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

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

Every point in a plane can be described by coordinates. Two fundamental systems are the Cartesian (rectangular) frame and the polar frame. Understanding both, and their interconversion, is essential for solving a wide range of A-Level mathematics problems.

平面上的每一个点都可以用坐标来描述。两种基本坐标系是笛卡尔（直角）坐标系和极坐标系。理解这两种坐标系及其相互转换，对于解决 A-Level 数学中的大量问题至关重要。

1. The Cartesian Frame | 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 signed horizontal distance from the vertical axis, and \(y\) is the signed vertical distance from the horizontal axis.

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。一个点记为 \((x, y)\)，其中 \(x\) 是到纵轴的有向水平距离，\(y\) 是到横轴的有向垂直距离。

The origin is the point \((0,0)\). The axes divide the plane into four quadrants, and the signs of \(x\) and \(y\) identify the quadrant. Lines, curves and regions are expressed by equations such as \(y = mx + c\), \(x^2 + y^2 = r^2\), or \(y = f(x)\).

原点是 \((0,0)\)。两条轴将平面分成四个象限，\(x\) 和 \(y\) 的符号决定点所在的象限。直线、曲线和区域用方程表示，例如 \(y = mx + c\)、\(x^2 + y^2 = r^2\) 或 \(y = f(x)\)。

Cartesian coordinates are intuitive for straight lines, parabolas and many algebraic curves, and they are the default frame for graphing functions.

笛卡尔坐标对于直线、抛物线及许多代数曲线非常直观，而且是绘制函数图像的默认坐标系。

2. The Polar Frame | 极坐标系

The polar frame locates a point by its distance from a fixed point (the pole, usually the origin) and the angle from a fixed direction (the initial line, usually the positive \(x\)-axis). A point is written as \((r, \theta)\), where \(r\) is the radial distance and \(\theta\) is the polar angle, measured in radians.

极坐标系利用一个固定点（极点，通常为原点）和一条固定方向（极轴，通常为 \(x\) 轴正方向）来确定点的位置。一个点记为 \((r, \theta)\)，其中 \(r\) 是径向距离，\(\theta\) 是极角，以弧度为单位。

By convention, \(r\) can be negative: the point \((-r, \theta)\) is the same as \((r, \theta + \pi)\). The angle \(\theta\) is not unique; adding \(2\pi\) gives the same point. Usually we choose \(r \ge 0\) and \(0 \le \theta < 2\pi\) as the principal range.

根据约定，\(r\) 可以为负：点 \((-r, \theta)\) 与 \((r, \theta + \pi)\) 相同。角度 \(\theta\) 不唯一；加上 \(2\pi\) 得到同一点。通常我们选择 \(r \ge 0\) 且 \(0 \le \theta < 2\pi\) 作为主值范围。

Polar coordinates are especially useful for curves with circular or rotational symmetry, such as circles centred at the pole, spirals, and rose curves.

极坐标特别适合具有圆形或旋转对称性的曲线，例如以极点为中心的圆、螺线以及玫瑰线。

3. Conversion Between Cartesian and Polar | 笛卡尔与极坐标互化

The relationship between the two frames is derived from right-angled trigonometry. If a point has Cartesian coordinates \((x, y)\) and polar coordinates \((r, \theta)\), then:

两种坐标系之间的关系由直角三角形三角函数推导得出。如果一点的笛卡尔坐标为 \((x, y)\)，极坐标为 \((r, \theta)\)，那么：

x = r cos θ, y = r sin θ

These equations convert polar to Cartesian. For example, the polar point \((2, \pi/3)\) has Cartesian coordinates:

这些公式将极坐标转换为笛卡尔坐标。例如，极坐标点 \((2, \pi/3)\) 的笛卡尔坐标为：

x = 2 cos(π/3) = 1, y = 2 sin(π/3) = √3

To convert from Cartesian to polar, use:

从笛卡尔坐标转换为极坐标，使用：

r² = x² + y², tan θ = y / x

When calculating \(\theta\), you must consider the quadrant of \((x,y)\). The formula \(\theta = \tan^{-1}(y/x)\) only gives the correct angle for \(x > 0\). For other quadrants, add or subtract \(\pi\).

计算 \(\theta\) 时，必须考虑 \((x,y)\) 所在的象限。公式 \(\theta = \tan^{-1}(y/x)\) 仅对 \(x > 0\) 给出正确角度。对于其他象限，需要加或减 \(\pi\)。

4. Cartesian Equations to Polar Form | 笛卡尔方程化为极坐标形式

Many curves are easier to express in one frame than the other. To convert a Cartesian equation into polar form, substitute \(x = r \cos θ\) and \(y = r \sin θ\), then simplify.

许多曲线在某种坐标系中更容易表示。要将笛卡尔方程化为极坐标形式，代入 \(x = r \cos θ\) 和 \(y = r \sin θ\)，然后化简。

Example: the circle \(x^2 + y^2 = 4\). Substituting \(r^2\) for \(x^2 + y^2\) gives \(r^2 = 4\), hence \(r = 2\) (for \(r \ge 0\)).

例如：圆 \(x^2 + y^2 = 4\)。用 \(r^2\) 代替 \(x^2 + y^2\)，得到 \(r^2 = 4\)，因此 \(r = 2\)（当 \(r \ge 0\)）。

Example: the line \(y = x\) becomes \(r \sin θ = r \cos θ\), so \(\tan θ = 1\), giving \(\theta = \pi/4\) or \(5\pi/4\).

例如：直线 \(y = x\) 变为 \(r \sin θ = r \cos θ\)，因此 \(\tan θ = 1\)，得到 \(\theta = \pi/4\) 或 \(5\pi/4\)。

Another example: \(x^2 + y^2 = 2ax\). Substituting yields \(r^2 = 2a r \cos θ\), so \(r = 2a \cos θ\) (assuming \(r \neq 0\)). This is a circle with centre \((a,0)\) and radius \(|a|\).

另一个例子：\(x^2 + y^2 = 2ax\)。代入得到 \(r^2 = 2a r \cos θ\)，因此 \(r = 2a \cos θ\)（假设 \(r \neq 0\)）。这是一个圆心在 \((a,0)\)、半径为 \(|a|\) 的圆。

5. Polar Equations to Cartesian Form | 极坐标方程化为笛卡尔形式

Conversely, a polar equation can be converted to Cartesian form using \(r^2 = x^2 + y^2\), \(r \cos θ = x\), \(r \sin θ = y\), and often \(r = \sqrt{x^2 + y^2}\).

反过来，极坐标方程可利用 \(r^2 = x^2 + y^2\)、\(r \cos θ = x\)、\(r \sin θ = y\) 以及 \(r = \sqrt{x^2 + y^2}\) 化为笛卡尔形式。

Example: the polar curve \(r = 6 \cos θ\). Multiply both sides by \(r\): \(r^2 = 6 r \cos θ\). Thus \(x^2 + y^2 = 6x\). Completing the square gives \((x-3)^2 + y^2 = 9\), a circle centre \((3,0)\), radius 3.

例如：极坐标曲线 \(r = 6 \cos θ\)。两边同乘 \(r\)：\(r^2 = 6 r \cos θ\)。因此 \(x^2 + y^2 = 6x\)。配方得 \((x-3)^2 + y^2 = 9\)，即圆心为 \((3,0)\)、半径为 3 的圆。

Example: \(r = 4 \sin θ\). Multiply by \(r\): \(r^2 = 4 r \sin θ\), so \(x^2 + y^2 = 4y\), or \(x^2 + (y-2)^2 = 4\).

例如：\(r = 4 \sin θ\)。两边同乘 \(r\)：\(r^2 = 4 r \sin θ\)，即 \(x^2 + y^2 = 4y\)，或 \(x^2 + (y-2)^2 = 4\)。

These conversions are invaluable for identifying the shape represented by a given polar equation.

这些转换对于识别给定极坐标方程所表示的形状非常有价值。

6. Plotting Polar Curves | 绘制极坐标曲线

To sketch a polar curve \(r = f(θ)\), evaluate \(r\) for representative angles, then plot the corresponding points. Useful angles are multiples of \(\pi/6\) and \(\pi/4\).

要绘制极坐标曲线 \(r = f(θ)\)，取有代表性的角度计算 \(r\)，然后描点。常用的角度是 \(\pi/6\) 和 \(\pi/4\) 的倍数。

Consider the circle \(r = 2 \cos θ\). For \(\theta = 0\), \(r = 2\), giving the point \((2,0)\). For \(\theta = \pi/3\), \(r = 1\), giving \((1, \pi/3)\). For \(\theta = \pi/2\), \(r = 0\), the pole.

考虑圆 \(r = 2 \cos θ\)。当 \(\theta = 0\) 时，\(r = 2\)，得到点 \((2,0)\)。当 \(\theta = \pi/3\) 时，\(r = 1\)，得到点 \((1, \pi/3)\)。当 \(\theta = \pi/2\) 时，\(r = 0\)，即极点。

For negative \(r\), plot the point at angle \(\theta + \pi\) with distance \(|r|\). For instance, in \(r = 2 \cos θ\), when \(\theta = 2\pi/3\), \(\cos θ = -1/2\), so \(r = -1\). This point is equivalent to \((1, 5\pi/3)\).

对于负 \(r\)，在角度 \(\theta + \pi\) 方向、距离 \(|r|\) 处描点。例如，在 \(r = 2 \cos θ\) 中，当 \(\theta = 2\pi/3\) 时，\(\cos θ = -1/2\)，所以 \(r = -1\)。该点等价于 \((1, 5\pi/3)\)。

Continue for \(\theta \in [0, 2\pi]\) and connect the points smoothly. For \(r = 2 \cos θ\), the trace completes a full circle as \(\theta\) runs from 0 to \(\pi\).

继续对 \(\theta \in [0, 2\pi]\) 取值并平滑连接各点。对于 \(r = 2 \cos θ\)，当 \(\theta\) 从 0 到 \(\pi\) 时，轨迹已构成完整的圆。

7. Standard Polar Curves | 常见极坐标曲线

Several families of curves appear frequently in A-Level examinations. Recognising them allows rapid sketching and checking of results.

有几类曲线在 A-Level 考试中经常出现。辨认它们有助于快速作图并检查结果。

Circles through the pole:

经过极点的圆：

\(r = a\) — circle centred at the pole, radius \(a\).
\(r = 2a \cos θ\) — circle centre \((a,0)\), radius \(a\).
\(r = 2a \sin θ\) — circle centre \((0,a)\), radius \(a\).

\(r = a\) — 圆心在极点、半径为 \(a\) 的圆。
\(r = 2a \cos θ\) — 圆心在 \((a,0)\)、半径为 \(a\) 的圆。
\(r = 2a \sin θ\) — 圆心在 \((0,a)\)、半径为 \(a\) 的圆。

Lines through the pole:

经过极点的直线：

\(\theta = \alpha\) — a half-line making angle \(\alpha\) with the initial line.

\(\theta = \alpha\) — 与极轴成角 \(\alpha\) 的射线。

Spirals and flowers:

螺线及玫瑰线：

\(r = a θ\) — Archimedean spiral.
\(r = a \cos(k θ)\) or \(r = a \sin(k θ)\) — rose curves with \(k\) petals if \(k\) is odd, \(2k\) petals if \(k\) is even.

\(r = a θ\) — 阿基米德螺线。
\(r = a \cos(k θ)\) 或 \(r = a \sin(k θ)\) — 玫瑰线，\(k\) 为奇数时花瓣数为 \(k\)，\(k\) 为偶数时花瓣数为 \(2k\)。

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

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

极坐标曲线 \(r = f(\theta)\) 在 \(\theta = \alpha\) 与 \(\theta = \beta\) 之间围成的面积由积分给出：

A = ½ ∫_α^β r² dθ

This formula comes from summing infinitesimal sectors of a circle, each with area \(\frac{1}{2}r^2\,\mathrm{d}\theta\). It is analogous to the area under a Cartesian curve, but uses sectors instead of rectangles.

该公式来源于对无穷小扇形面积 \(\frac{1}{2}r^2\,\mathrm{d}\theta\) 的求和。它类似于笛卡尔曲线下的面积，只不过用的是扇形而不是矩形。

Example: find the area inside the circle \(r = 2 \cos θ\). The circle is traced as \(\theta\) runs from \(-\pi/2\) to \(\pi/2\), or equivalently 0 to \(\pi\). Using symmetry from 0 to \(\pi/2\) and doubling:

例如：求圆 \(r = 2 \cos θ\) 内部的面积。圆在 \(\theta\) 从 \(-\pi/2\) 到 \(\pi/2\) 时被完整描出，也等价于从 0 到 \(\pi\)。利用对称性取 0 到 \(\pi/2\) 再乘 2：

A = 2 × ½ ∫₀^π/2 (2 cos θ)² dθ = ∫₀^π/2 4 cos² θ dθ = π

This matches the area of a circle of radius 1: \(\pi \times 1^2 = \pi\).

这与半径为 1 的圆面积 \(\pi \times 1^2 = \pi\) 一致。

When finding areas between two polar curves, subtract the inner area from the outer area, ensuring the correct limits are used.

当求两条极坐标曲线之间的面积时，用外曲线面积减去内曲线面积，注意使用正确的积分上下限。

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

The slope of a tangent to a polar curve \(r = f(\theta)\) is obtained from the parametric forms \(x = r \cos θ\), \(y = r \sin θ\). The gradient is:

极坐标曲线 \(r = f(\theta)\) 的切线斜率可由参数形式 \(x = r \cos θ\)、\(y = r \sin θ\) 求得。其斜率为：

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

This expression is derived using the product rule on \(y = r \sin θ\) and \(x = r \cos θ\), then dividing \(dy/dθ\) by \(dx/dθ\).

该表达式通过对 \(y = r \sin θ\) 和 \(x = r \cos θ\) 使用乘积法则求导，然后用 \(dy/dθ\) 除以 \(dx/dθ\) 得到。

At the pole (\(r=0\)), the tangent line has angle \(\theta\) given by \(f(\theta)=0\). For example, in \(r = 2 \cos θ\), \(r=0\) when \(\theta = \pi/2\), and the tangent there is vertical.

在极点（\(r=0\)）处，切线的角度 \(\theta\) 由 \(f(\theta)=0\) 给出。例如，在 \(r = 2 \cos θ\) 中，当 \(\theta = \pi/2\) 时 \(r=0\)，该处的切线是竖直的。

10. Symmetry in Polar Coordinates | 极坐标中的对称性

Symmetry can simplify graphing and integration. A polar curve has:

对称性可以简化作图和积分。极坐标曲线具有：

symmetry about the initial line if \(r(\theta) = r(-\theta)\) or \(r(\theta) = -r(\pi-\theta)\);
symmetry about the vertical axis if \(r(\theta) = r(\pi-\theta)\);
symmetry about the pole if \(r(\theta) = -r(\theta)\) (translation by \(\pi\)).

关于极轴对称，如果 \(r(\theta) = r(-\theta)\) 或 \(r(\theta) = -r(\pi-\theta)\)；
关于竖直轴对称，如果 \(r(\theta) = r(\pi-\theta)\)；
关于极点对称，如果 \(r(\theta) = -r(\theta)\)（即旋转 \(\pi\)）。

For instance, \(r = 2 \cos θ\) is symmetric about the initial line because \(\cos(-\theta) = \cos θ\). This is why the circle appears centred on the \(x\)-axis.

例如，\(r = 2 \cos θ\) 关于极轴对称，因为 \(\cos(-\theta) = \cos θ\)。这就是该圆圆心位于 \(x\) 轴上的原因。

Using symmetry, it is often possible to compute areas or arc lengths over a smaller interval and then multiply by the number of symmetric parts.

利用对称性，通常可以在较小的区间内计算面积或弧长，然后乘以对称部分的数量。

11. Arc Length in Polar Form | 极坐标下的弧长

The length of an arc of a polar curve \(r = f(\theta)\) from \(\theta = \alpha\) to \(\theta = \beta\) is:

极坐标曲线 \(r = f(\theta)\) 从 \(\theta = \alpha\) 到 \(\theta = \beta\) 的弧长为：

L = ∫_α^β √( r² + (dr/dθ)² ) dθ

This formula is derived from the Cartesian arc length element \(ds = \sqrt{(dx)^2 + (dy)^2}\), expressed in polar coordinates.

该公式由笛卡尔弧长微元 \(ds = \sqrt{(dx)^2 + (dy)^2}\) 在极坐标中表达而得。

Example: the circumference of the circle \(r = a\) (a constant). Here \(dr/dθ = 0\), so the integral becomes \(\int_0^{2\pi} a \, dθ = 2\pi a\), as expected.

例如：圆 \(r = a\)（常值）的周长。这里 \(dr/dθ = 0\)，因此积分变为 \(\int_0^{2\pi} a \, dθ = 2\pi a\)，符合预期。

12. Choosing the Right Frame | 选择合适的坐标系

Some problems are naturally suited to one frame. Use Cartesian coordinates when dealing with straight lines, quadratic curves, or translating between function forms. Use polar coordinates when the problem involves circular motion, radial distances, angles, or curves defined by distance from a point.

某些问题天然适合某一种坐标系。处理直线、二次曲线或函数形式转换时使用笛卡尔坐标。当问题涉及圆周运动、径向距离、角度，或由到某点距离定义的曲线时，使用极坐标。

In AQA A-Level Mathematics, polar coordinates appear in the second year Further Mathematics syllabus. You are expected to convert equations, sketch curves, find areas and arc lengths, and understand tangents and symmetry.

在 AQA A-Level 数学中，极坐标出现在进阶数学第二年的考纲中。要求你能够转换方程、绘制曲线、求面积和弧长，并理解切线与对称性。

Mastering both frames and switching flexibly between them is a key skill. Practice by taking a familiar Cartesian equation, converting it to polar, sketching both, and checking they represent the same set of points.

掌握两种坐标系并灵活切换是一项关键技能。练习方法：取一个熟悉的笛卡尔方程，转换为极坐标，分别绘图，并检查它们是否表示相同的点集。

Ultimately, the choice of frame is about clarity. The same curve may look complicated in one system and beautifully simple in the other.

归根结底，选择坐标系是为了清晰。同一条曲线在一种系统中可能看起来很复杂，而在另一种系统中却优美简洁。

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