A Simple and Efficient Method for Modeling Constant Mean Curvature Surfaces

A constant mean curvature (CMC) surface is a critical point of surface area with respect to variations that preserve the volume bounded by the surface. We present a simple and elegant method for constructing a triangle mesh approximation to a CMC surface by minimizing the discretized surface area subject to a fixed volume and other constraints such as fixed boundary curves. The key idea is an optimization procedure in which the surface area functional is replaced by a least squares energy functional that is easier to minimize and whose critical points are uniformly parameterized surfaces or, more generally, surfaces parameterized in accord with an arbitrary density function. A good initial solution estimate is not required. In fact, the input mesh quality may be almost arbitrarily poor. The user-specified volume is taken to be a shape-control parameter, and since the volume constraint need not be satisfied precisely, it is simply treated as an additional equation in the least squares system. The nonlinear least squares problem is treated by a trust region method. Test results demonstrate the effectiveness of the procedure.