1. Lection 1: Introduction Computational Geometry Prof.Dr.Th.Ottmann 1 History: Proof-based, algorithmic, axiomatic geometry, computational geometry today Problem fields An example: Computing the convex hull: 1.the “naive approach” 2.Graham‘s Scan 3.Lower bound Design, analysis, and implementation of geometrical algorithms Lecture 1: Introduction

2. Lection 1: Introduction Computational Geometry Prof.Dr.Th.Ottmann 2 Problem fields Typical questions Geometrical objects: points, lines, surfaces Techniques Applications

3. Lection 1: Introduction Computational Geometry Prof.Dr.Th.Ottmann 3 Finding the nearest fast-food restaurant

4. Lection 1: Introduction Computational Geometry Prof.Dr.Th.Ottmann 4 Partitioning the plane into areas of equal nearest neighbors

5. Lection 1: Introduction Computational Geometry Prof.Dr.Th.Ottmann 5 Art gallery problem How many stationary guards are needed to guard the room?

6. Lection 1: Introduction Computational Geometry Prof.Dr.Th.Ottmann 6 Watchmen routes Compute the optimal watchman route for a mobile guard

7. Lection 1: Introduction Computational Geometry Prof.Dr.Th.Ottmann 7 Visibility problems Hidden-line-elimination Visible surface computation

8. Lection 1: Introduction Computational Geometry Prof.Dr.Th.Ottmann 8 Intersection problems Given a set of line segments, rectangles, polygons,...: Compute all pairs of intersecting Objects.

9. Lection 1: Introduction Computational Geometry Prof.Dr.Th.Ottmann 9 Geometric objects: Points, lines, …

10. Lection 1: Introduction Computational Geometry Prof.Dr.Th.Ottmann 10 Different algorithms for points Minimum spanning tree

11. Lection 1: Introduction Computational Geometry Prof.Dr.Th.Ottmann 11 Different algorithms for points Delauney triangulation

12. Lection 1: Introduction Computational Geometry Prof.Dr.Th.Ottmann 12 Different algorithms for points Convex hull

13. Lection 1: Introduction Computational Geometry Prof.Dr.Th.Ottmann 13 Voronoi Region

14. Lection 1: Introduction Computational Geometry Prof.Dr.Th.Ottmann 14 Voronoi Diagram

15. Lection 1: Introduction Computational Geometry Prof.Dr.Th.Ottmann 15 Geometric search Closest pair Is it possible to close the gap between  (n log n) and O(n²)? Asymptotic bounds are relevant!

16. Lection 1: Introduction Computational Geometry Prof.Dr.Th.Ottmann 16 Difference between n, n log n and n² 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.

17. Lection 1: Introduction Computational Geometry Prof.Dr.Th.Ottmann 17 Application domains Computer graphics: 2- and 3-dimensional Robotics, CAD, CAM VLSI design Database systems, GIS Molecular modelling,....

18. Lection 1: Introduction Computational Geometry Prof.Dr.Th.Ottmann 18 Geographical information systems UNI-Offspring sofion Documentation, analysis, and maintenance of gas, water and sewage pipes and telecommunications lines

19. Lection 1: Introduction Computational Geometry Prof.Dr.Th.Ottmann 19 Robotics Laserscan robot Localisation and path-finding in unknown environments. Example of an On-line scenario of geometrical algorithms