1. Computational Geometry The study of algorithms for combinatorial, topological, and metric problems concerning sets of points, typically in Euclidean space. Representative areas of research include geometric search, convexity, proximity, intersection, and linear programming. Online Computing Dictionary CS691G Computational Geometry - UMass Amherst

2. Discrete Geometry Covering Tiling Packing CS691G Computational Geometry - UMass Amherst

3. Video Games CS691G Computational Geometry - UMass Amherst

4. What we saw… • Walking through large model • Collisions • Dynamic simulation • (Compare with automated movie generation) CS691G Computational Geometry - UMass Amherst

5. What to look for… • Algorithms • Complexity • Data structures • Geometric primitives CS691G Computational Geometry - UMass Amherst

6. Proximity Queries CS691G Computational Geometry - UMass Amherst

7. Dynamic Simulation CS691G Computational Geometry - UMass Amherst

8. Dynamic Simulation CS691G Computational Geometry - UMass Amherst

9. Multi-Player Games CS691G Computational Geometry - UMass Amherst

10. Multi-Player Games • Some players might be computer generated (animations) • Distributed state representation CS691G Computational Geometry - UMass Amherst

11. Motion Planning CS691G Computational Geometry - UMass Amherst

12. Kinetic Data Structures CS691G Computational Geometry - UMass Amherst

13. Georgy Voronoi 1868-1908 The Post Office Problem • Which is the closest post office to every house? (Don Knuth) • Given nsites in the plane • Subdivision of planebased on proximity CS691G Computational Geometry - UMass Amherst

14. Voronoi Diagram CS691G Computational Geometry - UMass Amherst

15. Descartes in 1644: Gravitational Influence of stars René Descartes 1596-1650 CS691G Computational Geometry - UMass Amherst

16. Distribution of McDonalds in SF CS691G Computational Geometry - UMass Amherst

17. Soap Bubble in a Frame CS691G Computational Geometry - UMass Amherst

18. Honeycomb CS691G Computational Geometry - UMass Amherst

19. Dragonfly’s Wing CS691G Computational Geometry - UMass Amherst

20. Graphic by D'Arcy Thompson CS691G Computational Geometry - UMass Amherst

21. Installation by Scott Snibbe, 1998 CS691G Computational Geometry - UMass Amherst

22. Anthropology and Archeology -- Identify the parts of a region under the influence of different Neolithic clans, chiefdoms, ceremonial centers, or hill forts. Astronomy -- Identify clusters of stars and clusters of galaxies (Here we saw what may be the earliest picture of a Voronoi diagram, drawn by Descartes in 1644, where the regions described the regions of gravitational influence of the sun and other stars.) Biology, Ecology, Forestry -- Model and analyze plant competition ("Area potentially available to a tree", "Plant polygons") Cartography -- Piece together satellite photographs into large "mosaic" maps CrystallographyandChemistry -- Study chemical properties of metallic sodium ("Wigner-Seitz regions"); Modelling alloy structures as sphere packings ("Domain of an atom") Finite Element Analysis -- Generating finite element meshes which avoid small angles Geography -- Analyzing patterns of urban settlements Geology -- Estimation of ore reserves in a deposit using information obtained from bore holes; modelling crack patterns in basalt due to contraction on cooling Geometric Modeling -- Finding "good" triangulations of 3D surfaces Marketing -- Model market of US metropolitan areas; market area extending down to individual retail stores Mathematics -- Study of positive definite quadratic forms ("Dirichlet tessellation", "Voronoi diagram") Metallurgy -- Modelling "grain growth" in metal films Meteorology -- Estimate regional rainfall averages, given data at discrete rain gauges ("Thiessen polygons") Pattern Recognition -- Find simple descriptors for shapes that extract 1D characterizations from 2D shapes ("Medial axis" or "skeleton" of a contour) Physiology -- Analysis of capillary distribution in cross-sections of muscle tissue to compute oxygen transport ("Capillary domains") Robotics -- Path planning in the presence of obstacles Statistics and Data Analysis -- Analyze statistical clustering ("Natural neighbors" interpolation) Zoology -- Model and analyze the territories of animals Uses for Voronoi Diagram CS691G Computational Geometry - UMass Amherst

23. Voronoi Graph • Voronoi region Vor(p) (p in set S) • the set of points on the plane that are closer to p than to any othe rpoint in S • Voronoi Graph VOR(S) • dual to voronoi region graph • two points are adjacent if their voronoi regions have common contiguous boundary (segment) CS691G Computational Geometry - UMass Amherst

24. Voronoi Graph • Voronoi Graph in the rectilinear plane • Rectilinear distance: p = (x, y); p’=(x’,y’) Voronoi region of b ab b a bc c ac CS691G Computational Geometry - UMass Amherst

25. THEOREM: For any set S of points in the plane, MST is subgraph of the Voronoi Graph VG(S) PROOF Let an edge XY between two points X and Y does not belong to the Voronoi graph VG(S). We will show that there is an X-Y- path in VG(S) which contains edges e1, e2,…,ek each shorter than XY, this will imply that XY not belong to MST. Indeed, for each edge eI (I=1,…,k) there is an MST path pI connecting ends of eI consisting of MST edges each no longer than eI. The path obtained by concatenating paths p1,…,pk connects X to Y and contains MST edges each shorter than XY. Thus XY does not belong to MST. Now we will find such X-Y-path e1, e2,…,ek in VG(S) CS691G Computational Geometry - UMass Amherst

26. Delaunay Triangulation (1934) Boris Nikolaevich Delone (1890 - 1980) Dual of Voronoi (graph theoretic, topological, combinatorial) CS691G Computational Geometry - UMass Amherst

27. Delaunay Triangulation Properties • maximizes minimum angle in each triangle • minimizes maximum radius of circumcircle and enclosing circle • minimizes sum of inscribed radii • many more… CS691G Computational Geometry - UMass Amherst

28. Finite Element Analysis CS691G Computational Geometry - UMass Amherst