MINIMAL-SURFACES AND SOBOLEV GRADIENTS

We treat the problem of computing triangle-based piecewise linear appr oximations to parametric minimal surfaces in R(3). More specifically, given a triangulation T of the unit square Omega and a function f(0) f rom the nodes of T into R(3), We Seek a function f from the nodes of T into R(3) such that f agrees with f(0) on the boundary of Omega, and the triangulated surface area corresponding to the image of f is minim al. We employ a descent method in which, at each iteration, the gradie nt of the surface area functional is computed with respect to an inner product that depends on the current approximation to f. Test results: show that, starting with extremely poor initial estimates, a few desc ent iterations produce approximations in the vicinity of the solution. We also introduce a new characterization of minimal surfaces that eli minates the potential problem of triangle areas approaching zero. In p lace of the surface area functional, we minimize a functional whose cr itical points are uniformly parameterized minimal surfaces. This not o nly results in rapid convergence of the descent method, but also simpl ifies the expressions for gradients and Hessians.