How to Make Good Meshes

1
How to Make Good Meshes

• John C. Hart
• CS 319
• Advanced Topics in Computer Graphics

2
Meshing

• Properties
• Manifold
• Delaunay
• Representations
• Winged-Edge
• Quad-Edge
• Half-Edge
• Double-Linked-Face-List

Guibas Stolfi. Primitives for the Manipulation
of General Subdivisionsand the Computation of
Voronoi Diagrams. TOG 4(2), Apr. 1985, pp.
74-123.
3
Manifolds

• Let M be a surface in 3-D
• M is a manifold iff a neighborhood of any point x
  in M is topologically equivalent to the unit disk
• Topologically equivalence allows deformation that
  does not rip, tear or poke holes
• Also called homeomorphism
• deformation is 1-1, onto and bicontinuous
• Need not be differentiable

good
bad
4
Piecewise Linear Manifolds

• Closed mesh
• No dangling faces
• Watertight
• no holes in the surface
• no boundary
• orientable
• Edges only bound two faces
• Eulers formula
• V E F 2(C G)
• C of components
• G genus ( of donut holes)

good
bad
5
Delaunay Triangulation

• Any number of ways to connect scattered points in
  plane with edges
• Creates a good looking mesh
• Triangle face ABC is in Delaunay triangulation
  iff circumcircle of every face contains no other
  vertex
• Also Delauney iff every edge has some circle
  passing through its endpoints that contains no
  other vertex
• Rhymes with baloney

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Building Delaunay Triangulations

• Let L and R be two sets of vertices.
• Let TL, TR, and T be the Delaunay triangulations
  of L, R and L ? R.
• Any edge in T whose endpoints are both in L is in
  TL.
• No new edges added
• TL not a subset of T
• Any edge in T that is not also in TL nor TR has
  one endpoint in L and one endpoint in R.
• New edges between L and R
• TL, TR not subsets of T

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CircumcircleTest

• The point P is in the circumcircle of triangle
  ABC iff D(A,B,C,P) gt 0
• Assume A,B,C,P cocircular with center x,y and
  radius r
• Then (xA x)2 (yA y)2 r2 and likewise for
  B,C and P
• Which is equivalent to2xxA2yyA(xA2yA2)(x2y2
  r2) 0and likewise for B,C and P
• Equations are linearly dependent

z x2 y2
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Indexed Face Set
(x3,y3,z3)
(x2,y2,z2)

• Popular file format
• VRML, Wavefront, etc.
• Ordered list of vertices
• Prefaced by v (Wavefront)
• Spatial coordinates x,y,z
• Index given by order
• List of polygons
• Prefaced by f (Wavefront)
• Ordered list of vertex indices
• Length of sides
• Orientation given by order

(x1,y1,z1)
(x0,y0,z0)
v x0 y0 z0 v x1 y1 z1 v x2 y2 z2 v x3 y3 z3 f 0
1 2 f 1 3 2
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Other Attributes
v x0 y0 z0 v x1 y1 z1 v x2 y2 z2 vn a0 b0 c0 vn
a1 b1 c1 vn a2 b2 c2 vt u0 v0 vt u1 v1 vt u2 v2
(x2,y2,z2)(a2,b2,c2) (u2,v2)

• Vertex normals
• Prefixed w/ vn (Wavefront)
• Contains x,y,z of normal
• Not necessarily unit length
• Not necessarily in vertex order
• Indexed as with vertices
• Texture coordinates
• Prefixed with vt (Wavefront)
• Not necessarily in vertex order
• Contains u,v surface parameters
• Faces
• Uses / to separate indices
• Vertex / normal / texture
• Normal and texture optional
• Can eliminate normal with //

(x0,y0,z0) (a0,b0,c0) (u0,v0)
(x1,y1,z1)(a1,b1,c1) (u1,v1)
f 0/0/0 1/1/1 2/2/2
f 0/0/0 1/0/1 2/0/2
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Winged-Edge

• Edge pointers
• to two endpoint vertices
• to two faces that share edge
• to four edges emanating from its endpoints
• Faces, vertices contain pointer to one edge
• Faces outlines by walking edges
• Mesh information really contained in edge
  datastructure

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Quad-Edge Operators
e Dest
e Lnext

• Edge e orientation and direction
• Orientation is surface normal
• Direction is from origin vertex (e Org) to
  destination vertex (e Dest)
• Sym operator toggles direction
• e Org e Sym Dest
• Flip operator toggles orientation
• e Left e Flip Right
• e Onext next edge (ccw) with same origin
• e Left e Onext Right
• e Left e Lnext Left

e Right
e Left
e
e Onext
e Org
Sym
e Sym
e
Flip
e Sym Flip
e Flip
12
Dual Meshes

• Dual of a mesh exchanges faces and vertices
• Place new vertices at centroid of faces
• Edges in dual cross original edges
• at right angles
• at midpoints
• but might not actually cross
• E.g. Dual of a Delaunay triangulation is a
  Voronoi diagram
• Voronoi faces denote regions closer to
  corresponding Delaunay vertex than any other
  Delaunay vertex

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Dual Edge Operator
e
e Dual

• Dual operator rotates edge 90 degrees clockwise
• Definition
• e Dual Dual e
• e Sym Dual e Dual Sym
• e Flip Dual e Dual Flip Sym
• e Lnext Dual e Dual Onext-1
• Dual mesh is inside out and flips its edges
• Rot operation avoids flipping
• e Rot e Flip Dual
• e Rot Rot e Sym

e Dual
e
e
e Rot
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Fun w/ Quad Edge
e Lnext
e Dest

• e Rot4 e
• e Rot Onext Rot Onext e
• e Rot2 ? e (but e Sym)
• e Flip2 e
• e Flip Onext Flip Onext e
• e Flip Onextn ? e for any n
• e Flip Rot Flip Rot e
• e Flip-1 e Flip
• e Rot-1 e Rot3 e Flip Rot Flip
• e Dual e Flip Rot
• e Onext-1 e Rot Onext Rot e Flip Onext Flip

e Right
e Left
e
e Onext
e Org
e Dual
e
e
e Rot
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Implementation

• All we need is
• Rot
• Flip
• Onext
• Edge reference lte,r,fgt
• e Rotr Flipf
• r 0..3
• f 0..1 (above/below bit)
• Quarter record er contains
• Data (geometry)
• Next ? e Rotr Onext

lte,r,fgt Rot lte, (r12f)4, fgt lte,r,fgt Flip
lte, r, (f1)2gt lte,r,fgt Onext lte(rf)4.Nextgt
Rotf Flipf
lte,r,0gt Onext lter.Nextgt lte,r,1gt Onext
lter1.Nextgt Rot Flip ( lte,r,0gt Rot Onext Rot
Flip) ( lte,r,0gt Onext-1 Flip)
If meshes always orientable lte,rgt Rot
lte,r1gt lte,rgt Onext er.Next lte,rgt Sym
lte,r2gt