NUMERICAL CONFORMAL MAPPING USING CROSS-RATIOS AND DELAUNAY TRIANGULATION

We propose a new algorithm for computing the Riemann mapping of the un it disk to a polygon, also known as the Schwarz{Christoffel transforma tion. The new algorithm, CRDT (for cross-ratios of the Delaunay triang ulation), is based on cross-ratios of the prevertices, and also on cro ss-ratios of quadrilaterals in a Delaunay triangulation of the polygon . The CRDT algorithm produces an accurate representation of the Rieman n mapping even in the presence of arbitrary long, thin regions in the polygon, unlike any previous conformal mapping algorithm. We believe t hat CRDT solves all difficulties with crowding and global convergence, although these facts depend on conjectures that we have so far not be en able to prove. We demonstrate convergence with computational experi ments. The Riemann mapping has applications in two-dimensional potenti al theory and mesh generation. We demonstrate CRDT on problems in long , thin regions in which no other known algorithm can perform comparabl y.