Surfaces Overview

The Scientific Graphics Project is concerned primarily with surfaces in R3 (three dimensional euclidean space). The surfaces are organized into three major categories: Minimal Surfaces, CMC (Constant Mean Curvature) Surfaces, and Level Surfaces. This breakdown is more pragmatic than logical. The first two categories reflect properties possessed by the surfaces: Minimal and CMC surfaces have the same mean curvature at every point. Minimal surfaces, having zero mean curvature at every point, are a subset of CMC surfaces. The last category reflects a method of representing surfaces: Level surfaces are the solution to a single valued function of three variables. Typically, specialized methods are used to generate models of of CMC and minimal surfaces. Most of the examples in the minimal surface library exploit theory specific to minimal surfaces developed by Riemann, Weierstrass, and others. These three categories overlap. There are, however, level surface representations for several simple minimal surfaces, such as the plane, catenoid, and helicoid, and for several simple CMC surfaces, such as the sphere and cylinder.

Surface Characteristics

Surface characteristics can be divided into two types: local and global. Local characteristics are associated with points on the surface, and can be discovered by examining the local neighborhoods of points. Global characteristics are associated with the surface as a whole and cannot be determined strictly by looking at local neighborhoods. The local characteristics of a surface are typically much easier to deduce from its mathematical description than are its global characteristics.

Local characteristics include:

Global characteristics include:

Surface Generation Methods

Surface generation methods can be broadly classified into three categories:

The parametric method of generating minimal surfaces exploits various maps