Polar Coordinates | 极坐标

📚 Polar Coordinates | 极坐标

Polar coordinates describe points in the plane by their distance from a fixed origin and the angle from a fixed initial line. This alternative to Cartesian coordinates makes many curves with circular or rotational symmetry much simpler to express and is essential for AQA A-level polar curve sketching and area problems.

极坐标通过点到固定原点的距离以及从固定初始线转过的角度来描述平面上的点。这种直角坐标的替代方法使许多具有圆形或旋转对称性的曲线表达更加简洁，也是 AQA A-level 极坐标曲线绘制与面积问题的基础。

1. What Are Polar Coordinates? | 什么是极坐标？

In polar coordinates, a point P is written as (r, θ), where r is the radial distance from the pole O and θ is the angle measured anticlockwise from the initial line. The pole is the origin, and the initial line is usually the positive x-axis.

在极坐标中，点 P 记为 (r, θ)，其中 r 是从极点 O 到该点的径向距离，θ 是从初始线按逆时针方向测量的角度。极点就是原点，初始线通常是正 x 轴。

Negative values of r are interpreted by measuring along the opposite direction to the angle θ, so (r, θ) and (−r, θ + π) represent the same point. Polar coordinates are not unique: adding 2π to θ gives the same point.

负的 r 值理解为沿着角度 θ 的反方向测量，因此 (r, θ) 与 (−r, θ + π) 表示同一个点。极坐标并不唯一：给 θ 加上 2π 后仍然表示同一个点。

2. Converting Between Polar and Cartesian Coordinates | 极坐标与直角坐标互化

The forward conversion uses x = r cos θ and y = r sin θ. For example, the polar point (4, π/3) becomes (2, 2√3) in Cartesian coordinates.

正向转换使用 x = r cos θ 和 y = r sin θ。例如，极坐标点 (4, π/3) 转换为直角坐标为 (2, 2√3)。

x = r cos θ, y = r sin θ

The reverse conversion finds r = √(x² + y²) and uses tan θ = y/x. You must then choose θ in the correct quadrant by looking at the signs of x and y.

反向转换先求 r = √(x² + y²)，并用 tan θ = y/x。随后必须根据 x 和 y 的符号选择正确象限中的 θ。

r = √(x² + y²), tan θ = y/x

3. Choosing the Correct Angle | 选择正确的角度

A calculator’s arctan function only gives the principal angle, usually between −π/2 and π/2. For points in the second or third quadrant, add π to the calculator value.

计算器的 arctan 函数只给出主值，通常在 −π/2 到 π/2 之间。对于第二或第三象限的点，需要在计算器结果上加 π。

θ = arctan(y/x) for x > 0; θ = arctan(y/x) + π for x < 0

If x = 0, use θ = π/2 for positive y and θ = −π/2 for negative y. Always draw a small diagram showing the point relative to the initial line to avoid angle errors.

如果 x = 0，则 y 为正时 θ = π/2，y 为负时 θ = −π/2。始终画一个显示点相对初始线位置的简图，以避免角度错误。

4. Sketching Polar Curves | 绘制极坐标曲线

To sketch a polar curve r = f(θ), build a table of θ values, calculate r, and plot points along the corresponding rays. Use exact angles such as 0, π/6, π/4, π/3, π/2 and their multiples.

要绘制极坐标曲线 r = f(θ)，可以建立 θ 值表、计算 r，并沿相应的射线描点。使用精确角度，如 0、π/6、π/4、π/3、π/2 及其倍数。

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