1. Computational Geometry Definition and Application Areas

2. Computational Geometry is a subfield of the Design and Analysis of Algorithms

3. Computational Geometry is a subfield of the Design and Analysis of Algorithms deals with efficient data structures and algorithms for geometric problems

4. Computational Geometry is a subfield of the Design and Analysis of Algorithms deals with efficient data structures and algorithms for geometric problems is only about 30 years old

5. Computational Geometry is a subfield of the Design and Analysis of Algorithms deals with efficient data structures and algorithms for geometric problems is only about 30 years old started out by developing solid theoretical foundations, but became more and more applied over the last years

6. Application Areas Computer Graphics Computer-aided design / manufacturing Telecommunication Geology Architecture Geographic Information Systems VLSI design (chip layout)...

7. Surface Reconstruction Digitizing 3-dimensional objects Stanford Bunny

8. Surface Reconstruction Step 1: Scan the object (3d laser scanner) set of points in R 3

9. Surface Reconstruction Step 2: Create a triangulation set of triangles in R 3

10. Surface Reconstruction Step 3: process the triangulation (rendering) smooth surface in R 3

11. Surface Reconstruction Major Computational Geometry task:  Create a “good” triangulation

12. In this Course: Good and bad triangulations in R ≥ 2 bad triangulation (long and skinny triangles)

13. In this Course: Good and bad triangulations in R ≥ 2 good triangulation (no small angles, almost regular triangles)

14. Collision detection Check whether two (possibly complicated) 3d objects intersect!

15. Collision detection Bounding volume heuristic:  Approximate the objects by simple ones that enclose them (bounding volumes)

16. Collision detection Bounding volume heuristic:  Approximate the objects by simple ones that enclose them (bounding volumes)  popular bounding volumes: boxes, spheres, ellipsoids,...

17. Collision detection Bounding volume heuristic:  Approximate the objects by simple ones that enclose them (bounding volumes)  popular bounding volumes: boxes, spheres, ellipsoids,...  if bounding volumes don’t intersect, the objects don’t intersect, either

18. Collision detection Bounding volume heuristic:  Approximate the objects by simple ones that enclose them (bounding volumes)  popular bounding volumes: boxes, spheres, ellipsoids,...  if bounding volumes don’t intersect, the objects don’t intersect, either  only if bounding volumes intersect, apply more expensive intersection test(s)

19. In this Course: Smallest enclosing ball Given: finite point set in R d

20. In this Course: Smallest enclosing ball Given: finite point set in R d Wanted: the smallest ball that contains all the points

21. In this Course: Smallest enclosing ball Given: finite point set in R d Wanted: the smallest ball that contains all the points popular free software (also some commercial licenses sold): http://www.inf.ethz.ch/personal/gaertner/miniball.html

22. Boolean Operations Given two (2d,3d) shapes, compute their...

23. Boolean Operations ubiquituous in computer-aided design

24. In this Course: Arrangements of lines

25. Link to Boolean Operations:

26. In this Course: Arrangements of lines Link to Boolean Operations: Arrangement “contains” all the unions / differences / intersections

27. Boolean Operations Important point: no roundoff errors!

28. Boolean Operations Important point: no roundoff errors!

29. Boolean Operations Important point: no roundoff errors!

30. Boolean Operations Exact computation correct: intersection is 1-dimensional

31. Boolean Operations Naive floating-point computation wrong: intersection is 2-dimensional

32. In this Course: CGAL Computational Geometry Algorithms Library

33. In this Course: CGAL Computational Geometry Algorithms Library C++ library of efficient data structures and algorithms for many computational geometry problems

34. In this Course: CGAL Computational Geometry Algorithms Library C++ library of efficient data structures and algorithms for many computational geometry problems exists since 1996 (Michael Hoffmann and Bernd Gärtner are members of the editorial board)

35. In this Course: CGAL Computational Geometry Algorithms Library C++ library of efficient data structures and algorithms for many computational geometry problems exists since 1996 (Michael Hoffmann and Bernd Gärtner are members of the editorial board) supports roundoff-free exact computations