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.