1. Introduction to Computational Geometry Computational Geometry, WS 2007/08 Lecture 1 – Part II Prof. Dr. Thomas Ottmann Algorithmen & Datenstrukturen, Institut für Informatik Fakultät für Angewandte Wissenschaften Albert-Ludwigs-Universität Freiburg

2. Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann2 Overview Historicity –Proof-based geometry –Algorithmic geometry –Axiomatic geometry Computational geometry today Problems and applications Geometrical objects –Points –Lines –Surfaces Analyses and techniques

3. Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann3 Geometric Objects

4. Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann4 Areas of Nearest Neighbours Where is the nearest Starbucks?

5. Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann5 Art Gallery Problem How many stationary guards are needed?

6. Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann6 Watchman Route What is the optimal route for a mobile guard to take?

7. Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann7 Visibility Problems Which surfaces should be visible on screen? Scene Spatial-objects Screen-surface Deeper in the scene Closer to the screen-surface

8. Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann8 Intersection Problems Where do lines/rectangles/polygons… intersect? Given a set of line segments, Rectangles, polygons, etc. Compute all pairs of intersecting objects

9. Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann9 Algorithms Involving Points (2D) Minimum spanning tree (MST) Delaunay triangulation Convex hull

10. Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann10 Voronoi Region

11. Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann11 Voronoi Diagram

12. Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann12 Geometric Search Which is the closest pair of coordinates? Is it possible to close the gap between  (n log n) and O(n²)?  Asymptotic bounds are relevant!

13. Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann13 Runtime Efficiency Comparisons n n log n n² 2 10  10³ 10 2 10  10 4 2 20  10 6 2 20  10 6 20 2 20  2 10 7 2 40  10 12 Interactive Processing n log n algorithms n² algorithms n = 1000 yes ? n = 1000000 ? no Computational geometry has developed new types of algorithms which may solve basic geometric problems efficiently.

14. Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann14 Application Domains Computer graphics: 2- and 3-dimensional Robotics, CAD, CAM VLSI design Database systems, GIS Molecular modelling,....

15. Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann15 Geographical Information Systems UNI-Offspring Arc-*-Software Documentation, analysis, and maintenance of gas, water and sewage pipes and telecommunications lines

16. Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann16 Robotics Laserscan robot Localisation and path-finding in unknown environments. Example of an On-line scenario of geometrical algorithms