Many problems in optimal geometry require specialized software systems because there is often no explicit parametrization of a desired minimal surface. Here we take note of some of the specific software systems being used in minimal surface research.

One tactic for generating minimal surfaces is to evolve a given
initial surface to minimize *energy*, such as surface tension.
Ken Brakke's Surface Evolver
[2] is an interactive program that evolves a
surface toward minimal energy by a gradient descent method. The energy
in the Evolver can be a combination of factors such as
surface tension, gravitational
energy, squared mean curvature, user-defined surface integrals, or
knot energies. The user can interactively modify the surface to
change its properties or to keep the evolution well-behaved. The
Evolver was originally written for one and two dimensional surfaces,
but it can handle higher dimensional surfaces with some restrictions
on the features available. A limitation on the Evolver is the
requirement that it be given an initial combinatorial structure.

Another approach is taken by the University of Massachusetts MESH system, which generates triangulations of parametric surfaces defined by conformal mappings from a two-dimensional domain to a three dimensional range. It deals correctly even with highly non-uniform mappings, for example where points on the domain map to infinity on the range, employing an incremental process starting at the origin of the domain and repeatedly adding new triangles to the perimeter of a growing region.

Ulrich Pinkall's geometrical graphics group at the Technical University of Berlin has chosen the software product AVS as its primary visualization tool. This system allows the researcher to chain together independent modules into complex computational networks. With a family of standard and user-generated modules, this group conducts research on the visualization of optimal surfaces, such as H-surfaces (surfaces of constant mean curvature), and on discrete dynamical models for quantum systems. This work has led to new results such as the discovery of the simplest soliton.

A group at the University of Bonn led by Konrad Polthier has created its own visualization environment, known as GRAPE, for their research into minimal surfaces and related differential systems. GRAPE (GRAphical Programming Environment) reflects an object-oriented approach that encourages users to create specific geometric objects, and includes features that expedite the creation of animations.

Thu Sep 21 19:17:33 CDT 1995