Your SlideShare is downloading. ×
0
Computational geometry
Computational geometry
Computational geometry
Computational geometry
Computational geometry
Computational geometry
Computational geometry
Computational geometry
Computational geometry
Computational geometry
Computational geometry
Computational geometry
Computational geometry
Computational geometry
Computational geometry
Computational geometry
Computational geometry
Computational geometry
Computational geometry
Computational geometry
Computational geometry
Computational geometry
Computational geometry
Computational geometry
Computational geometry
Computational geometry
Computational geometry
Computational geometry
Computational geometry
Computational geometry
Computational geometry
Computational geometry
Computational geometry
Computational geometry
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

123

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
123
On Slideshare
0
From Embeds
0
Number of Embeds
0
Actions
Shares
0
Downloads
11
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

×