1 / 29

An Introduction to Computational Geometry

An Introduction to Computational Geometry. Joseph S. B. Mitchell Stony Brook University. Mini-Course Info. Joe Mitchell (Math 1-109) jsbm@ams.sunysb.edu (631-632-8366) http://www.ams.sunysb.edu/~jsbm/nasa/tutorial.html Recommended Texts:

luce
Download Presentation

An Introduction to Computational Geometry

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. An Introduction to Computational Geometry Joseph S. B. Mitchell Stony Brook University

  2. Mini-Course Info • Joe Mitchell (Math 1-109) jsbm@ams.sunysb.edu (631-632-8366) • http://www.ams.sunysb.edu/~jsbm/nasa/tutorial.html • Recommended Texts: • Computational Geometry, by Mark de Berg, Marc van Kreveld, Mark Overmars and OtfriedSchwartzkopf (2nd edition) • Computational Geometry in C, by Joe O’Rourke (2nd Edition) • Surveys/Reference books: • CRC Handbook of Discrete and Computational Geometry, editted by Goodman and O’Rourke (2nd edition) • Handbook of Computational Geometry, editted by Sack and Urrutia

  3. Sessions and Topics • Session 1: • Introduction • Convex Hulls in 2D, 3D • A variety of algorithms • Session 2: • Searching for intersections: Plane sweep • Halfspace intersections, low-dimensional linear programming (LP) • Range Search

  4. Sessions and Topics • Session 3: • Triangulation • Proximity: Voronoi/Delaunay diagrams • Point location search • Session 4: • Arrangements • Duality and applications • Visibility graphs, shortest paths

  5. Sessions and Topics • Session 5: • Spatial partitioning, optimization, load balancing • Binary Space Partitions • Clustering trajectories • Session 6: • GeoSect: Overview and algorithms • 3D partitioning • Flow-respecting partitioning

  6. Arrangements Zone(L) L Given a set S of n lines in the plane Arrangement, A(S), is the S-induced decomposition of the plane into cells (2-faces), edges (1-faces), and vertices (0-faces)

  7. Constructing A(S) • Goal: Build a DCEL of the planar graph of vertices/edges of A(S) • Complexity: O(n2 ) faces, edges, vertices • Incremental construction: • Zone Theorem: The zone, Z(L), of line L has total complexity O(n) • Proof: Induction • Add lines one by one; cost of adding line Liis O(i), so total time is O(n2 )

  8. More Arrangements • Huge subfield: Combinatorics/algorithms • Arrangements of • Segments (Bentley-Ottmann sweep) • Circles • Curves • Higher dimensions: Surfaces • Applications: Motion planning, optimization, Voronoi diagrams, pattern matching, fundamental to many algorithms • Further motivation: Point-line duality

  9. Duality Key Property: p is above/on/below L iff L* is above/on/below p* r* P* r q p L* L q* Point p=(px,py) and line L: y=mx+bin the “primal”  Line p*: y=pxx-pyand point L*=(m,-b)

  10. Duality A really nice demo (MIT) Applet Another nice (simple) demo (Andrea) Applet

  11. Applications of Duality Degeneracy testing: Any 3 collinear points? Stabbing line segments Min-area triangle Halfplane range search Ham sandwich cuts Computing discrepancy Sorting points around each other point: visibility graphs

  12. Geometric Shortest Paths • Given: • (Outer) Polygon P • Polygonal obstacles • Points s and t • Find: • shortest s-t path • avoiding obstacles • Robotics motion planning

  13. Application: Weather Avoidance

  14. Basic Optimal Paths Visibility Graph Local optimality: taut string Naïve algorithm: O(n3) Better algorithms: O(n2 log n) sweep O(n2 ) duality BEST: Output-sensitive alg: O(|E|+n log n)

  15. Visibility Graphs Visibility Graph applet Shows also the topological sweep of an arrangement, animated.

  16. Example: Shortest Path in Simple Polygon Triangulate P Determine the “sleeve” defined by the triangles in a path in the dual tree, from start to goal WUSTL applet

  17. Shortest Path Map Decomposition of the free space Shortest Paths go through same vertices Construct in time: O(n log n); size O(n)

  18. Optimal Path Map: Control Law Fix s Vary t

  19. Hazardous Weather Avoidance

  20. Hazardous Weather Avoidance

  21. Optimal Paths: Weighted Regions Different costs/nmi in different regions (e.g. weather intensity) Local optimality: Snell’s Law of Refraction Can also apply grid-based searching

  22. Grid-Based Searches

  23. Other Optimal Path Problems • 3D shortest paths: NP-hard • But good approximations • Turn constraints: • Min-# turns: Link distance • Bounds on turn angles (radius of curvature) • “Thick” paths (for air lanes)

  24. Turn-Constrained Weighted Costs: Weather Route Generator (WRG) fix Departure space DFW

  25. Weather Avoidance Algorithms for En Route Aircraft Sector 3 Flows

  26. Routing MaxFlow Find maximum number of air lanes through the mincut

  27. Example: Robot Motion Planning Shows Minkowski sum, visibility graph, shortest path for a disk, triangle, etc. applet

  28. Example: Robot Motion Planning Voronoi method of robot motion planning applet

  29. Probabilistic Road Maps Movie (Stanford)

More Related