Those are the slides for the book: Visual Computing: Geometry, Graphics, and Vision. by Frank Nielsen (2005) http://www.sonycsl.co.jp/person/nielsen/visualcomputing/ http://www.amazon.com/Visual-Computing-Geometry-Graphics-Charles/dp/1584504277

1. © Frank Nielsen 2011
INF555
Fundamentals of 3D
Lecture 8: Introduction to meshes
Les maillages
Frank Nielsen
nielsen@lix.polytechnique.fr
23 Novembre 2011

2. © Frank Nielsen 2011
Polygons

3. © Frank Nielsen 2011
Polygons: Star-shaped decomposition

4. © Frank Nielsen 2011
Polygons: Star-shaped decomposition
Art gallery, illumination problems, robots'race, etc.
Place guards...

5. © Frank Nielsen 2011
3D : Orienting the fact edges for outer normals
Polyhedron, convex polyhedra

6. © Frank Nielsen 2011
Platonic solids: 5 convex polyhedra
Identical faces
(group of symmetry)

7. © Frank Nielsen 2011
Uniform polyhedra
rhombicosidodecahedronrhombicuboctahedr
on
●faces=regular polygons (not necessarily the same),
●isometry mapping of its vertices (=symmetry)

8. © Frank Nielsen 2011
75 uniform finite polyhedra/ if you like origami
Science Museum, London

9. © Frank Nielsen 2011
Meshes: Notations ● vertices
● edges
● faces

10. © Frank Nielsen 2011
Meshes: Connectivity (= structuring)

11. © Frank Nielsen 2011
Meshes: Non-orientable surfaces
Möbius band Klein bottle

12. © Frank Nielsen 2011
Textured meshes: Realistic computer graphics

13. © Frank Nielsen 2011
Osculating circles and curvature
Curvature is the inverse of the radius of the osculating circle.

14. © Frank Nielsen 2011
Mesh: Sectional curvatures and principal directions
Minimum curvature Maximum curvature
Directions are perpendicular to each other
sectional curvatures.
Intersection of a surface S with a plane containing point p and its normal:
2D curve that can be analyzed using the osculating circles.

15. © Frank Nielsen 2011
Gaussian curvature:Gaussian curvature:
Mean curvature:
Mesh: Gaussian and mean curvatures
→ In Riemannian geometry, many ways to define curvatures (and torsions)

16. © Frank Nielsen 2011
Mesh: Integral Gaussian curvature/angle excess
(Deviation from flatness)

17. © Frank Nielsen 2011
Mesh: Ingredients of topology
Euler's formula is a topological invariant:
Closed triangulated manifold:

18. © Frank Nielsen 2011
Mesh topology: Genus, polyhedra with holes
(topology=global property)
Genus 0
Genus1
Genus 3

19. © Frank Nielsen 2011
Mesh: Primal/Dual graph representations

20. © Frank Nielsen 2011
Mesh: Primal/Dual graph representations
Primal Voronoi/Dual Delaunay triangulation

21. © Frank Nielsen 2011
Simplicial complexes: Building blocks (LEGO-type)

22. © Frank Nielsen 2011
Sketching meshes: Pen computing
Meshes for the masses.
→ Difficult to design
http://www.sonycsl.co.jp/person/nielsen/PT/vteddy/vteddy-desc.html

23. © Frank Nielsen 2011
Triangulation meshes:
Always possible in 2D (but difficult)
NOT Always possible in 3D!!! (require additional Steiner points)
Schonhardt’s polyhedron Chazelle’s polyhedron
Untetrahedralizable Objects

24. © Frank Nielsen 2011
Meshes: Procedural modeling/ city(buildings)

25. © Frank Nielsen 2011
L-system (Lindenmayer) process
Fibonacci's sequences
Grammar, language, parsing, etc.

26. © Frank Nielsen 2011
L-system (Lindenmayer) process / LOGO

27. © Frank Nielsen 2011
Object Oriented Graphics Library (OOGL) / OFF format
Geomview.org viewer
Data-structures for meshes: Indexed face list
(This is the PPM equivalent for 3D objects)

28. © Frank Nielsen 2011
Data-structures for meshes: Indexed face list
→ If we change vertex coordinates, all faces updated simultaneously !

29. © Frank Nielsen 2011
Optimizing bandwidth: Triangle/quad strips
Compress mesh vertices
→ Compress mesh connectivity
Bunny with 150 strips

30. © Frank Nielsen 2011
Many data-structures for meshes....
Winged edges
Half edges
Quad edges
Winged edges
Half edges
Quad edges

31. © Frank Nielsen 2011
(Discrete) Laplacian smoothing on meshes

32. © Frank Nielsen 2011
Surface subdivision: Refining
→ Limit surface is smooth

33. © Frank Nielsen 2011
Surface subdivision: Dr Loop scheme
New vertex creation Vertex relocation
Simplified to

34. © Frank Nielsen 2011
Limit position, smoothness properties

35. © Frank Nielsen 2011
Kobbelt's subdivision (+edge flipping)

36. © Frank Nielsen 2011
Mesh subdivision: Combinatorial explosion !

37. © Frank Nielsen 2011
Remeshing
Decimating mesh
Mesh simplification

38. © Frank Nielsen 2011
Progressive mesh representations
Level of details/streaming...

39. © Frank Nielsen 2011
Parameterization and texture mapping
Image texture is 2D

40. © Frank Nielsen 2011
Parameterization and texture mapping
Conformal mapping
(preserve angles)
Minimize distortion
Atlas

41. © Frank Nielsen 2011
H- and V- representations of polytopes
Half-spaces representation
Vertex representation (convex hull)
http://www.math.tu-berlin.de/polymake/index.html