Parametric Surfaces

The parametric method of representing surfaces uses a function to map some portion of R2 (the domain) to a patch of the surface in R3. Because any position in the plane, and thus any position on the surface patch, can be uniquely given by two coordinates, the surface is said to be parameterized by those coordinates.

A single surface patch may cover an entire surface, or it may represent only a portion of that surface. In the latter case a larger portion of the surface may be assembled using replicas of one or multiple patches.

A simple way to generate surfaces parametrically is to create a rectangular grid over some portion of R2 and evaluate the function at each point on the grid. Quadrilateral or triangular facets are readily recovered from the grid. Numerous software packages, such as ^Mathematica, ^JavaView, and VPS support the generation of surface patches using this approach.

A less restricted but more complicated way of generating surfaces parametrically is used by Mesh. Mesh treats domain coordinates as complex numbers. This enables it to use the Weierstrass Formula, which gives surface coordinates as the integral of an expression involving two functions (f and g) whose domains are some portion of the complex plane (C). Mesh uses programmable domain shapes, and employs an adaptive mesh generation algorithm, which varies the size of the triangles in the domain to achieve desired mesh density over the surface.