Computational Geometry   Definition and Application Areas
Computational Geometry   is a subfield of the Design and Analysis of    Algorithms
Computational Geometry   is a subfield of the Design and Analysis of    Algorithms   deals with efficient data structure...
Computational Geometry   is a subfield of the Design and Analysis of    Algorithms   deals with efficient data structure...
Computational Geometry   is a subfield of the Design and Analysis of    Algorithms   deals with efficient data structure...
Application Areas   Computer Graphics   Computer-aided design / manufacturing   Telecommunication   Geology   Archite...
Surface Reconstruction   Digitizing 3-dimensional objects        Stanford Bunny
Surface Reconstruction   Step 1: Scan the object (3d laser scanner)        set of points in R3
Surface Reconstruction   Step 2: Create a triangulation     set of triangles in R3
Surface Reconstruction   Step 3: process the triangulation (rendering)     smooth surface in R3
Surface Reconstruction   Major Computational Geometry task:       Create a “good” triangulation
In this Course:Good and bad triangulations in R ≥ 2  bad triangulation  (long and skinny  triangles)
In this Course:Good and bad triangulations in R ≥ 2  good triangulation  (no small angles,  almost regular  triangles)
Collision detectionCheck whethertwo (possiblycomplicated) 3dobjects intersect!
Collision detection   Bounding volume heuristic:       Approximate the objects by simple ones that        enclose them (...
Collision detection   Bounding volume heuristic:       Approximate the objects by simple ones that        enclose them (...
Collision detection   Bounding volume heuristic:       Approximate the objects by simple ones that        enclose them (...
Collision detection   Bounding volume heuristic:       Approximate the objects by simple ones that        enclose them (...
In this Course:Smallest enclosing ball   Given: finite point set in R d
In this Course:Smallest enclosing ball   Given: finite point set in R d   Wanted: the smallest ball that contains all th...
In this Course:Smallest enclosing ball   Given: finite point set in R d   Wanted: the smallest ball that contains all th...
Boolean Operations   Given two (2d,3d) shapes, compute their...
Boolean Operations   ubiquituous in computer-aided design
In this Course: Arrangements of lines
In this Course: Arrangements of lines   Link to Boolean Operations:
In this Course: Arrangements of lines   Link to Boolean Operations:   Arrangement “contains” all                         ...
Boolean Operations   Important point: no roundoff errors!
Boolean Operations   Important point: no roundoff errors!
Boolean Operations   Important point: no roundoff errors!
Boolean Operations   Exact computation                        correct: intersection is 1-dimensional
Boolean Operations   Naive floating-point computation                         wrong: intersection is 2-dimensional
In this Course:CGAL   Computational Geometry Algorithms Library
In this Course:CGAL   Computational Geometry Algorithms Library   C++ library of efficient data structures and    algori...
In this Course:CGAL   Computational Geometry Algorithms Library   C++ library of efficient data structures and    algori...
In this Course:CGAL   Computational Geometry Algorithms Library   C++ library of efficient data structures and    algori...
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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

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