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Published byVirginia Beasley Modified over 6 years ago

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

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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!

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

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

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

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

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

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

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

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

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

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

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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

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

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

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

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

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

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1st Meeting Industrial Geometry Computational Geometry 18 Thank you

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