List of Coordinate Charts - Favorite Surfaces

Favorite Surfaces

Here are some charts which (with appropriate metric tensors) can be used in the stated classes of Riemannian and semi-Riemannian surfaces:

• isothermal chart
• Radially symmetric surfaces:
  • polar chart
• Surfaces embedded in E3:
  • Monge chart
• Certain minimal surfaces:
  • asymptotic chart (see also asymptotic line)

Here are some charts on some of the most useful Riemannian surfaces (note that there is some overlap, since many charts of S2 have closely analogous charts on H2; in such cases, both are discussed in the same article):

• Euclidean plane E2:
  • Cartesian chart
  • Maxwell chart

• Sphere S2:
  • polar chart (arc length radial chart)
  • stereographic chart
  • central projection chart
  • axial projection chart
  • Mercator chart

• Hyperbolic plane H2:
  • polar chart
  • stereographic chart (Poincaré model)
  • upper half space chart (another Poincaré model)
  • central projection chart (Klein model)
  • Mercator chart

Favorite semi-Riemannian surface:

• AdS₂ (or S1,1) and dS₂ (or H1,1):
  • central projection
  • equatorial trig

Note: the difference between these two surfaces is in a sense merely a matter of convention, according to whether we consider either the cyclic or the non-cyclic coordinate to be timelike; in higher dimensions the distinction is less trivial.