1 / 8

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.

tamma
Download Presentation

Getting A Speeding Ticket

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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.

More Related