Efficient Computation of Planar Triangulations

In 19] we introduced a new algorithm for computing planar triangulations of faceted surfaces for surface parameter-ization. Our algorithm computes a mapping that minimizes the distortion of the surface metric structures (lengths, angles, etc.). Compared with alternative approaches, the algorithm provides a signiicant improvement in robustness and applicability; it can handle more complicated surfaces and it does not require a convex or predeened planar domain boundary. However, our algorithm involves the solution of a constraint minimization problem. The potential high cost in solving the optimization problem has given rise to concerns about the applicability of the method, especially for very large problems. This paper is concerned with the eecient solution of the linear systems that arise when Newton's method is applied to the constraint minimization problem. In small to moderate size models the linear systems can be solved eeciently with a sparse direct method. We give examples from computations with the SuperLU package 6]. For larger models we have to use preconditioned iterative methods. We develop a new preconditioner that takes into account the structure of the problem. Some preliminary experimental results are shown that indicate the eeectiveness of this approach.