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# Computational geometry

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### Computational geometry

1. 1. Computational Geometry Definition and Application Areas
2. 2. Computational Geometry is a subfield of the Design and Analysis of Algorithms
3. 3. Computational Geometry is a subfield of the Design and Analysis of Algorithms deals with efficient data structures and algorithms for geometric problems
4. 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. 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. 6. Application Areas Computer Graphics Computer-aided design / manufacturing Telecommunication Geology Architecture Geographic Information Systems VLSI design (chip layout) ...
7. 7. Surface Reconstruction Digitizing 3-dimensional objects Stanford Bunny
8. 8. Surface Reconstruction Step 1: Scan the object (3d laser scanner) set of points in R3
9. 9. Surface Reconstruction Step 2: Create a triangulation set of triangles in R3
10. 10. Surface Reconstruction Step 3: process the triangulation (rendering) smooth surface in R3
11. 11. Surface Reconstruction Major Computational Geometry task:  Create a “good” triangulation
12. 12. In this Course:Good and bad triangulations in R ≥ 2 bad triangulation (long and skinny triangles)
13. 13. In this Course:Good and bad triangulations in R ≥ 2 good triangulation (no small angles, almost regular triangles)
14. 14. Collision detectionCheck whethertwo (possiblycomplicated) 3dobjects intersect!
15. 15. Collision detection Bounding volume heuristic:  Approximate the objects by simple ones that enclose them (bounding volumes)
16. 16. Collision detection Bounding volume heuristic:  Approximate the objects by simple ones that enclose them (bounding volumes)  popular bounding volumes: boxes, spheres, ellipsoids,...
17. 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. 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. 19. In this Course:Smallest enclosing ball Given: finite point set in R d
20. 20. In this Course:Smallest enclosing ball Given: finite point set in R d Wanted: the smallest ball that contains all the points
21. 21. In this Course:Smallest enclosing ball Given: finite point set in R d Wanted: the smallest ball that contains all the pointspopular free software (also some commerciallicenses sold): http://www.inf.ethz.ch/personal/gaertner/miniball.html
22. 22. Boolean Operations Given two (2d,3d) shapes, compute their...
23. 23. Boolean Operations ubiquituous in computer-aided design
24. 24. In this Course: Arrangements of lines
25. 25. In this Course: Arrangements of lines Link to Boolean Operations:
26. 26. In this Course: Arrangements of lines Link to Boolean Operations: Arrangement “contains” all the unions / differences / intersections
27. 27. Boolean Operations Important point: no roundoff errors!
28. 28. Boolean Operations Important point: no roundoff errors!
29. 29. Boolean Operations Important point: no roundoff errors!
30. 30. Boolean Operations Exact computation correct: intersection is 1-dimensional
31. 31. Boolean Operations Naive floating-point computation wrong: intersection is 2-dimensional
32. 32. In this Course:CGAL Computational Geometry Algorithms Library
33. 33. In this Course:CGAL Computational Geometry Algorithms Library C++ library of efficient data structures and algorithms for many computational geometry problems
34. 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. 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