📚 Cartesian and Polar Frames of Reference | 笛卡尔坐标系与极坐标系
Choosing the right coordinate system can turn a complicated curve into a simple equation. In AQA A-Level Mathematics, you need to move fluently between Cartesian coordinates (x, y) and polar coordinates (r, θ), and use polar curves for sketching, tangents and area calculations.
选择合适的坐标系可以把复杂的曲线变成简单的方程。在 AQA A-Level 数学中，你需要熟练地在笛卡尔坐标 (x, y) 与极坐标 (r, θ) 之间转换，并能利用极坐标曲线进行图像绘制、切线计算和面积求解。
1. What Is a Frame of Reference? | 什么是参考坐标系
A frame of reference is a rule that labels each point in the plane. In Cartesian form, the label is an ordered pair (x, y) giving signed horizontal and vertical distances from a fixed origin O. In polar form, the label is an ordered pair (r, θ), where r is the distance from O and θ is the angle measured anticlockwise from the positive x-axis.
参考坐标系是给平面上每个点标注坐标的规则。笛卡尔形式用有序数对 (x, y) 表示相对于固定原点 O 的有符号水平和垂直距离。极坐标形式用有序数对 (r, θ) 表示，其中 r 是点到 O 的距离，θ 是从正 x 轴逆时针测量的角度。
Cartesian coordinates give a unique label for every point. Polar coordinates do not: adding 2π to θ gives the same direction, and replacing r by −r with an angle shift of π also gives the same point. In AQA questions, be aware that a polar equation may trace the same curve more than once over a given interval.
笛卡尔坐标对每个点的表示是唯一的。极坐标则不唯一：θ 加上 2π 表示同一方向，把 r 取负并将 θ 平移 π 也表示同一个点。在 AQA 试题中，要注意一个极坐标方程在某个区间内可能重复描述同一条曲线。
2. Cartesian Coordinates as a Rectangular Grid | 作为矩形网格的笛卡尔坐标
e origin. These forms are especially useful when distances are naturally horizontal and vertical.
笛卡尔坐标轴把平面分成四个象限。方程 y = mx + c 表示直线，x² + y² = a² 表示以原点为圆心的圆。当距离自然呈现水平与垂直方向时，这种形式尤其有用。
For curves such as circles, ellipses and parabolas, the Cartesian frame often makes intercepts and transformations easier to see. However, for curves that grow or rotate around a fixed point, Cartesian equations can become very complicated.
对于圆、椭圆和抛物线等曲线，笛卡尔坐标系通常能让截距和变换更直观。然而，对于绕固定点旋转或扩展的曲线，笛卡尔方程可能会变得非常复杂。
3. Polar Coordinates: Radial Distance and Angle | 极坐标系：径向距离与角度
angle from the initial line to OP. As in all AQA calculus work, θ is measured in radians.
固定点 O 称为极点，正 x 轴称为极轴。点 P 的极坐标为 (r, θ)，其中 r = OP，θ 是从极轴到 OP 的角度。在所有 AQA 微积分问题中，θ 均以弧度为单位。
By convention, r is usually taken to be non-negative, but AQA questions may use negative r. A negative r means the point lies on the opposite ray: (−r, θ) is the same point as (r, θ + π).
按照惯例，r 通常取非负值，但 AQA 试题中可能会出现负的 r。负 r 表示点位于相反射线上：( −r, θ ) 与 ( r, θ + π ) 表示同一个点。
4. Converting from Polar to Cartesian | 从极坐标转换为笛卡尔坐标
OP, the polar point (r, θ) has Cartesian coordinates given by x = r cos θ and y = r sin θ.
根据 OP 的直角三角形关系，极坐标点 (r, θ) 的笛卡尔坐标为 x = r cos θ 和 y = r sin θ。
x = r cos θ, y = r sin θ
For example, the point (4, π/3) gives x = 4 cos(π/3) = 2 and y = 4 sin(π/3) = 2√3, so the Cartesian form is (2, 2√3).
例如，点 (4, π/3) 对应的 x = 4 cos(π/3) = 2，y = 4 sin(π/3) = 2√3，因此笛卡尔坐标为 (2, 2√3)。
5. Converting from Cartesian to Polar | 从笛卡尔坐标转换为极坐标
angent alone is not enough: you must choose θ in the correct quadrant by checking the signs of x and y.
将 (x, y) 转换为极坐标时，使用 r² = x² + y² 和 tan θ = y/x。但仅凭正切值不够：必须根据 x 和 y 的符号选择 θ 所在象限。
r² = x² + y², tan θ = y/x
| x sign | y sign | Quadrant | θ range |
|---|---|---|---|
| positive | positive | First | 0 < θ < π/2 |
| negative | positive | Second | π/2 < θ < π |
| negative | negative | Third | π < θ < 3π/2 |
| positive | negative | Fourth | 3π/2 < θ < 2π |
For example, convert (−1, √3) to polar form. Here r² = 1 + 3 = 4, so r = 2, and tan θ = −√3 gle is in the second quadrant, so θ = 2π/3. Therefore the polar coordinates are (2, 2π/3).
例如，将 (−1, √3) 转换为极坐标。这里 r² = 1 + 3 = 4，所以 r = 2，tan θ = −√3。由于 x < 0 且 y > 0，角度在第二象限，因此 θ = 2π/3。所以极坐标为 (2, 2π/3)。
6. Common Polar Curves | 常见极坐标曲线
A polar equation can produce shapes that are awkward in Cartesian form. Knowing the standard forms below helps you recognise curves quickly in AQA exam questions.
极坐标方程可以产生笛卡尔形式下很难处理的曲线。掌握以下标准形式有助于你在 AQA 考试中快速识别曲线。
| Polar equation | Cartesian form | Shape |
|---|---|---|
| r = a | x² + y² = a² | Circle centred at O, radius a |
| θ = α | y = x tan α | Line through O at angle α |
| r = 2a cos θ | (x − a)² + y² = a² | Circle centre (a, 0), radius a |
| r = a(1 + cos θ) | not usually required | Cardioid |
id. Its Cartesian equation is very messy, so polar form is preferred for sketching and area work.
例如，r = a(1 + cos θ) 是心形线。它的笛卡尔方程非常复杂，因此在绘图和面积计算中优先使用极坐标形式。
7. Symmetry and Curve Sketching | 对称性与曲线绘制
s if replacing θ by π − θ gives an equivalent equation. Symmetry about the pole occurs if replacing r by −r gives an equivalent equation.
绘制极坐标曲线前先使用对称性
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