Your SlideShare is downloading. ×
Computational geometry
Upcoming SlideShare
Loading in...5
×

Thanks for flagging this SlideShare!

Oops! An error has occurred.

×
Saving this for later? Get the SlideShare app to save on your phone or tablet. Read anywhere, anytime – even offline.
Text the download link to your phone
Standard text messaging rates apply

Computational geometry

113

Published on

Published in: Design
0 Comments
0 Likes
Statistics
Notes
  • Be the first to comment

  • Be the first to like this

No Downloads
Views
Total Views
113
On Slideshare
0
From Embeds
0
Number of Embeds
0
Actions
Shares
0
Downloads
10
Comments
0
Likes
0
Embeds 0
No embeds

Report content
Flagged as inappropriate Flag as inappropriate
Flag as inappropriate

Select your reason for flagging this presentation as inappropriate.

Cancel
No notes for slide

Transcript

  • 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 detectionCheck whethertwo (possiblycomplicated) 3dobjects 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 pointspopular free software (also some commerciallicenses 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

×