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1st Meeting Industrial Geometry Computational Geometry ---- Some Basic Structures 1st IG-Meeting.

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Presentation on theme: "1st Meeting Industrial Geometry Computational Geometry ---- Some Basic Structures 1st IG-Meeting."— Presentation transcript:

1 1st Meeting Industrial Geometry Computational Geometry ---- Some Basic Structures 1st IG-Meeting

2 1st Meeting Industrial Geometry Computational Geometry 1 Intention  Computational Geometry: Algorithmic treatment of simple geometric objects  Vast body of results  Ideally: People see problem solved ☺  At least: Hope that some material useful (first positive results from Otmar, Helmut) Encourage further application!

3 1st Meeting Industrial Geometry Computational Geometry 2 Voronoi Diagrams Structure sought that captures proximity information among objects Regions of influence Size small, easy computation

4 1st Meeting Industrial Geometry Computational Geometry 3 Voronoi Diagrams Point location Nearest neighbor pairs clustering points Minimum spanning tree

5 1st Meeting Industrial Geometry Computational Geometry 4 Voronoi Diagrams Gabriel graph Nearest neighborhood graph

6 1st Meeting Industrial Geometry Computational Geometry 5 Voronoi Diagrams Vertices: Largest empty circle (point density) Edges: Robot trails (motion planning)

7 1st Meeting Industrial Geometry Computational Geometry 6 Variants Power diagram Union of disks

8 1st Meeting Industrial Geometry Computational Geometry 7 Variants Medial axis and straight skeleton of a polygon

9 1st Meeting Industrial Geometry Computational Geometry 8 Variants City Voronoi diagram Sites + network

10 1st Meeting Industrial Geometry Computational Geometry 9 Triangulations Delaunay triangulation from Voronoi diagram Optimizes: smallest angle largest covering circle mean inradius

11 1st Meeting Industrial Geometry Computational Geometry 10 Triangulations Delaunay does NOT optimize: largest angle (quadr.) largest vertex degree (NP) total edge length (?)

12 1st Meeting Industrial Geometry Computational Geometry 11 Triangulations Minimum weight triangulation large connected subgraph in linear expected time

13 1st Meeting Industrial Geometry Computational Geometry 12 Variants Relax triangulation by controlling face shape Pseudo-triangulation each face has exactly 3 angles smaller than π

14 1st Meeting Industrial Geometry Computational Geometry 13 Variants Visibility in polygons (shortest paths) Maintaining free space (collision detection) ↓

15 1st Meeting Industrial Geometry Computational Geometry 14 Variants Easier to adapt: Edge flips are - more general - less local

16 1st Meeting Industrial Geometry Computational Geometry 15 Variants Lifting property: Surface for a triangulation

17 1st Meeting Industrial Geometry Computational Geometry 16 Variants Pseudo-triangulations are liftable, too Surface representation using pseudo-triangles

18 1st Meeting Industrial Geometry Computational Geometry 17 Variants Pseudo-tetrahedra in 3D (bend solid to convex…)

19 1st Meeting Industrial Geometry Computational Geometry 18 Thank you


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