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Getting A Speeding Ticket

Getting A Speeding Ticket. Mesh Generation. 2D Point Set. Delaunay Triangulation. Delaunay Tetrahedralization. 3D Point Set. 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.

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Getting A Speeding Ticket

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  1. Getting A Speeding Ticket

  2. Mesh Generation 2D Point Set Delaunay Triangulation Delaunay Tetrahedralization 3D Point Set

  3. 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 Rdforms a triangular mesh (Delaunay Triangulation). • (Edelsbrunner & Seidel, 1986) 2D points projected onto a 3D paraboloid

  4. Lower Hull Extraction Algorithm: • Compute convex hull • Search for point Pmax with maximum distance from (d+1)th axis • Construct tangent (hyper) plane at Pmax • Find tangent plane’s z intercept (optimal viewpoint) • Extract all facets visible from optimal viewpoint • Project facets in Rd+1 to Rd space Paraboloid Complexity: Ω(nd/2) (O’Rourke, 1998) Optimal Viewpoint

  5. Results: Hyperplane-Intersection as Optimal Viewpoint • Works well for structured meshes with good aspect ratio 2D Mesh obtained from 3D Convex Hull Optimal Viewpoint

  6. Results: Hyperplane-Intersection as Optimal Viewpoint • Meshes verified for higher dimensions 3D Mesh obtained from 4D convex hull Exact Solution using MATLAB

  7. Results: • Method fails when a facet is very thin. Horizon Facets h b Low Aspect Ratio

  8. 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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