3 3 D Surface Parameterization D Surface Parameterization Olga Sorkine, May 2005 Olga Sorkine, May 2005 Part One Part One Parameterization and Partition Parameterization and Partition Some slides borrowed from Pierre Alliez and Craig Gotsman What is a parameterization? What is a parameterization? S S ⊆ ⊆ R R 3 3 - - given surface given surface D D ⊆ ⊆ R R 2 2 - - parameter domain parameter domain s s : D : D → → S 1 S 1 - - 1 and onto 1 and onto ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ = ) , ( ) , ( ) , ( ) , ( v u z v u y v u x v u s Example Example – – flattening the earth flattening the earth
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3D Surface Parameterization Part One Parameterization and
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Parameterization and PartitionParameterization and Partition
Some slides borrowed from Pierre Alliez and Craig Gotsman
What is a parameterization?What is a parameterization?
S S ⊆⊆ RR3 3 -- given surfacegiven surface
D D ⊆⊆ RR22 -- parameter domainparameter domain
ss : D : D →→ S 1S 1--1 and onto 1 and onto
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛=
),(),(),(
),(vuzvuyvux
vus
Example Example –– flattening the earthflattening the earth
Isoparametric curves Isoparametric curves on the surfaceon the surface
One parameter fixed, one varies:One parameter fixed, one varies:
Family 1 (varying u): Family 1 (varying u): LLv0 v0 ((uu)) = = ss((uu, v, v00))
Family 2 (varying v): Family 2 (varying v): MMu0 u0 ((vv)) = = ss((vv00, , vv))
Analytic example:Analytic example:
Parameters: Parameters: u = u = x, v = yx, v = y
D = D = [[––1,11,1]]××[[––1,1]1,1]. .
z = zz = z((x,yx,y)) = = ––((xx22+y+y22))
ss((x,yx,y)) = = ((x, y, zx, y, z((x,yx,y))))
-1
1
π α
h
Another example:Another example:
Parameters: Parameters: αα, h, h
D = D = [0,[0,ππ]]××[[––1,1]1,1]
xx((αα, h, h)) = = coscos((αα))
yy((αα, h, h)) = h= h
zz((αα, h, h)) = sin= sin((αα))
Triangular MeshTriangular Mesh
•• Standard Standard discretediscrete 3D surface representation 3D surface representation in Computer Graphics in Computer Graphics –– piecewise linearpiecewise linear
•• Mesh GeometryMesh Geometry: list of vertices (3D points of : list of vertices (3D points of the surface)the surface)
•• Mesh Connectivity or TopologyMesh Connectivity or Topology: description : description of the facesof the faces
Dealing with distortion and Dealing with distortion and nonnon--disk topologydisk topology
ProblemsProblems: : 1) Parameterization of complex surfaces 1) Parameterization of complex surfaces
introduces introduces distortion. distortion. 2) Only topological disk can be embedded.2) Only topological disk can be embedded.
SolutionSolution: : partitionpartition the mesh into several the mesh into several patches and/or introduce patches and/or introduce seams (cuts)seams (cuts), , parameterize each patch independently.parameterize each patch independently.
Introducing seams (cuts)Introducing seams (cuts) ProblemsProblems•• Discontinuity of parameterizationDiscontinuity of parameterization•• Visible artifacts in texture mappingVisible artifacts in texture mapping•• Require special treatmentRequire special treatment
–– Vertices along seams have several (u,v) Vertices along seams have several (u,v) coordinatescoordinates
–– Problems in Problems in mipmip--mappingmapping
Make seams short and hide them
SummarySummary
•• ““GoodGood”” parameterization = nonparameterization = non--distortingdistorting–– Angles and area preservationAngles and area preservation–– Continuous parameterization of complex surfaces Continuous parameterization of complex surfaces
cannot avoid distortion.cannot avoid distortion.
•• ““GoodGood”” partition/cut:partition/cut:–– Large patches, minimize seam lengthLarge patches, minimize seam length–– Align seams with features (=hide them)Align seams with features (=hide them)