Surfaces

Two-dimensional surfaces from algebra, differential geometry and topology. Classic surfaces as well as new examples and counter-examples. Immersed surfaces, implicit surfaces, topological shapes. The surfaces are specified as a simplicial mesh immersed in a euclidean space Rn.

A a surface in R3 can locally be parametrized by a map

F(u,v) = (x(u, v),y(u, v),z(u, v)) : D ->  R3

of a domain D in R2 where x, y, z are the coordinate functions.

References

  1. M.P. do Carmo Differential Geometry of Curves and Surfaces Prentice- Hall Englewood Cliffs, NJ, 1976
  2. Gerd Fischer Mathematical Models Vieweg Verlag 1986.
  3. George Francis A Topological Picture Book Springer Verlag 1987.
  4. Alfred Gray Modern Differential Geometry of Curves and Surfaces CRC Press 1994.
  5. M. Schilling Catalog Mathematischer Modelle für den höheren mathematischen Unterricht Leipzig, 1911.

Technical Note

As a guide, meshes should have no holes, no degenerate triangles and elements, no duplicate vertices. Surfaces should have meshes with an adjacency relation.