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