IB Mathematics: Introduction to Cylindrical and Polar Coordinates | IB数学：柱面极坐标系入门

📚 IB Mathematics: Introduction to Cylindrical and Polar Coordinates | IB数学：柱面极坐标系入门

In the IB Mathematics curriculum, Cartesian coordinates are only one way to locate a point. We also study coordinate systems that follow circular symmetry. Polar coordinates describe a point by its distance from a fixed origin and its direction from a fixed line. Cylindrical coordinates extend this idea into three dimensions by adding a vertical coordinate. These systems appear in graph sketching, complex numbers, and volume integration, and they can turn difficult-looking equations into simple ones.

在 IB 数学课程中，直角坐标只是定位一个点的方式之一。我们还会学习具有圆对称性的坐标系。极坐标用点到固定原点的距离和方向来描述平面上的点；柱面坐标在此基础上增加垂直高度，将极坐标推广到三维空间。这些系统在函数图像、复数与体积积分中都会出现，能够把看似复杂的方程变成简单的结构。

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1. Why Do We Need New Coordinate Systems? | 为什么需要新坐标系?

In the usual Cartesian plane, a point is described by (x, y), where x and y measure horizontal and vertical distances from two perpendicular axes. This system is excellent for straight lines, rectangles, and many algebraic curves. However, when a problem involves rotation, circles, or circular motion, Cartesian equations can become unnecessarily complicated.

在通常的直角坐标平面中，一个点用 (x, y) 描述，其中 x 和 y 分别表示到两条互相垂直轴的水平距离和竖直距离。这套系统非常适用于直线、矩形和许多代数曲线。然而，当问题涉及旋转、圆或圆周运动时，直角坐标方程往往会变得不必要的复杂。

For example, a particle moving on the circle x² + y² = a² can be described by x = a cos θ and y = a sin θ. The angle θ and the radius a are more natural than x and y. Polar coordinates make the angular motion explicit. Cylindrical coordinates do the same for three-dimensional objects such as cylinders, cones, and rotating solids.

例如，粒子在圆 x² + y² = a² 上运动时，可以用 x = a cos θ、y = a sin θ 描述。角度 θ 和半径 a 比 x、y 更自然。极坐标让角运动变得明确。柱面坐标则对圆柱、圆锥和旋转体等三维对象起到了类似作用。

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2. Defining Polar Coordinates | 极坐标的定义

To set up a polar coordinate system in the plane, choose a fixed point O called the pole, and a fixed ray Ox called the polar axis. The polar axis is usually drawn as the positive x-axis of a Cartesian grid. For any point P, let r be the distance from O to P. Let θ be the directed angle measured counterclockwise from the polar axis to the line OP. Then the ordered pair (r, θ) is a set of polar coordinates for P.

为了在平面内建立极坐标系，先取一个固定点 O 称为极点，再取一条固定射线 Ox 称为极轴。极轴通常画成直角坐标中的正 x 轴。对于任意点 P，设 r 为 O 到 P 的距离，设 θ 为从极轴逆时针旋转到直线 OP 的有向角度。那么有序数对 (r, θ) 就是 P 的一组极坐标。

The angle θ is normally measured in radians. If r is positive, P lies on the ray in direction θ. If r is negative, we move in the opposite direction, so the point (-r, θ) is the same as (r, θ + π). This convention is useful in many IB questions.

角 θ 通常用弧度表示。若 r 为正，点 P 位于方向 θ 的射线上；若 r 为负，则沿相反方向移动，因此点 (-r, θ) 与 (r, θ + π) 相同。这一约定在许多 IB 题目中非常有用。

x = r cos θ, y = r sin θ

r² = x² + y², tan θ = y/x

These conversion formulas connect polar coordinates to Cartesian coordinates.

以上公式将极坐标与直角坐标联系起来。

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3. Plotting Points and Multiple Representations | 点的作图与多重表示

When plotting a polar point, first draw the ray that makes angle θ with the polar axis. Then move r units along that ray if r is positive. If r is negative, move |r| units in the opposite direction. This is straightforward, but remember that every point has infinitely many different polar coordinate representations.

绘制极坐标点时，先画出与极轴成 θ 角的射线。若 r 为正，沿该射线移动 r 个单位；若 r 为负，则向相反方向移动 |r| 个单位。这个过程很直接，但要注意每个点都有无限多种极坐标表示。

Because angles repeat after a full turn, any point can be written as (r, θ + 2πk) for any integer k. It can also be written with negative r as (-r, θ + π + 2πk). For example, the point (2, π/4) is the same as (2, 9π/4) and is also the same as (-2, 5π/4).

由于角度旋转一整圈后会重复，任意点都可以写成 (r, θ + 2πk)，其中 k 为任意整数；也可以写成 (-r, θ + π + 2πk)。例如，点 (2, π/4) 等同于 (2, 9π/4)，也等同于 (-2, 5π/4)。

In IB problems, choose the representation that makes the equation or the graph easiest to analyse.

在 IB 题目中，应选择让方程或图像最容易分析的表示形式。

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4. Converting Between Cartesian and Polar Coordinates | 直角坐标与极坐标的互化

To convert from polar to Cartesian, use x = r cos θ and y = r sin θ. To convert from Cartesian to polar, use r² = x² + y² and tan θ = y/x. However, the equation tan θ = y/x alone is not enough, because tangent is positive in both the first and third quadrants. You must check the signs of x and y to choose the correct angle.

从极坐标化直角坐标时，使用 x = r cos θ，y = r sin θ。从直角坐标化极坐标时，使用 r² = x² + y²，tan θ = y/x。注意，仅凭 tan θ = y/x 并不够，因为正切在第一象限和第三象限都为正。必须根据 x、y 的正负确定正确角度。

Example 1: The Cartesian point (1, √3) has r = √(1² + (√3)²) = 2 and tan θ = √3. Since the point is in the first quadrant, θ = π/3. Thus the polar coordinates are (2, π/3).

例 1：直角坐标点 (1, √3) 满足 r = √(1² + (√3)²) = 2，tan θ = √3。由于该点在第一象限，θ = π/3。因此极坐标为 (2, π/3)。

Example 2: The Cartesian point (-1, -1) has r = √2 and tan θ = 1. Both coordinates are negative, so the point is in the third quadrant, giving θ = 5π/4. Therefore one polar representation is (√2, 5π/4).

例 2：直角坐标点 (-1, -1) 满足 r = √2，tan θ = 1。因为两个坐标都为负，点在第三象限，所以 θ = 5π/4。因此一组极坐标为 (√2, 5π/4)。

Example 3: Convert the polar equation r = 2 cos θ into Cartesian form. Multiply both sides by r to obtain r² = 2r cos θ. Then x² + y² = 2x, which can be written as (x – 1)² + y² = 1. This is a circle with centre (1, 0) and radius 1.

例 3：将极坐标方程 r = 2 cos θ 化为直角坐标方程。两边同时乘以 r，得到 r² = 2r cos θ。于是 x² + y² = 2x，即 (x – 1)² + y² = 1。这是以 (1, 0) 为圆心、半径为 1 的圆。

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5. Common Polar Curves | 常见极坐标曲线

Polar equations often create patterns that are difficult to describe in Cartesian form. The table below lists several standard polar curves that appear in IB-style questions.

极坐标方程常常生成用直角坐标难以描述的

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