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 R3

9. Surface Reconstruction • Step 2: Create a triangulation set of triangles in R3

10. Surface Reconstruction • Step 3: process the triangulation (rendering) smooth surface in R3

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. In this Course: Arrangements of lines • 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