Cartesian and Polar Frames of Reference | 笛卡尔坐标系与极坐标系

📚 Cartesian and Polar Frames of Reference | 笛卡尔坐标系与极坐标系

A frame of reference is a coordinate system used to locate points in space. In A-Level mathematics, the two most important frames are the Cartesian (rectangular) frame and the polar frame. Each offers a different way to represent positions, curves, and geometric relationships.

参考系是用于确定空间中点位置的坐标系。在A-Level数学中，最重要的两种参考系是笛卡尔（直角）坐标系和极坐标系。它们提供了不同的方式来描述位置、曲线和几何关系。

1. The Cartesian Coordinate System | 笛卡尔坐标系

The Cartesian frame uses two perpendicular axes, usually labelled \(x\) and \(y\). A point is written as \((x, y)\), where \(x\) is the horizontal distance from the origin and \(y\) is the vertical distance.

笛卡尔坐标系使用两条互相垂直的轴，通常标记为\(x\)轴和\(y\)轴。一个点写为\((x, y)\)，其中\(x\)是到原点的水平距离，\(y\)是到原点的垂直距离。

For example, the point \((3, 4)\) is located 3 units to the right of the origin and 4 units above it. The coordinates can be positive, negative, or zero.

例如，点\((3, 4)\)位于原点右侧3个单位、上方4个单位处。坐标可以为正、负或零。

The axes divide the plane into four quadrants, numbered anticlockwise from the positive \(x\)-axis.
直线、圆和函数图像都可以用笛卡尔方程表示，例如 \(y = 2x + 1\).

坐标轴将平面分成四个象限，从正\(x\)轴开始逆时针编号。
Lines, circles and function graphs can all be written as Cartesian equations, e.g. \(y = 2x + 1\).

2. The Polar Coordinate System | 极坐标系

The polar frame uses a fixed point called the pole (the origin) and a fixed ray called the polar axis (usually the positive \(x\)-axis). A point is written as \((r, \theta)\), where \(r\) is the distance from the pole and \(\theta\) is the angle measured anticlockwise from the polar axis.

极坐标系使用一个称为极点（原点）的固定点，以及一条称为极轴（通常取正\(x\)轴）的固定射线。一个点写为\((r, \theta)\)，其中\(r\)是到极点的距离，\(\theta\)是从极轴逆时针测量的角度。

In polar coordinates, \(r\) can be negative. A negative \(r\) means that the point lies in the opposite direction to the ray at angle \(\theta\). For instance, \((-2, \frac{\pi}{3})\) is the same point as \((2, \frac{4\pi}{3})\).

在极坐标中，\(r\) 可以为负。负的 \(r\) 表示点位于角度 \(\theta\) 射线的相反方向。例如，\((-2, \frac{\pi}{3})\) 与 \((2, \frac{4\pi}{3})\) 表示同一个点。

Cartesian	Polar
\((x, y)\)	\((r, \theta)\)
Origin \((0, 0)\)	Pole \(r = 0\)
Axes fixed and straight	Polar axis and circle directions

3. Converting Between Cartesian and Polar Coordinates | 笛卡尔坐标与极坐标的互化

To convert from polar to Cartesian, use the relationships:

从极坐标转换为笛卡尔坐标，使用以下关系：

\(x = r \cos \theta, \quad y = r \sin \theta\)

For example, the polar point \((4, \frac{\pi}{3})\) has Cartesian coordinates:

例如，极坐标点 \((4, \frac{\pi}{3})\) 的笛卡尔坐标为：

\(x = 4 \cos \frac{\pi}{3} = 4 \times \frac{1}{2} = 2, \quad y = 4 \sin \frac{\pi}{3} = 4 \times \frac{\sqrt{3}}{2} = 2\sqrt{3}\)

To convert from Cartesian to polar, use:

从笛卡尔坐标转换为极坐标，使用：

\(r = \sqrt{x^{2} + y^{2}}, \quad \theta = \tan^{-1}\left(\frac{y}{x}\right)\)

When using \(\tan^{-1}\), pay attention to the quadrant. A calculator gives a principal value, so add \(\pi\) if \(x < 0\).
The angle \(\theta\) is usually given in radians in A-Level questions, but degrees may be accepted.

使用 \(\tan^{-1}\) 时要注意点所在的象限。计算器给出主值，若 \(x < 0\) 则需要加上 \(\pi\)。
角度 \(\theta\) 在A-Level题目中通常使用弧度，但度也可能被接受。

4. Plotting Curves in Polar Coordinates | 在极坐标系中绘制曲线

A curve in polar form is given by an equation relating \(r\) and \(\theta\), such as \(r = f(\theta)\). To sketch the curve, plot points for various values of \(\theta\) and join them smoothly.

极坐标曲线由联系 \(r\) 和 \(\theta\) 的方程给出，例如 \(r = f(\theta)\)。要绘制曲线，取一系列 \(\theta\) 值计算对应的 \(r\)，再平滑连接各点。

For example, consider the curve \(r = 2\) with \(\theta\) from 0 to \(2\pi\). This gives a circle centred at the pole with radius 2.

例如，考虑曲线 \(r = 2\)，其中 \(\theta\) 从0到 \(2\pi\)。这给出了圆心在极点、半径为2的圆。

Another example is the spiral \(r = \theta\). As \(\theta\) increases, the distance from the pole grows steadily, producing a spiral.

另一个例子是螺线 \(r = \theta\)。随着 \(\theta\) 增加，到极点的距离稳定增长，形成一条螺旋线。

Symmetry can help sketch polar curves. If \(r(\theta) = r(-\theta)\), the curve is symmetric about the polar axis.
If \(r(\theta) = r(\pi – \theta)\), the curve is symmetric about the vertical line \(\theta = \frac{\pi}{2}\).

对称性可用于简化绘制。若 \(r(\theta) = r(-\theta)\)，则曲线关于极轴对称。
若 \(r(\theta) = r(\pi – \theta)\)，则曲线关于直线 \(\theta = \frac{\pi}{2}\) 对称。

5. Standard Polar Curves | 常见极坐标曲线

Certain polar equations produce well-known curves. You should be able to recognise and sketch them.

某些极坐标方程会生成著名曲线。你应该能够识别并绘制它们。

Equation	Shape	Key Feature
\(r = a\)	Circle	Centre at pole, radius \(a\)
\(\theta = \alpha\)	Straight line through pole	Direction angle \(\alpha\)
\(r = a \cos \theta\)	Circle passing through pole	Diameter \(a\) along polar axis
\(r = a \sin \theta\)	Circle passing through pole	Diameter \(a\) along vertical axis
\(r = a(1 \pm \cos \theta)\)	Cardioid	Heart-shaped with a cusp at pole
\(r = a \cos (n\theta)\)	Rose curve	\(n\) petals if \(n\) is odd; \(2n\) petals if \(n\) is even

For example, \(r = 2\cos(3\theta)\) produces a rose with 3 petals. The petals point in directions determined by solving \(\cos(3\theta) = 0\).

例如，\(r = 2\cos(3\theta)\) 生成一个具有3片花瓣的玫瑰曲线。花瓣的方向通过解 \(\cos(3\theta) = 0\) 来确定。

6. Area in Polar Coordinates | 极坐标下的面积

The area enclosed by a polar curve \(r = f(\theta)\) between \(\theta = \alpha\) and \(\theta = \beta\) is given by:

极坐标曲线 \(r = f(\theta)\) 在 \(\theta = \alpha\) 到 \(\theta = \beta\) 之间围成的面积为：

\(A = \frac{1}{2} \int_{\alpha}^{\beta} r^{2} \, d\theta\)

This formula comes from summing the areas of small sectors. If the curve is closed, choose \(\alpha\) and \(\beta\) so that the whole region is covered.

此公式来源于将微小扇形的面积相加。如果曲线是闭合的，选择 \(\alpha\) 和 \(\beta\) 使整个区域被覆盖。

Make sure \(r\) is non-negative over the interval, or split the integral at points where \(r = 0\).
For symmetric curves, you can calculate one part and multiply by the number of symmetric parts.

确保 \(r\) 在积分区间内非负，或者在 \(r = 0\) 处拆分积分。
对于对称曲线，可只计算其中一部分，再乘以对称部分的个数。

7. Tangents in Polar Coordinates | 极坐标中的切线

For a polar curve \(r = f(\theta)\), the gradient of the tangent at a point can be found using:

对于极坐标曲线 \(r = f(\theta)\)，其上一点处的切线斜率可以通过以下公式计算：

\(\frac{dy}{dx} = \frac{\frac{dr}{d\theta} \sin \theta + r \cos \theta}{\frac{dr}{d\theta} \cos \theta – r \sin \theta}\)

This is derived from \(x = r\cos\theta\) and \(y = r\sin\theta\) using the chain rule. You may be asked to find the equation of a tangent at a given value of \(\theta\).

这是由 \(x = r\cos\theta\) 和 \(y = r\sin\theta\) 通过链式法则推导出来的。你可能会被要求在给定的 \(\theta\) 值处求切线方程。

Sometimes you only need to find where the tangent is horizontal or vertical. Set the numerator equal to zero for horizontal, and the denominator equal to zero for vertical.

有时你只需找到切线水平或垂直的位置。令分子为零得水平切线，令分母为零得垂直切线。

8. Applications of Cartesian and Polar Frames | 笛卡尔与极坐标系的应用

Cartesian coordinates are ideal for describing straight lines, distances between points, and functions such as polynomials and exponential functions.

笛卡尔坐标适合描述直线、两点之间的距离以及多项式、指数函数等函数。

Polar coordinates are more natural in situations involving circular motion, waves, fields, and problems with rotational symmetry.

极坐标在涉及圆周运动、波动、场以及具有旋转对称性的问题中更为自然。

Physics: projectile motion often uses Cartesian components; circular orbits are easier in polar coordinates.
Engineering: polar coordinates are used in radar, robot arm positioning, and antenna radiation patterns.
Navigation: a bearing and distance form a polar pair, while maps use grid (Cartesian) references.

物理：抛体运动常用笛卡尔分量；圆周轨道用极坐标更方便。
工程：极坐标用于雷达、机械臂定位和天线辐射方向图。
导航：方位角加距离构成极坐标对，而地图使用网格（笛卡尔）参考。

9. Worked Example: Conversion | 例题：坐标转换

Convert the Cartesian point \((-1, \sqrt{3})\) into polar coordinates with \(r > 0\) and \(0 \le \theta < 2\pi\).

将笛卡尔点 \((-1, \sqrt{3})\) 转换为极坐标，要求 \(r > 0\) 且 \(0 \le \theta < 2\pi\)。

First, compute \(r\):

首先计算 \(r\)：

\(r = \sqrt{(-1)^{2} + (\sqrt{3})^{2}} = \sqrt{1 + 3} = 2\)

Next, find \(\theta\). Since \(x < 0\) and \(y > 0\), the point lies in the second quadrant. The reference angle satisfies \(\tan \alpha = \frac{y}{|x|} = \sqrt{3}\), so \(\alpha = \frac{\pi}{3}\). Hence:

接着求 \(\theta\)。因为 \(x < 0\) 且 \(y > 0\)，点位于第二象限。参考角满足 \(\tan \alpha = \frac{y}{|x|} = \sqrt{3}\)，所以 \(\alpha = \frac{\pi}{3}\)。因此：

\(\theta = \pi – \frac{\pi}{3} = \frac{2\pi}{3}\)

Thus the polar coordinates are \((2, \frac{2\pi}{3})\).

因此极坐标为 \((2, \frac{2\pi}{3})\)。

10. Worked Example: Area of a Cardioid | 例题：心形线的面积

Find the area enclosed by the cardioid \(r = 2(1 + \cos \theta)\).

求心形线 \(r = 2(1 + \cos \theta)\) 围成的面积。

The curve is symmetric about the polar axis and closes when \(\theta\) runs from 0 to \(2\pi\). Therefore the area is:

该曲线关于极轴对称，且当 \(\theta\) 从0到 \(2\pi\) 时闭合。因此面积为：

\(A = \frac{1}{2} \int_{0}^{2\pi} [2(1 + \cos \theta)]^{2} \, d\theta\)

Expand the integrand:

展开被积函数：

\(A = 2 \int_{0}^{2\pi} (1 + 2\cos \theta + \cos^{2}\theta) \, d\theta\)

Using \(\cos^{2}\theta = \frac{1 + \cos 2\theta}{2}\), we obtain:

利用 \(\cos^{2}\theta = \frac{1 + \cos 2\theta}{2}\)，得到：

\(A = 2 \int_{0}^{2\pi} \left( \frac{3}{2} + 2\cos \theta + \frac{1}{2}\cos 2\theta \right) d\theta\)

The integrals of \(\cos \theta\) and \(\cos 2\theta\) over a full period are zero. Hence:

\(\cos \theta\) 和 \(\cos 2\theta\) 在整个周期上的积分为零。因此：

\(A = 2 \times \frac{3}{2} \times 2\pi = 6\pi\)

11. Common Mistakes and Tips | 常见错误与技巧

Forgetting the quadrant when using \(\tan^{-1}\) can give the wrong angle. Always plot the point mentally.
Using degrees and radians inconsistently in the same calculation leads to errors. Stick to radians in calculus.
Missing negative \(r\) values when sketching polar curves. Always test both positive and negative \(r\) if relevant.

忘记象限 使用 \(\tan^{-1}\) 时可能得到错误角度。先在心中画出点的位置。
在同一计算中混用度与弧度 会导致错误。在微积分中统一使用弧度。
绘制极坐标曲线时忽略负的 \(r\) 值。在相关情况下，应同时测试正和负的 \(r\)。

Remember that a point in polar coordinates is not unique. Adding \(2\pi\) to \(\theta\) or changing the sign of \(r\) and adding \(\pi\) to \(\theta\) gives the same point.

请记住，极坐标中的点并不唯一。将 \(\theta\) 加上 \(2\pi\)，或者将 \(r\) 取负并将 \(\theta\) 加上 \(\pi\)，都会得到同一个点。

12. Summary | 总结

Cartesian and polar frames are two fundamental ways to locate points in the plane. The Cartesian frame uses perpendicular axes and is excellent for linear and rectangular patterns. The polar frame uses distance and angle, which simplifies circular and rotational problems.

笛卡尔坐标系与极坐标系是平面内确定点的两种基本方式。笛卡尔坐标系使用垂直轴，非常适合直线和矩形模式。极坐标系使用距离和角度，能够简化圆形和旋转问题。

You must be able to convert between the two systems, sketch standard polar curves, compute areas using integration, and handle gradients. Mastery of both frames gives you the flexibility to choose the most convenient approach for any problem.

你必须能够在两种坐标系之间转换，绘制标准极坐标曲线，利用积分计算面积，并处理切线斜率。掌握这两种坐标系将使你在解题时能够灵活选择最方便的方法。

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Cartesian and Polar Frames of Reference | 笛卡尔坐标系与极坐标系