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AMS 345/CSE 355 Computational Geometry

Learn the algorithmic study of efficient geometric problem-solving methods. Topics include point-in-polygon tests, segment intersections, Voronoi diagrams, triangulation, sensor placements, guarding problems, shortest paths, convex hulls, collision detection, and more.

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AMS 345/CSE 355 Computational Geometry

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  1. AMS 345/CSE 355 Computational Geometry Lecture 1: Introduction Joe Mitchell

  2. Course Info • Joe Mitchell (Math 1-109) jsbm@ams.sunysb.edu (2-8366) • http://www.ams.sunysb.edu/~jsbm/courses/345/ams345.html • Texts: • Discrete and Computational Geometry by Devadoss and O’Rourke • Computational Geometry in C, by Joe O’Rourke (2nd Edition) • Grades: 40% midterm+40% final+20% hw Midterm: in class, tentatively Oct 17 Final: 5:30-6:45pm, Mon, Dec 11 Second-chance (optional): 6:45-8:00pm, Dec 11 Optional second-chance midterm: Replaces midterm score with 0.8xhigh + 0.2xlow

  3. Homeworks • Approx 8 assignments • Tentatively due: 9/19, 9/26, 10/3, 10/10, 10/31, 11/7, 11/14, 11/21 • Submit HW on time • I drop the lowest hw score • Each student is allowed ONE slightly late HW, as long as it is submitted before solutions posted(inform me by email) • HW will be called in, near the beginning of class; you will pass it forward; do not bring it to the front of the class (unless you arrive late and then bring it at the END of class, to minimize disruption)

  4. HW Policy You may discuss problems with other students in this class, but you must write up your hw completely on your own; if you do work with other student(s), you must declare this on the cover of your own hw, giving names of your collaborators. (Your writings must be independent: Do not look at another writeup, either of classmate or of anything you find on internet when writing your own solution. To do otherwise is a case of Academic Dishonesty and is subject to University policy through CASA

  5. Teaching Assistants • Phil Ammirato (philammirato@gmail.com) • Office hours (in Harriman 010): Tues, 4:00-5:00; Thurs, 10:00-11:00 • Sixin Li (sixin210@hotmail.com) • Office hours (in Harriman 010): Mon, 1:30-3:30

  6. Background • Geometry: vectors, dot/cross product • AMS301: Counting, graphs, recursion • Permutations, combinations • Planar graphs, DFS, Euler • F(n) ≤ 2F(n/2) + CnSolve: F(n)=O(n log n) • Algorithms: read/parse a C program, big-Oh notation (O(n), O(n log n),O(n2), O(n log n))

  7. What is CG? • The algorithmic and mathematical study of efficient methods to solve geometric problems

  8. Example 1 • Point-in-polygon test: Is point q inside simple polygon P? Naïve: O(n) per test CG: O(log n) q P n-gon Applet: O’Rourke

  9. Example 2 • Segment intersection: Given n line segments in the plane, determine: • Does some pair intersect? (DETECT) • Compute all points of intersection (REPORT) Naïve: O(n2) Applet: CG: O(n log n) detect, O(k+n log n) report

  10. Example 3 • Post office problem: Voronoi diagrams • http://www.cs.cornell.edu/Info/People/chew/Delaunay.html

  11. Example 4 • Triangulation of polygons http://www.cosy.sbg.ac.at/~held/projects/triang/triang.html

  12. Ear-Clipping Triangulation Ear-clipping applet

  13. Example 5 • Sensor placement, “guarding”

  14. The Problem Determine a small set of guards to see all of a given polygon P 5 guards suffice to cover P (what about 4 guards? 3?)

  15. Vertex Guarding a Simple Polygon • Vertex guarding applet 11 yellow vertices 11 blue vertices 16 white vertices Place guards at yellow (or blue) vertices: at most n/3 vertex guards (here, n=38)

  16. Mobile Robotic Guard Subject to: stay inside polygonal domain P Visit all visibility polygons Watchman Route Problem

  17. Example 6 • Shortest path for a mobile robot

  18. Application: Weather Avoidance

  19. 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)

  20. Visibility Graphs Shows also the topological sweep of an arrangement, animated.

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

  22. Convex Hull of Points • Graham scan applet • Jarvis march applet

  23. QuickCD: Collision Detection

  24. The 2-Box Cover Problem • Find “smallest” (tightest fitting) pair of bounding boxes • Motivation: • Best outer approximation • Bounding volume hierarchies

  25. Bounding Volume Hierarchy BV-tree: Level 0 k-dops

  26. BV-tree: Level 1 14-dops 6-dops 18-dops 26-dops

  27. BV-tree: Level 2

  28. BV-tree: Level 5

  29. BV-tree: Level 8

  30. Large Convex Inner Approx • “Grow” k-dops from selected seed points: collision detection (QuickCD), response

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

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

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