Plane

1. Cartesian coordinates
2. Polar coordinates

1. Cartesian coordinates
2. Spherical coordinates
3. Cylindrical coordinates

Sphere $S^2$

1. Stereographic projection
2. Mercator projection
3. ..

Miscellaneous

1. Barycentric coordinates
2. Homogeneous coordinates $(x,\,y,\,t)$ in plane, $(x,\,y,\,z,\,t)$ in

Differential geometry

1. Normal coordinates in Riemann geometry
2. Darboux coordinates in symplectic geometry
3. Gaussian coordinates
4. Fermi coordinates for a submanifold
5. Boundary coordinates

Coordinate transformations

1. affine transformations
2. continuous transformations in algebraic topology (homotopy theory)
3. Lorentz group transformations
4. Cartesian-to-polar coordinate transformations
5. Cartesian-to-spherical coordinate transformations
6. A.
7. B.