Magnetism and Minimal Surfaces -- a Different Tool for Surface Design

Abstract

The design of free form surfaces is usually based on NURBS and it works well to quickly get shapes that a designer intends to create. Such surfaces then have desired properties like given border lines and C1 or C2 continuity along lines where several surfaces touch. Our approach is to create surfaces with certain physical properties that designers often need. Given a closed or not closed border line, can we then find an elastic surface (comparable with a rubber surface) with the property requiring that in each point the tension is equally distributed? This is simplified spoken the condition for a minimal surface. Our solution does not use any differential equations but rather the following idea: We start from a patch that may be planar or part of a cylinder or any easy to define surface. This patch is tesselated in such a way that the vertices have roughly equal distances. Each point is considered to be magnetic. Now we start a converging real-time-iteration that allows the points to move according to the rules of magnetism. Border lines or parts of them may be fixed and manipulated. The corresponding algorithm is adapted from earlier algorithms by Fruchterman et al. The result is an approximation to a minimal surface that is defined by the fixed border lines. The advantage of such a surface design is twofold: First, the problem is hard to solve exactly by means of differential equations, and second the algorithm works interactively in real time. This means that the designer can change shapes almost as quickly as with conventional free form surfaces. Finally, the surface is already suitably triangulated.