Mesh Generation 2D Point Set Delaunay Triangulation 3D Point Set Delaunay Tetrahedralization
Theory A point set in R d can be projected onto a paraboloid in R d+1. The convex hull in R d+1 will contain ALL points. The lower convex hull will contain only triangular faces. The projection of the lower hull back onto R d forms a triangular mesh (Delaunay Triangulation). (Edelsbrunner & Seidel, 1986) 2D points projected onto a 3D paraboloid
Lower Hull Extraction Algorithm: 1.Compute convex hull 2.Search for point Pmax with maximum distance from (d+1)th axis 3.Construct tangent (hyper) plane at Pmax 4.Find tangent plane’s z intercept (optimal viewpoint) 5.Extract all facets visible from optimal viewpoint 6.Project facets in R d+1 to R d space Paraboloid Optimal Viewpoint Complexity: Ω (n d/2 ) (O’Rourke, 1998)
Results: Hyperplane-Intersection as Optimal Viewpoint Works well for structured meshes with good aspect ratio Optimal Viewpoint 2D Mesh obtained from 3D Convex Hull
Results: Hyperplane-Intersection as Optimal Viewpoint Meshes verified for higher dimensions 3D Mesh obtained from 4D convex hullExact Solution using MATLAB
Results: Method fails when a facet is very thin. Low Aspect Ratio Horizon Facets h b
Analysis LAR Triangle + Optimal Viewpoint = 4 nearly co-planar points Coplanar points are treated as invisible. If AR < 10 -4, points are numerically coplanar When this happens, choose a lower optimal viewpoint.