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© Frank Nielsen 2011
INF555
Fundamentals of 3DLecture 8: Introduction to meshes
Les maillages
Frank [email protected]
23 Novembre 2011
© Frank Nielsen 2011
Polygons
© Frank Nielsen 2011
Polygons: Star-shaped decomposition
© Frank Nielsen 2011
Polygons: Star-shaped decomposition
Art gallery, illumination problems, robots'race, etc.Place guards...
© Frank Nielsen 2011
3D : Orienting the fact edges for outer normals
Polyhedron, convex polyhedra
© Frank Nielsen 2011
Platonic solids: 5 convex polyhedra
Identical faces(group of symmetry)
© Frank Nielsen 2011
Uniform polyhedra
rhombicosidodecahedronrhombicuboctahedron
●faces=regular polygons (not necessarily the same), ●isometry mapping of its vertices (=symmetry)
© Frank Nielsen 2011
75 uniform finite polyhedra/ if you like origami
Science Museum, London
© Frank Nielsen 2011
Meshes: Notations ● vertices● edges● faces
© Frank Nielsen 2011
Meshes: Connectivity (= structuring)
© Frank Nielsen 2011
Meshes: Non-orientable surfaces
Möbius band Klein bottle
© Frank Nielsen 2011
Textured meshes: Realistic computer graphics
© Frank Nielsen 2011
Osculating circles and curvature
Curvature is the inverse of the radius of the osculating circle.
© 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.
© Frank Nielsen 2011
Gaussian curvature:Gaussian curvature:
Mean curvature:
Mesh: Gaussian and mean curvatures
→ In Riemannian geometry, many ways to define curvatures (and torsions)
© Frank Nielsen 2011
Mesh: Integral Gaussian curvature/angle excess(Deviation from flatness)
© Frank Nielsen 2011
Mesh: Ingredients of topology
Euler's formula is a topological invariant:
Closed triangulated manifold:
© Frank Nielsen 2011
Mesh topology: Genus, polyhedra with holes(topology=global property)
Genus 0
Genus1Genus 3
© Frank Nielsen 2011
Mesh: Primal/Dual graph representations
© Frank Nielsen 2011
Mesh: Primal/Dual graph representations
Primal Voronoi/Dual Delaunay triangulation
© Frank Nielsen 2011
Simplicial complexes: Building blocks (LEGO-type)
© Frank Nielsen 2011
Sketching meshes: Pen computingMeshes for the masses.→ Difficult to design
http://www.sonycsl.co.jp/person/nielsen/PT/vteddy/vteddy-desc.html
© 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
© Frank Nielsen 2011
Meshes: Procedural modeling/ city(buildings)
© Frank Nielsen 2011
L-system (Lindenmayer) process
Fibonacci's sequences
Grammar, language, parsing, etc.
© Frank Nielsen 2011
L-system (Lindenmayer) process / LOGO
© 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)
© Frank Nielsen 2011
Data-structures for meshes: Indexed face list
→ If we change vertex coordinates, all faces updated simultaneously !
© Frank Nielsen 2011
Optimizing bandwidth: Triangle/quad strips
Compress mesh vertices→ Compress mesh connectivity
Bunny with 150 strips
© Frank Nielsen 2011
Many data-structures for meshes....
Winged edgesHalf edges Quad edges
Winged edgesHalf edges Quad edges
© Frank Nielsen 2011
(Discrete) Laplacian smoothing on meshes
© Frank Nielsen 2011
Surface subdivision: Refining→ Limit surface is smooth
© Frank Nielsen 2011
Surface subdivision: Dr Loop scheme
New vertex creation Vertex relocation
Simplified to
© Frank Nielsen 2011
Limit position, smoothness properties
© Frank Nielsen 2011
Kobbelt's subdivision (+edge flipping)
© Frank Nielsen 2011
Mesh subdivision: Combinatorial explosion !
© Frank Nielsen 2011
RemeshingDecimating meshMesh simplification
© Frank Nielsen 2011
Progressive mesh representations
Level of details/streaming...
© Frank Nielsen 2011
Parameterization and texture mapping
Image texture is 2D
© Frank Nielsen 2011
Parameterization and texture mapping
Conformal mapping(preserve angles)
Minimize distortion
Atlas
© 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