinates (x, y) are fundamental for sketching and analysis. From a right-angled triangle, we have:

极坐标 (r, θ) 与直角坐标 (x, y) 之间的关系是作图和分析的基础。由直角三角形可得：

x = r cos θ, y = r sin θ

Conversely, the polar representation can be found using:

反过来，由直角坐标求极坐标可用：

r² = x² + y², tan θ = y / x

These formulas allow us to translate an equation in r and θ into one in x and y, and vice versa. For example, r = 2a cos θ becomes x² + y² = 2ax, which is a circle centred at (a, 0) with radius a.

这些公式允许我们把关于 r 和 θ 的方程与关于 x 和 y 的方程互相转换。例如，r = 2a cos θ 可化为 x² + y² = 2ax，即圆心在 (a, 0)、半径为 a 的圆。

3. Plotting Polar Curves | 绘制极坐标曲线

To sketch r = f(θ), we can create a table of values. Choose angles θ from 0 to 2π (or a suitable interval), compute corresponding r, then plot each point (r, θ). Negative values of r indicate that the point is placed in the opposite direction of θ, i.e. at angle θ + π with positive distance |r|.

要绘制 r = f(θ)，可以制作数值表。选取 θ 从 0 到 2π（或合适的区间），计算对应的 r，然后标出每个点 (r, θ)。当 r 为负值时，点位于 θ 的反方向，即角度 θ + π、距离 |r| 处。

Another useful method is to analyse the function f(θ): where r is maximum or zero, how r changes with θ, and whether the curve is symmetric. Marking these key points helps produce an accurate sketch quickly.

另一种有用的方法是分析函数 f(θ)：确定 r 的最大值或零点，观察 r 随 θ 的变化趋势，以及曲线是否具有对称性。标出这些关键点有助于快速作出准确的草图。

4. Circles and Lines in Polar Form | 极坐标下的圆与直线

Certain simple equations give famous curves. A circle passing through the origin with centre on the polar axis has equation r = 2a cos θ, where a is the radius. If the centre lies on the line perpendicular to the polar axis, the equation becomes r = 2a sin θ.

某些简单方程对应著名曲线。经过原点且圆心在极轴上的圆的方程为 r = 2a cos θ，其中 a 为半径。若圆心位于垂直于极轴的直线上，方程为 r = 2a sin θ。

A straight line through the pole is given by θ = α for a fixed angle α. A line not passing through the pole can be written as r cos(θ − α) = d, where d is the perpendicular distance from the pole to the line and α is the direction of the normal.

通过极点的直线方程为 θ = α（α 为固定角）。不经过极点的直线可写成 r cos(θ − α) = d，其中 d 是极点到该直线的垂直距离，α 是法线方向角。

5. Limaçons and Cardioids | 蜗形线与心形线

Curves of the form r = a + b cos θ or r = a + b sin θ are called limaçons (or snail curves). When a = b, the curve is a cardioid, which resembles a heart. For example, r = a(1 + cos θ) produces a cardioid with a cusp at the pole.

形如 r = a + b cos θ 或 r = a + b sin θ 的曲线称为蜗形线。当 a = b 时，曲线就是心形线，外形像心脏。例如，r = a(1 + cos θ) 生成在极点处有一个尖点的心形线。

The shape of a limaçon depends on the ratio a/b. If a/b > 1, the curve has no inner loop; if a/b = 1, it is a cardioid; if 0 < a/b < 1, the curve has an inner loop; if a/b < 0, the loop may appear in the opposite direction depending on sine or cosine.

蜗形线的形状取决于比值 a/b。若 a/b > 1，曲线没有内环；若 a/b = 1，则为心形线；若 0 < a/b < 1，曲线带有内环；若 a/b < 0，内环可出现在相反方向，具体取决于正弦或余弦。

6. Rose Curves | 玫瑰线

Rose curves have equations of the form r = a cos(nθ) or r = a sin(nθ). If n is a positive integer, the curve consists of petals. When n is odd, there are exactly n petals; when n is even, there are 2n petals.

玫瑰线的方程为 r = a cos(nθ) 或 r = a sin(nθ)。若 n 为正整数，曲线由花瓣构成。当 n 为奇数时，花瓣数为 n；当 n 为偶数时，花瓣数为 2n。

For example, r = a cos(3θ) gives a three-petal rose, while r = a sin(2θ) gives a four-petal rose. The maximum value of r on each petal is a, occurring at evenly spaced angles.

例如，r = a cos(3θ) 产生三瓣玫瑰线，而 r = a sin(2θ) 产生四瓣玫瑰线。每个花瓣上 r 的最大值为 a，出现在等间隔的角度处。

7. Spirals | 螺旋线

An Archimedean spiral has equation r = aθ, where a is a positive constant. As θ increases, r increases linearly, creating a spiral that spreads outward uniformly. It crosses the polar axis at multiples of 2π.

阿基米德螺旋线的方程为 r = aθ，其中 a 为正常数。随着 θ 增大，r 线性增大，形成均匀向外扩展的螺旋。它在极轴上每隔 2π 相交一次。

A logarithmic spiral has equation r = ae^{bθ}. Here the distances between successive turns grow geometrically. This type of spiral appears in nature, for example in nautilus shells.

对数螺旋线的方程为 r = ae^{bθ}。相邻圈层之间的距离按几何级数增长。这种螺旋出现在自然界中，例如鹦鹉螺的外壳。

8. Symmetry of Polar Curves | 极坐标曲线的对称性

Symmetry helps reduce the amount of plotting needed. If the equation is unchanged when θ is replaced by −θ, the curve is symmetric about the polar axis (x-axis). If replacing r by −r (or θ by θ + π) leaves the equation unchanged, the curve is symmetric about the pole (origin).

对称性可减少绘制工作量。若将 θ 替换为 −θ 后方程不变，则曲线关于极轴（x 轴）对称。若将 r 替换为 −r（或 θ 替换为 θ + π）后方程不变，则曲线关于极点（原点）对称。

If replacing θ by π − θ leaves the equation unchanged, the curve is symmetric about the line θ = π/2 (the y-axis). Recognising these symmetries allows us to sketch only part of the curve and then reflect appropriately.

若将 θ 替换为 π − θ 后方程不变，则曲线关于直线 θ = π/2（即 y 轴）对称。认识这些对称性允许我们只绘制部分曲线，然后作相应反射。

9. Tangents to Polar Curves | 极坐标曲线的切线

To find the slope of the tangent to r = f(θ) at a point (r, θ), we use the chain rule with the parametric equations x = r cos θ and y = r sin θ. The derivative dy/dx is given by:

要求 r = f(θ) 在点 (r, θ) 处切线的斜率，可利用参数方程 x = r cos θ 和 y = r sin θ 及链式法则。导数 dy/dx 由下式给出：

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

This formula is extremely useful. At a point where the tangent is horizontal or vertical, we set dy/dθ = 0 or dx/dθ = 0 respectively. At the pole (r = 0), the tangent direction is given by the solutions of f(θ) = 0.

这个公式非常有用。在切线水平或垂直的点，分别令 dy/dθ = 0 或 dx/dθ = 0。在极点（r = 0）处，切线方向由 f(θ) = 0 的解给出。

10. Area Enclosed by a Polar Curve | 极坐标曲线围成的面积

The area enclosed by a polar curve r = f(θ) between θ = α and θ = β is found using the integral:

极坐标曲线 r = f(θ) 从 θ = α 到 θ = β 围成的面积用积分计算：

A = ½ ∫_α^β r² dθ

This formula comes from summing up areas of small sectors of a circle, each with radius r and angle dθ. Care must be taken when r takes negative values, as the area contribution is still based on r². For curves with loops, calculate the area of one loop and multiply by the number of identical loops.

该公式来自对许多半径为 r、角度为 dθ 的微小扇形面积的求和。当 r 取负值时需小心，面积贡献仍基于 r²。对于有环的曲线，可先计算一个环的面积，再乘以相同环的数量。

11. Worked Example: The Cardioid r = 1 + cos θ | 例题：心形线 r = 1 + cos θ

Let us sketch and find the area inside the cardioid defined by r = 1 + cos θ for 0 ≤ θ ≤ 2π. First, note that the curve is symmetric about the polar axis because cos(−θ) = cos θ. Key values: at θ = 0, r = 2; at θ = π/2, r = 1; at θ = π, r = 0. The curve has a cusp at the pole.

我们来绘制心形线 r = 1 + cos θ 在 0 ≤ θ ≤ 2π 内的图形并求其面积。首先，注意到曲线关于极轴对称，因为 cos(−θ) = cos θ。关键值：θ = 0 时 r = 2；θ = π/2 时 r = 1；θ = π 时 r = 0。曲线在极点处有一个尖点。

Using the area formula, the total area is:

利用面积公式，总面积为：

A = ½ ∫₀^2π (1 + cos θ)² dθ = ½ ∫₀^2π (1 + 2 cos θ + cos² θ) dθ

Using the identity cos² θ = (1 + cos 2θ)/2, we obtain:

利用恒等式 cos² θ = (1 + cos 2θ)/2，得到：

A = ½ [ 3θ/2 + 2 sin θ + (sin 2θ)/4 ]₀^2π = 3π/2

So the area enclosed by the cardioid is 3π/2 square units.

因此该心形线围成的面积为 3π/2 平方单位。

12. Key Points for Exams | 考试要点总结

Always convert between polar and Cartesian using x = r cos θ, y = r sin θ, and r² = x² + y².
当需要转换时，务必使用 x = r cos θ, y = r sin θ 以及 r² = x² + y²。
Know the standard forms: circles (r = 2a cos θ), cardioids (r = a(1 ± cos θ)), rose curves (r = a cos(nθ)), spirals (r = aθ).
熟悉标准形式：圆 r = 2a cos θ、心形线 r = a(1 ± cos θ)、玫瑰线 r = a cos(nθ)、螺旋线 r = aθ。
Check symmetry before plotting: about the polar axis, the pole, or the line θ = π/2.
作图前检查对称性：关于极轴、极点或直线 θ = π/2 对称。
Use the tangent formula carefully; remember dy/dx is not r’s derivative.
切线公式使用时要小心；dy/dx 不是 r 的导数。
Area is A = ½ ∫ r² dθ; pay attention to limits and loops.
面积公式为 A = ½ ∫ r² dθ；注意积分限和环。
For negative r, plot in the opposite direction of θ.
当 r 为负时，沿 θ 的反方向绘制点。

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Representation of Curves in Polar Form | 极坐标形式的曲线表示