Curvilinear coordinates
（重定向自Curvilinear coordinate system）
![Curvilinear (top), affine (right), and Cartesian (left) coordinates in two-dimensional space](/Images/godic/202501/07/Curvilinear.svg1804.png")
![Fig. 1 - Coordinate surfaces, coordinate lines, and coordinate axes of general curvilinear coordinates.](/Images/godic/202501/07/General_curvilinear_coordinates_1.svg1804.png")
![Fig. 2 - Coordinate surfaces, coordinate lines, and coordinate axes of spherical coordinates. Surfaces: r - spheres, θ - cones, φ - half-planes; Lines: r - straight beams, θ - vertical semicircles, φ - horizontal circles;
Axes: r - straight beams, θ - tangents to vertical semicircles, φ - tangents to horizontal circles](/Images/godic/202501/07/Spherical_coordinate_elements.svg1804.png")
![A vector v (red) represented by
• a vector basis (yellow, left: e1, e2, e3), tangent vectors to coordinate curves (black) and
• a covector basis or cobasis (blue, right: e1, e2, e3), normal vectors to coordinate surfaces (grey)
in general (not necessarily orthogonal) curvilinear coordinates (q1, q2, q3). Note the basis and cobasis do not coincide unless the coordinate system is orthogonal.[1]](/Images/godic/202501/07/Vector_1-form.svg1804.png")
In geometry, curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved. These coordinates may be derived from a set of Cartesian coordinates by using a transformation that is locally invertible (a one-to-one map) at each point. This means that one can convert a point given in a Cartesian coordinate system to its curvilinear coordinates and back. The name curvilinear coordinates, coined by the French mathematician Lamé, derives from the fact that the coordinate surfaces of the curvilinear systems are curved.