Global Characteristics

Global surface characteristics are associated with entire surfaces, and are contrasted with local characteristics, which are associated with points. The Global characteristics of a surface may be difficult to determine from its mathematical description, as is evident from the difficulty of constructing formal proofs of the global property of embeddeness for several recently discovered surfaces listed in the minimal surface library.

Embeddedness

A surface is embedded if it does not have any self-intersections. One with self-intersections is said to be immersed. Whereas implicitly defined surfaces are inherently embedded, parametrically defined surfaces frequently are not.

Total Curvature and Euler Number

The total gaussian curvature of a surface is the integral of the gauss map over the surface. According to the Gauss-Bonnet theorem, the total curvature is a multiple of its Euler characteristic (or Euler number). The Euler characteristic is given by the formula V - E + F, where V is the number of vertices, E is the number of edges, and F is the number of faces. This formula can be applied to surfaces by drawing a vertex-edge graph on the surface which divides it into entirely simply-connected faces (where edges and faces are taken to mean curved arcs and patches on the surface, not flat elements).

Genus and Euler Number

The genus of a surface is a topological property giving the number of 'holes' or 'handles' in it. More precisely, it is the maximum number of closed nonintersecting curves along which one could cut the surface without producing multiple pieces. The genus, like the total curvature, can be computed using the Euler characteristic.

Symmetry

Symmetries in surfaces may be imposed by their definition or discovered through their analysis. Arguments based on symmetries are frequently used in embeddedness proofs. Symmetries can also be exploited by surface generation methods, which need only compute a fundamental piece (one with no symmetries) of the surface and then copy and transform the piece (possibly in display hardware) to generate the entire surface.

Skeletal Graphs

Skeletal graphs have the same symmetry as their surfaces. Although they do not enjoy a rigorous mathematical theory, they can be useful tools for characterizing families of surfaces, particluarly in the context of modelling phenomena for which only ad-hoc surface models exist.