Getting A Speeding Ticket
Mesh Generation
2D Point Set Delaunay Triangulation
3D Point Set Delaunay Tetrahedralization
Theory
• A point set in Rd can be projected onto a paraboloid in Rd+1.
• The convex hull in Rd+1 will contain ALL points.
• The lower convex hull will contain only triangular faces.
• The projection of the lower hull back onto Rd forms a triangular mesh (Delaunay Triangulation).
(Edelsbrunner & Seidel, 1986)
2D points projected onto a 3D paraboloid
Lower Hull Extraction
Algorithm:1. Compute convex hull2. Search for point Pmax with maximum distance
from (d+1)th axis3. Construct tangent (hyper) plane at Pmax 4. Find tangent plane’s z intercept (optimal
viewpoint)5. Extract all facets visible from optimal viewpoint6. Project facets in Rd+1 to Rd space
Paraboloid
Optimal Viewpoint
Complexity: Ω(nd/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 hull Exact Solution using MATLAB
Results:• Method fails when a facet is very thin.
Low Aspect Ratio
Horizon Facets
bh Ratio Aspect
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.
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