1. 242-535 ADA: 14. Intro to CG1 Objective o give a non-technical overview of Computational geometry, concentrating on its main application areas Algorithm Design and Analysis (ADA) 242-535, Semester 1 2014-2015 14. Introduction to Computational Geometry

2. 242-535 ADA: 14. Intro to CG2 1.What is Computational Geometry? 2.Uses in Computer Graphics 3.Uses in Robotics 4.Uses in GIS 5.Uses in CAD/CAM 6.A TextbookOverview

3. 1. What is Computational Geometry? The systematic study of algorithms and data structures for geometric objects, with a focus on exact algorithms that are asymptotically fast.

4. 242-535 ADA: 14. Intro to CG4 CG in Context Theoretical Computer Science Applied Computer Science Applied Math Geometry Computational Geometry Efficient Geometric Algorithms Design Analyze Apply

5. 2. Uses in Computer Graphics  Intersect geometric primitives (lines, polygons, polyhedra, etc.)  Determine primitives lying in a region.  Hidden surface removal – determine the visible part of a 3D scene while discard the occluded part from a view point.  Deal with moving objects and detect collisions.

6. 242-535 ADA: 14. Intro to CG6 Is point q inside simple polygon P? Point in Polygon Testing P n-gon q Naïve: O(n) per test CG: O(log n)

7. 242-535 ADA: 14. Intro to CG7 Given n line segments in the plane, determine: o Does some pair intersect? (DETECT) o Compute all points of intersection (REPORT) Segment Intersection Naïve: O(n 2 ) CG: O(n log n) detect, O(k+n log n) report

8. 242-535 ADA: 14. Intro to CG8 Find “smallest” (tightest fitting) pair of bounding boxes Motivation: o Best outer approximation o Bounding volume hierarchies The 2-Box Cover Problem

9. Triangulation of Polygons

10. 242-535 ADA: 14. Intro to CG10 Collision Detection

11. 3. Uses in Robotics  Motion planning  Grasping  Parts orienting  Optimal placement

12. Proximity Closest coffee shop in PSU? Voronoi diagram Delaunay triangulation

13. 242-535 ADA: 14. Intro to CG13 A Voronoi diagram is a way of dividing space into smaller regions. A set of points (called seeds, sites, or "coffee shops") is specified beforehand and for each seed there will be a corresponding region consisting of all points closer to that seed than to any other. The regions are called Voronoi cells. Closely related to Delaunay triangulation A Voronoi Diagram

14. Voronoi Diagrams in Nature Dragonfly wing Honeycomb Constrained soap bubbles Giraffe pigmentation

15. 242-535 ADA: 14. Intro to CG15 A Delaunay triangulation for a set points results in a series of triangles connecting those points. A circle drawn through the three points in a triangle will contain no other points. Delaunay Triangulation Delaunay triangulation

16. Path Planning Robot How can a robot find a short route to the destination that avoids all obstacles?

17. 242-535 ADA: 14. Intro to CG17 Mobile Robotic Guard Watchman Route Problem

18. Determine the smallest number of cameras needed to see all of a given area. 5 cameras are enough to see everywhere (what about 4 cameras? 3?) How Many Cameras? viewable area for this camera

19. 4. Uses in GIS Storage of geographical data (contours of countries, height of mountains, course of rivers, population, roads, electricity lines, etc.)  Large amount of data – requiring efficient algorithms.  Geographic data storage (e.g., map of roads for car positioning or computer display).  Interpolation between nearby sample data points  Overlay of multiple maps.

20. 5. Uses in CAD/CAM  Intersection, union, and decomposition of objects.  Testing on product specifications.  Design for assembly – modeling and simulation of assembly.  Testing design for feasibility.

21. 242-535 ADA: 14. Intro to CG21 Bounding Volume Hierarchy BV-tree: Level 0

22. 242-535 ADA: 14. Intro to CG22 BV-tree: Level 1

23. 242-535 ADA: 14. Intro to CG23 BV-tree: Level 2

24. 242-535 ADA: 14. Intro to CG24 BV-tree: Level 5

25. 242-535 ADA: 14. Intro to CG25 BV-tree: Level 8

26. 242-535 ADA: 14. Intro to CG26 Computational Geometry in C o Joseph O’Rourke, Cambridge University Press, 2nd ed.,1998 5. A Textbook http://cs.smith.edu/~orourke/ books/compgeom.html