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Unformatted text preview: CSE 4101/5101 Computational Geometry Prof. Andy Mirzaian Overview 2 3 Landscape of Computational Geometry Applications : Graphics Robotics Vision GIS CAD VLSI Pattern Recognition Optimization Transportation Statistics . . . Geometric Tools : Convex Hull Space subdivision Arrangements Voronoi / Delaunay Diagram Triangulations Geometric Transforms Duality . . . Algorithmic Tools : general incremental divide&conquer space sweep topological sweep prune and search random sampling locus approach multidimensional search dynamization . . . Data Structures : general interval trees range trees segment trees priority search trees Kd trees fractional cascading persistent D.S. . . . Analysis Tools : general amortization DavenportSchinzel . . . Implementation Issues : Degeneracy (symbolic perturbation) Robustness (inexact arithmetic) . . . 4 Example 1: Convex Hull 5 Example 1: Convex Hull 6 Example 2: Point set triangulation 7 Example 2: Point set triangulation 8 Nonsimple Simple Convex Example 3: Simple Polygon Polygon: A closed curve in the plane consisting of finitely many straight segments. Simple Polygon: A connected nonselfcrossing polygon. Convex Polygon: A simple polygon with no interior angle exceeding 180. 9 Example 3: Simple Polygon Triangulation 10 Example 4: Planar Line Arrangement 11 Example 4: Planar Line Arrangement 12 Example 5: Voronoi Diagram & Delaunay Triangulation 13 Nearest site proximity partitioning of the plane Example 5: Voronoi Diagram & Delaunay Triangulation 14 Delaunay Triangulation = Dual of the Voronoi Diagram. Example 5: Voronoi Diagram & Delaunay Triangulation 15 Delaunay triangles have the empty circle property. Example 5: Voronoi Diagram & Delaunay Triangulation 16 Example 5: Voronoi Diagram & Delaunay Triangulation 17 x z y z=x2+y2 Example 6: 2D Delaunay Triangulation via 3D Convex Hull 18 D A B C D A B C Applications : Graphics Multiwindow user systems E E Example 7: Hidden Surface Removal 19 KoebeAndreevThurstons Circle Packing Theorem : [1] Any planar graph can be drawn in the plane with vertices as centers of nonoverlapping circles, such that edge between two vertices the two circles touch. Applications : CAD, VLSI, Graphics, Example 8: Planar Graph Drawing 20 Geometric Preliminaries 21 Metric Space (S,d) S = a set of objects called points of the space, d: S2 called the distance metric Metric Axioms: (x,y,z S) 1. d(x,y) 0 (nonnegativity) 2. d(x,y) = 0 x=y (positive definiteness) 3. d(x,y) = d(y,x) (symmetry) 4. d(x,y) + d(y,z) d(x,z) (triangle inequality) 22 d = ddimensional real space is the set of dvectors x=(x1,x2, ,xd)...
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This note was uploaded on 02/13/2012 for the course CSE 4101 taught by Professor Mirzaian during the Winter '12 term at York University.
 Winter '12
 Mirzaian
 Data Structures

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