Delaunay Triangulation for Curved Surfaces

Surface triangular meshing plays an important role in the areas of computer graphics and engineering analysis. Traditionally, surface meshing is achieved by mapping meshes created in 2D parametric space onto surfaces. Care is taken in generating meshes in the parametric space and mapping them to surfaces because the transformation of geometry from the parameter space to the real space could be extended and twisted along some directions. Therefore, a good looking mesh in 2D parametric space could be very poor on a surface. In this paper, we present a meshing scheme which generate Delaunay type triangular surface mesh. For each triangle on a surface , its circumcircle is mapped into the parametric space, and the geometry of the mapped circle in the parametric space is approximated by an ellipse function. The triangulation in the parametric space is created and maintained using the property of empty circumellipse instead of empty circumcircle. Also surface curvature is used to control surface mesh density distribution. Therefore, the triangular mesh has a good approximation to the surface shape. The implementation of the approach results in very good-quality surface mesh.