Overview of Two-Dimensional Curvilinear Coordinates | 二维曲线坐标概述

📚 Overview of Two-Dimensional Curvilinear Coordinates | 二维曲线坐标概述

In the world of coordinate systems, the familiar Cartesian grid is only the starting point. When a problem involves circles, rotating frames, or curved boundaries, it is often simpler to label points by parameters that naturally follow the geometry. Two-dimensional curvilinear coordinates provide a powerful language for such situations, with applications ranging from physics to engineering.

在坐标系的天地里，熟悉的直角网格仅仅是一个起点。当问题涉及圆、旋转框架或弯曲边界时，用自然贴合几何形态的参数来标记点往往会更简单。二维曲线坐标为这类问题提供了有力的语言，其应用遍及物理与工程等领域。

1. General Idea: Parameterising the Plane | 一般思想：对平面进行参数化

Let (x, y) be the usual Cartesian coordinates. A curvilinear coordinate system is defined by choosing two new real parameters (u, v) and a smooth, invertible rule that connects (u, v) to (x, y).

设 (x, y) 为通常的直角坐标。一个曲线坐标系由选择两个新实参数 (u, v) 以及一个光滑可逆的映射规则确定，该规则将 (u, v) 与 (x, y) 联系起来。

x = x(u, v), y = y(u, v)

Because the mapping is invertible, we can also write u = u(x, y) and v = v(x, y), so the pair (u, v) uniquely labels every point in a region of the plane. The position vector is written as r(u, v) = (x(u, v), y(u, v)). If the Jacobian determinant of the transformation is nonzero, the mapping is locally one-to-one.

由于映射可逆，我们也可以写出 u = u(x, y) 和 v = v(x, y)，因此数对 (u, v) 能在平面的某个区域中唯一标记每一个点。位置矢量写为 r(u, v) = (x(u, v), y(u, v))。若变换的雅可比行列式非零，则映射是局部一一对应的。

2. Coordinate Lines | 坐标线

By fixing v = v₀ and allowing u to vary, we trace out one family of coordinate lines. Similarly, fixing u = u₀ and allowing v to vary produces another family. In general these lines are curved rather than straight.

令 v = v₀ 为常数而让 u 变化，可以得到一族坐标线；类似地，令 u = u₀ 为常数而让 v 变化，则得到另一族坐标线。一般来说，这些线是弯曲的而不是直线。

The tangent vectors to the two families are ∂r/∂u and ∂r/∂v. If these two tangent vectors are perpendicular at every point, the coordinate system is called orthogonal. Cartesian and polar coordinates are both orthogonal coordinate systems.

两族坐标线的切向量分别为 ∂r/∂u 和 ∂r/∂v。如果这两个切向量在每个点都相互垂直，则称该坐标系为正交坐标系。直角坐标与极坐标都是正交坐标系。

3. The Polar Coordinate System | 极坐标系

The most familiar curvilinear coordinates in two dimensions are polar coordinates (r, θ). The transformation from polar to Cartesian coordinates is:

二维空间中最常见的曲线坐标是极坐标 (r, θ)。从极坐标到直角坐标的变换为：

x = r cos θ, y = r sin θ

The inverse transformation is r = √(x² + y²) and θ = atan2(y, x). The coordinate lines are circles r = constant and rays θ = constant. The origin is a singular point, because the angle θ is undefined there.

逆变换为 r = √(x² + y²)，θ = atan2(y, x)。其坐标线分别是 r = 常数的圆和 θ = 常数的射线。原点是一个奇异点，因为角度 θ 在该处没有定义。

4. Scale Factors and the Line Element | 比例因子与线元

For general curvilinear coordinates

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