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TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAAAAAAA
CS 468, Spring 2013 Differential Geometry for Computer Science
Justin Solomon and Adrian Butscher
http://alice.loria.fr/index.php/publications.html?redirect=0&Paper=VSDM@2011&Author=levy
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Not suitable for implementation http://upload.wikimedia.org/wikipedia/commons/b/bc/Double_torus_illustration.png
µ R3µ R2
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What is a discrete surface? How do you store it?
http://graphics.stanford.edu/data/3Dscanrep/stanford-bunny-cebal-ssh.jpg http://www.stat.washington.edu/wxs/images/BUNMID.gif
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1. Each edge is incident to one or two faces
2. Faces incident to a vertex form a closed or open fan
http://www.cs.mtu.edu/~shene/COURSES/cs3621/SLIDES/Mesh.pdf
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1. Each edge is incident to one or two faces
2. Faces incident to a vertex form a closed or open fan
http://www.cs.mtu.edu/~shene/COURSES/cs3621/SLIDES/Mesh.pdf
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http://www.pmp-book.org/download/slides/Representations.pdf
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f(t) f(t+ h)
f
P
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f(t) f(t+ h)
f
P
f(t¤)
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O(h2)
f(t) f(t+ h)
f
P
f(t¤)
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Piecewise linear faces are reasonable building blocks.
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Simple to render
Arbitrary topology possible
Basis for subdivision, refinement
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Topology [tuh-pol-uh-jee]:
The study of geometric properties that remain invariant under certain transformations
http://dictionary.reference.com/browse/topology?s=t
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http://upload.wikimedia.org/wikipedia/commons/f/fb/Dolphin_triangle_mesh.png
Geometry: “This vertex is at (x,y,z).”
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http://upload.wikimedia.org/wikipedia/commons/f/fb/Dolphin_triangle_mesh.png
Topology: “These vertices are connected.”
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V = fv1; v2; : : : ; vng ½ Rn
E = fe1; e2; : : : ; ekg ½ V £ V
F = ff1; f2; : : : ; fmg ½ V £ V £ V
Easy to generalize to non-triangles
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Valence = 6
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V ¡E+F = ÂÂ= 2¡ 2g
g = 0 g = 1 g = 2
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V ¡E+F = ÂÂ= 2¡ 2g
g = 0 g = 1 g = 2
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Closed mesh: Easy estimates!
V ¡E+F = Â
“Each edge is adjacent to two faces. Each face has three edges.”
2E = 3F
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Closed mesh: Easy estimates!
V ¡ 1
2F = Â
“Each edge is adjacent to two faces. Each face has three edges.”
2E = 3F
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Closed mesh: Easy estimates!
F ¼ 2V
V ¡ 1
2F = Â
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General estimates
E ¼ 3V
F ¼ 2V
average valence ¼ 6
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http://www.cse.ohio-state.edu/~tamaldey/isotopic.html
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Normal field isn’t continuous
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Normal field isn’t continuous
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Must represent geometry and topology of surface.
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Triangle soup CS 468 2011 (M. Ben-Chen), other slides
x1 y1 z1 / x2 y2 z2 / x3 y3 z3
x1 y1 z1 / x2 y2 z2 / x3 y3 z3
x1 y1 z1 / x2 y2 z2 / x3 y3 z3
x1 y1 z1 / x2 y2 z2 / x3 y3 z3
x1 y1 z1 / x2 y2 z2 / x3 y3 z3
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Triangle soup CS 468 2011 (M. Ben-Chen), other slides
x1 y1 z1 / x2 y2 z2 / x3 y3 z3
x1 y1 z1 / x2 y2 z2 / x3 y3 z3
x1 y1 z1 / x2 y2 z2 / x3 y3 z3
x1 y1 z1 / x2 y2 z2 / x3 y3 z3
x1 y1 z1 / x2 y2 z2 / x3 y3 z3
glBegin(GL_TRIANGLES)
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Shared vertex structure CS 468 2011 (M. Ben-Chen), other slides
f 1 5 3
f 5 1 2
…
v 0.2 1.5 3.2
v 5.2 4.1 8.9
… v1
v5
v3 v2f1 f2
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for i=1 to n
for each vertex v
v = .5*v +
.5*(average of neighbors);
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Neighboring vertices to a vertex
Neighboring faces to an edge
Edges adjacent to a face
Edges adjacent to a vertex
…
Mostly localized
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Neighboring vertices to a vertex
Neighboring faces to an edge
Edges adjacent to a face
Edges adjacent to a vertex
…
Mostly localized
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Vertices Faces Half-edges
Structure tuned for meshes
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Oriented edge
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Vertex stores: • Arbitrary
outgoing halfedge
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Face stores: • Arbitrary
adjacent halfedge
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Halfedge stores: • Flip • Next • Face • Vertex
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https://sympa.inria.fr/sympa/arc/cgal-discuss/2008-05/msg00054.html
Iterate(v):
startEdge = v.out;
e = startEdge;
do
process(e.flip.from)
e = e.flip.next
while e != startEdge
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Vertex Dimension 0
Edge Dimension 1
Face Dimension 2
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Vertex Dimension 0
Edge Dimension 1
Face Dimension 2
@ @
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Vertex Dimension 0
Edge Dimension 1
Face Dimension 2
@ @
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Vertex Dimension 0
Edge Dimension 1
Face Dimension 2
@ @
@2 ´ 0
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http://www.ieeta.pt/polymeco/Screenshots/PolyMeCo_OneView.jpg
f : ! R
Map points to real numbers
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http://www.ieeta.pt/polymeco/Screenshots/PolyMeCo_OneView.jpg
Map vertices to real numbers
f 2 RjV j
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What is the integral of f? Z
M
f dA
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Use hat functions to interpolate http://brickisland.net/cs177/wp-content/uploads/2011/11/ddg_hat_function.svg
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Discrete version of 𝒅𝑨
i
vi
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Discrete version of 𝒅𝑨
i
vi
Z
i
f dA = fijij
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http://www.grasshopper3d.com/group/kangaroo/forum/topics/isosurface-dynamic-remeshing
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???
?
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e!Rot!Rot = e!Flip
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http://johnrichie.com/V2/richie/simp/images/types1.jpg
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http://johnrichie.com/V2/richie/simp/images/types1.jpg
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Complex data structures enable simpler traversal at cost of more bookkeeping.
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Implicit surfaces http://www.cs.umd.edu/class/spring2005/cmsc828v/papers/mpu_implicits.pdf ftp://ftp-sop.inria.fr/geometrica/alliez/signing.pdf
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Smoothed-particle hydrodynamics
http://www.itsartmag.com/features/cgfluids/ https://developer.nvidia.com/content/fluid-simulation-alice-madness-returns
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Cleanest: Design software http://www.cad-sourcing.com/wp-content/uploads/2011/12/free-cad-software.jpg
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Cleanest: Design software
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Volumetric extraction http://www.colin-studholme.net/software/rview/rvmanual/morphtool5.gif
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Volumetric extraction http://en.wikipedia.org/wiki/Marching_cubes
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Point clouds http://www.engineeringspecifier.com/public/primages/pr1200.jpg
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Well-behaved dual mesh http://en.wikipedia.org/wiki/Delaunay_triangulation
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Tangent plane Derive local triangulation from tangent projection
Restricted Delaunay Usual Delaunay strategy but in smaller part of 𝑅3
Inside/outside labeling Find inside/outside labels for tetrahedra
Empty balls Require existence of sphere around triangle with no other point
Delaunay Triangulation Based Surface Reconstruction: Ideas and Algorithms Cazals and Giesen 2004
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http://faculty.cs.tamu.edu/schaefer/teaching/689_Fall2006/poissonrecon.pdf
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TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAAAAAAA
CS 468, Spring 2013 Differential Geometry for Computer Science
Justin Solomon and Adrian Butscher
http://alice.loria.fr/index.php/publications.html?redirect=0&Paper=VSDM@2011&Author=levy