1. CS691G Computational Geometry – UMass Amherst – Ileana Streinu and Oliver Brock 1 CS691G Computational Geometry Ileana Streinu Oliver Brock Fall 2004

2. CS691G Computational Geometry – UMass Amherst – Ileana Streinu and Oliver Brock 2 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

3. CS691G Computational Geometry – UMass Amherst – Ileana Streinu and Oliver Brock 3 Discrete Geometry Packing Covering Tiling

4. CS691G Computational Geometry – UMass Amherst – Ileana Streinu and Oliver Brock 4 Computational Geometry Previously: design and analysis of geometric algorithms Overlapping and merging with discrete geometry Now: study of geometrical problems from a computational point of view Handbook of Discrete and Computational Geometry

5. CS691G Computational Geometry – UMass Amherst – Ileana Streinu and Oliver Brock 5 Goals Theoretical background –algorithms –data structures –analysis Practical experience –programming experience –CGAL –Cinderella

6. CS691G Computational Geometry – UMass Amherst – Ileana Streinu and Oliver Brock 6 Administrative Things Prerequisites: mathematical maturity, exposure to: algorithms, complexity, programming Grade: homeworks (33%), in-class presentation (33%), final project (33%) Late Policy: get permission prior to due date Web Site (from my home page)

7. CS691G Computational Geometry – UMass Amherst – Ileana Streinu and Oliver Brock 7 Connection to Applications Video Games Voronoi Diagrams Computer Graphics Folding

8. CS691G Computational Geometry – UMass Amherst – Ileana Streinu and Oliver Brock 8 Video Games

9. CS691G Computational Geometry – UMass Amherst – Ileana Streinu and Oliver Brock 9 What we saw… Walking through large model Collisions Dynamic simulation (Compare with automated movie generation)

10. CS691G Computational Geometry – UMass Amherst – Ileana Streinu and Oliver Brock 10 What to look for… Algorithms Complexity Data structures Geometric primitives

11. CS691G Computational Geometry – UMass Amherst – Ileana Streinu and Oliver Brock 11 Proximity Queries

12. CS691G Computational Geometry – UMass Amherst – Ileana Streinu and Oliver Brock 12 Dynamic Simulation

13. CS691G Computational Geometry – UMass Amherst – Ileana Streinu and Oliver Brock 13 Dynamic Simulation

14. CS691G Computational Geometry – UMass Amherst – Ileana Streinu and Oliver Brock 14 Multi-Player Games

15. CS691G Computational Geometry – UMass Amherst – Ileana Streinu and Oliver Brock 15 Multi-Player Games Some players might be computer generated (animations) Distributed state representation

16. CS691G Computational Geometry – UMass Amherst – Ileana Streinu and Oliver Brock 16 Motion Planning

17. CS691G Computational Geometry – UMass Amherst – Ileana Streinu and Oliver Brock 17 Kinetic Data Structures

18. CS691G Computational Geometry – UMass Amherst – Ileana Streinu and Oliver Brock 18 The Post Office Problem Which is the closest post office to every house? (Don Knuth) Given n sites in the plane Subdivision of plane based on proximity Georgy Voronoi 1868-1908

19. CS691G Computational Geometry – UMass Amherst – Ileana Streinu and Oliver Brock 19 Voronoi Diagram See Applet

20. CS691G Computational Geometry – UMass Amherst – Ileana Streinu and Oliver Brock 20 Shape Recognition in Computer Vision

21. CS691G Computational Geometry – UMass Amherst – Ileana Streinu and Oliver Brock 21 Uses for Voronoi Diagram 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 Crystallography and Chemistry -- 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 tesselation", "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

22. CS691G Computational Geometry – UMass Amherst – Ileana Streinu and Oliver Brock 22 Facts about Voronoi A site has an unbounded region if and only if it lies on the convex hull of all sites All Voronoi regions are convex Dual of Delaunay triangulation Questions: How fast can it be constructed? How many vertices does it have? What is the complexity of each cell?

23. CS691G Computational Geometry – UMass Amherst – Ileana Streinu and Oliver Brock 23 Applications Graphics: –Realistic Rendering –Radiosity Computation –Morphing Computational Biology: –Molecular Visualization –Protein Folding –Ligand Docking Computer Vision: –Reconstructing a 3d model from images

24. CS691G Computational Geometry – UMass Amherst – Ileana Streinu and Oliver Brock 24 Graphics: –Realistic Rendering –Radiosity Computation –Morphing Computational Biology: –Molecular Visualization –Protein Folding –Ligand Docking Computer Vision: –Reconstructing a 3d model from images Applications

25. CS691G Computational Geometry – UMass Amherst – Ileana Streinu and Oliver Brock 25 Left: picture Right: computer rendering

26. CS691G Computational Geometry – UMass Amherst – Ileana Streinu and Oliver Brock 26 Right: computer rendering -0.0035 0.034385 0.0719602 -0.003 0.0343985 0.0720802 -0.0025 0.0343985 0.0720802 -0.002 0.0344256 0.0723203 -0.0015 0.0344526 0.0725603 -0.01425 0.0345802 0.0675169 -0.01375 0.034688 0.0684772 -0.01325 0.0347284 0.0688374 -0.01275 0.0347554 0.0690774 -0.01225 0.0347958 0.0694375 -0.01175 0.0348362 0.0697976 -0.01125 0.0348767 0.0701578 -0.01075 0.0349036 0.0703978 -0.01025 0.0349171 0.0705179 -0.00975 0.034944 0.0707579 -0.00925 0.034971 0.070998 -0.00875 0.0349845 0.071118 -0.00825 0.0349979 0.0712381 -0.00775 0.0350114 0.0713581

27. CS691G Computational Geometry – UMass Amherst – Ileana Streinu and Oliver Brock 27

28. CS691G Computational Geometry – UMass Amherst – Ileana Streinu and Oliver Brock 28

29. CS691G Computational Geometry – UMass Amherst – Ileana Streinu and Oliver Brock 29 Graphics: –Realistic RenderingRealisticRendering –Radiosity Computation –Morphing Computational Biology: –Molecular Visualization –Protein Folding –Ligand Docking Computer Vision: –Reconstructing a 3d model from images Applications Graphics concepts: Light source Shadow, penumbra Occluder Culling Geometric keywords: Visibility edges/regions High-dimensional polytope

30. CS691G Computational Geometry – UMass Amherst – Ileana Streinu and Oliver Brock 30 Graphics: –Realistic rendering –Radiosity ComputationRadiosity Computation –Morphing Computational Biology: –Molecular Visualization –Protein Folding –Ligand Docking Computer Vision: –Reconstructing a 3d model from images Applications Graphics concepts: Scene Radiosity Form factor Geometric keywords: Visibility edges/regions Visibility complex: high- dimensional topological space Duality point-line

31. CS691G Computational Geometry – UMass Amherst – Ileana Streinu and Oliver Brock 31 Applications Graphics: –Realistic Rendering –Radiosity Computation –MorphingMorphing Computational Biology: –Molecular Visualization –Protein Folding –Ligand Docking Computer Vision: –Reconstructing a 3d model from images Video

32. CS691G Computational Geometry – UMass Amherst – Ileana Streinu and Oliver Brock 32

33. CS691G Computational Geometry – UMass Amherst – Ileana Streinu and Oliver Brock 33 Graphics: –Realistic Rendering –Radiosity Computation –Morphing Computational Biology: –Molecular VisualizationMolecular Visualization –Protein Folding –Ligand Docking Computer Vision: –Reconstructing a 3d model from images Applications Biology concepts: Atom, molecule, molecular surface Van der Waals radii Geometric keywords: Alpha-hull (convex hull) Topology of surface Dynamic changes

34. CS691G Computational Geometry – UMass Amherst – Ileana Streinu and Oliver Brock 34

35. CS691G Computational Geometry – UMass Amherst – Ileana Streinu and Oliver Brock 35 Graphics: –Realistic Rendering –Radiosity Computation –Morphing Computational Biology: –Molecular Visualization –Protein Structure Prediction and Protein Folding –Ligand Docking Computer Vision: –Reconstructing a 3d model from images Applications

36. CS691G Computational Geometry – UMass Amherst – Ileana Streinu and Oliver Brock 36 Proteins on computers Where we see structure, shape, connections, regions The computer sees only coordinates For example, this PXR protein & ligand is in the Protein Data Bank as…

37. CS691G Computational Geometry – UMass Amherst – Ileana Streinu and Oliver Brock 37 ATOM 2395 O HOH 1600 29.442 64.461 -1.726 1.00 66.79 8 ATOM 2396 O HOH 1601 19.427 85.921 -22.662 1.00 60.16 8 ATOM 2397 O HOH 1602 5.344 90.815 7.154 1.00 54.96 8 ATOM 2398 O HOH 1603 -14.216 50.571 5.561 1.00 54.96 8 ATOM 2399 O HOH 1604 5.533 45.964 0.404 1.00 62.55 8 ATOM 2400 O HOH 1605 -1.394 63.145 20.705 1.00 40.08 8 ATOM 2401 O HOH 1606 -2.578 54.566 22.874 1.00 57.40 8 ATOM 2402 O HOH 1607 3.600 69.196 22.807 1.00 54.51 8 ATOM 2403 O HOH 1608 6.139 65.007 -18.611 1.00 54.86 8 ATOM 2404 O HOH 1609 4.202 75.224 -27.568 1.00 58.04 8 ATOM 2405 O HOH 1610 -5.421 61.703 24.061 1.00 57.88 8 ATOM 2406 O HOH 1611 -11.943 45.372 11.041 1.00 62.72 8 END 2380 lines later… REMARK Written by O version 7.0.0 REMARK Sun Jan 21 15:24:51 2001 CRYST1 91.345 91.345 85.302 90.00 90.00 90.00 ORIGX1 1.000000 0.000000 0.000000 0.00000 ORIGX2 0.000000 1.000000 0.000000 0.00000 ORIGX3 0.000000 0.000000 1.000000 0.00000 SCALE1 0.010948 0.000000 0.000000 0.00000 SCALE2 0.000000 0.010948 0.000000 0.00000 SCALE3 0.000000 0.000000 0.011723 0.00000 ATOM 1 C GLY 142 -5.808 44.753 13.561 1.00 58.97 6 ATOM 2 O GLY 142 -5.723 45.523 14.515 1.00 59.54 8 ATOM 3 N GLY 142 -4.377 43.177 14.842 1.00 59.37 7 ATOM 4 CA GLY 142 -5.307 43.330 13.685 1.00 59.68 6 ATOM 5 N LEU 143 -6.324 45.108 12.387 1.00 58.87 7 ATOM 6 CA LEU 143 -6.839 46.455 12.152 1.00 58.50 6 ATOM 7 CB LEU 143 -6.483 46.907 10.736 1.00 57.90 6 ATOM 8 CG LEU 143 -5.849 48.290 10.555 1.00 57.77 6 ATOM 9 CD1 LEU 143 -4.599 48.411 11.407 1.00 56.51 6 ATOM 10 CD2 LEU 143 -5.505 48.492 9.090 1.00 56.92 6 ATOM 11 C LEU 143 -8.352 46.446 12.333 1.00 58.92 6 ATOM 12 O LEU 143 -9.046 45.640 11.714 1.00 59.85 8 ATOM 13 N THR 144 -8.862 47.341 13.174 1.00 58.88 7 ATOM 14 CA THR 144 -10.299 47.407 13.444 1.00 59.76 6

38. CS691G Computational Geometry – UMass Amherst – Ileana Streinu and Oliver Brock 38 Protein ATOM 1 C GLY 142 -5.808 44.753 13.561 1.00 58.97 6 ATOM 2 O GLY 142 -5.723 45.523 14.515 1.00 59.54 8 ATOM 3 N GLY 142 -4.377 43.177 14.842 1.00 59.37 7 ATOM 4 CA GLY 142 -5.307 43.330 13.685 1.00 59.68 6 ATOM 5 N LEU 143 -6.324 45.108 12.387 1.00 58.87 7 ATOM 6 CA LEU 143 -6.839 46.455 12.152 1.00 58.50 6 ATOM 7 CB LEU 143 -6.483 46.907 10.736 1.00 57.90 6 ATOM 8 CG LEU 143 -5.849 48.290 10.555 1.00 57.77 6 ATOM 9 CD1 LEU 143 -4.599 48.411 11.407 1.00 56.51 6 ATOM 10 CD2 LEU 143 -5.505 48.492 9.090 1.00 56.92 6 Sequence Structure

39. CS691G Computational Geometry – UMass Amherst – Ileana Streinu and Oliver Brock 39 Protein GLY -5.808 44.753 13.561 1.00 58.97 6 LEU -6.324 45.108 12.387 1.00 58.87 7 THR -6.839 46.455 12.152 1.00 58.50 8 Sequence Structure a sentence written over a 20-letter alphabet Sequence: Structure: GLY LEU THR LEU GLY ….. Geometry = coordinates for all the atoms

40. CS691G Computational Geometry – UMass Amherst – Ileana Streinu and Oliver Brock 40 Graphics: –Realistic Rendering –Radiosity Computation –Morphing Computational Biology: –Molecular Visualization –Protein Structure Prediction and Protein FoldingFolding –Ligand Docking Computer Vision: –Reconstructing a 3d model from images Applications

41. CS691G Computational Geometry – UMass Amherst – Ileana Streinu and Oliver Brock 41 Protein Folding Predict: Structure from Sequence From Vijay Pande’s Folding@Home page at Stanford

42. CS691G Computational Geometry – UMass Amherst – Ileana Streinu and Oliver Brock 42

43. CS691G Computational Geometry – UMass Amherst – Ileana Streinu and Oliver Brock 43 Graphics: –Realistic Rendering –Radiosity Computation –Morphing Computational Biology: –Molecular Visualization –Protein Structure Prediction and Protein FoldingFolding –Ligand Docking Computer Vision: –Reconstructing a 3d model from images Applications Polygon folding : Creases Boundary of polygon matched with itself Origami (paper) folding Linkage (robot arm – protein backbone) folding

44. CS691G Computational Geometry – UMass Amherst – Ileana Streinu and Oliver Brock 44 Graphics: –Realistic Rendering –Radiosity Computation –Morphing Computational Biology: –Molecular Visualization –Protein Structure Prediction and Protein Folding –Ligand Docking Computer Vision: –Reconstructing a 3d model from images Applications

45. CS691G Computational Geometry – UMass Amherst – Ileana Streinu and Oliver Brock 45 Forma Urbis Romae http://formaurbis.stanford.edu/

46. CS691G Computational Geometry – UMass Amherst – Ileana Streinu and Oliver Brock 46 Protein docking 3-dim puzzle

47. CS691G Computational Geometry – UMass Amherst – Ileana Streinu and Oliver Brock 47 Drug design Archaeology

48. CS691G Computational Geometry – UMass Amherst – Ileana Streinu and Oliver Brock 48 Computational Geometry Basic objects: points, lines, line segments, polygons, polygonal lines, embedded graphs Computed objects: convex hull, alpha hull, triangulation, arrangement, Voronoi diagram, Delauney triangulation. Variations: static, dynamic (discrete changes), kinetic (continuous motion) Wanted: good algorithms

49. CS691G Computational Geometry – UMass Amherst – Ileana Streinu and Oliver Brock 49 More video clips SoCG’04: http://give-lab.cs.uu.nl/socg04video/ SoCG’03 http://theory.lcs.mit.edu/~edemaine/SoCG 2003_multimedia/webproceedings/ http://theory.lcs.mit.edu/~edemaine/SoCG 2003_multimedia/webproceedings/