CS691G Computational Geometry

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CS691G Computational Geometry

• Ileana Streinu Oliver Brock
• Fall 2004

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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
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Discrete Geometry
Covering
Tiling
Packing
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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
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Goals

• Theoretical background
• algorithms
• data structures
• analysis
• Practical experience
• programming experience
• CGAL
• Cinderella

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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)

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Connection to Applications

• Video Games
• Voronoi Diagrams
• Computer Graphics
• Folding

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Video Games
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What we saw

• Walking through large model
• Collisions
• Dynamic simulation
• (Compare with automated movie generation)

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What to look for

• Algorithms
• Complexity
• Data structures
• Geometric primitives

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Proximity Queries
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Dynamic Simulation
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Dynamic Simulation
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Multi-Player Games
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Multi-Player Games

• Some players might be computer generated
  (animations)
• Distributed state representation

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Motion Planning
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Kinetic Data Structures
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The Post Office Problem

• Which is the closest post office to every house?
  (Don Knuth)
• Given n sites in the plane
• Subdivision of planebased on proximity

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Voronoi Diagram
See Applet
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Shape Recognition in Computer Vision
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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

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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 triangulationQuestions
• How fast can it be constructed?
• How many vertices does it have?
• What is the complexity of each cell?

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Applications

• Graphics
• Realistic Rendering
• Radiosity Computation
• Morphing
• Computational Biology
• Molecular Visualization
• Protein Folding
• Ligand Docking
• Computer Vision
• Reconstructing a 3d model from images

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Applications

• Graphics
• Realistic Rendering
• Radiosity Computation
• Morphing
• Computational Biology
• Molecular Visualization
• Protein Folding
• Ligand Docking
• Computer Vision
• Reconstructing a 3d model from images

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Left picture Right computer rendering
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-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
Right computer rendering
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Applications

• Graphics
• Realistic Rendering
• Radiosity Computation
• Morphing
• Computational Biology
• Molecular Visualization
• Protein Folding
• Ligand Docking
• Computer Vision
• Reconstructing a 3d model from images

• Graphics concepts
• Light source
• Shadow, penumbra
• Occluder
• Culling
• Geometric keywords
• Visibility edges/regions
• High-dimensional polytope

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Applications

• Graphics
• Realistic rendering
• Radiosity Computation
• Morphing
• Computational Biology
• Molecular Visualization
• Protein Folding
• Ligand Docking
• Computer Vision
• Reconstructing a 3d model from images

• Graphics concepts
• Scene
• Radiosity
• Form factor
• Geometric keywords
• Visibility edges/regions
• Visibility complex high-dimensional topological
  space
• Duality point-line

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Applications

• Graphics
• Realistic Rendering
• Radiosity Computation
• Morphing
• Computational Biology
• Molecular Visualization
• Protein Folding
• Ligand Docking
• Computer Vision
• Reconstructing a 3d model from images

Video
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Applications

• Graphics
• Realistic Rendering
• Radiosity Computation
• Morphing
• Computational Biology
• Molecular Visualization
• Protein Folding
• Ligand Docking
• Computer Vision
• Reconstructing a 3d model from images

• Biology concepts
• Atom, molecule, molecular surface
• Van der Waals radii
• Geometric keywords
• Alpha-hull (convex hull)
• Topology of surface
• Dynamic changes

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Applications

• 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

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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

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REMARK Written by O version 7.0.0 REMARK Sun Jan
21 152451 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
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Protein
Structure
Sequence
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
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Protein
Structure
Sequence
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
a sentence written over a 20-letter alphabet
GLY LEU THR LEU GLY ..
Structure
Geometry coordinates for all the atoms
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Applications

• 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

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Protein Folding
Predict Structure from Sequence
From Vijay Pandes Folding_at_Home page at Stanford
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Applications

• 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

• Polygon folding
• Creases
• Boundary of polygon matched with itself
• Origami (paper) folding
• Linkage (robot arm protein backbone) folding

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Applications

• 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

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Forma Urbis Romae http//formaurbis.stanford.edu/
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3-dim puzzle
Protein docking
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Archaeology
Drug design
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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

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More video clips

• SoCG04
• http//give-lab.cs.uu.nl/socg04video/
• SoCG03 http//theory.lcs.mit.edu/edemaine/SoCG20
  03_multimedia/webproceedings/