Title: Mesh%20Parameterization:%20Theory%20and%20Practice
1Mesh ParameterizationTheory and Practice
- Differential Geometry Primer
2Parameterization
- surface
- parameter domain
- mapping and
3Example Cylindrical Coordinates
4Example Orthographic Projection
5Example Stereographic Projection
6Example Mappings of the Earth
- usually, surface properties get distorted
orthographic 500 B.C.
stereographic 150 B.C.
Mercator 1569
Lambert 1772
conformal (angle-preserving)
equiareal (area-preserving)
7Distortion is (almost) Inevitable
- Theorema Egregium (C. F. Gauß)
- A general surface cannot be parameterized
without distortion. - no distortion conformal equiareal isometric
- requires surface to be developable
- planes
- cones
- cylinders
8What is Distortion?
- parameter point
- surface point
- small disk around
-
- image of under
-
- shape of
9Linearization
- Jacobian of
-
- tangent plane at
-
- Taylor expansion of
-
- first order approximation of
-
10Infinitesimal Dis(k)tortion
- small disk around
- image of under
-
- shape of
- ellipse
- semiaxes and
- behavior in the limit
-
11Linear Map Surgery
- Singular Value Decomposition (SVD) of
-
- with rotations and
- and scale factors (singular values)
12Notion of Distortion
- isometric or length-preserving
- conformal or angle-preserving
- equiareal or area-preserving
- everything defined pointwise on
13Example Cylindrical Coordinates
14Example Orthographic Projection
neither conformal nor equiareal
15Example Stereographic Projection
16Computing the Stretch Factors
- first fundamental form
- eigenvalues of
- singular values of
- and
17Measuring Distortion
- local distortion measure
- has minimum at
- isometric measure
- conformal measure
- overall distortion
18Piecewise Linear Parameterizations
- piecewise linear atomic maps
- distortion constant per triangle
- overall distortion
19Linear Methods
- the terms and
are quadratic in the parameter points - Dirichlet energy
- Conformal energy
- minimization yields linear problem
Pinkall Polthier 1993Eck et al. 1995
Lévy et al. 2002Desbrun et al. 2002
20Linear Methods
- both result in barycentric mappings with
discrete harmonic weights for interior vertices - Dirichlet maps require to fix all boundary
vertices - Conformal maps only two
- result depends on this choice
- best choice ? Mullen et al. 2008
- both maps not necessarily bijective
21Non-linear Methods
- MIPS energy
-
- Area-preserving MIPS
-
Hormann Greiner 2000
Degener et al. 2003
22Non-linear Methods
- Green-Lagrange deformation tensor
-
- Stretch energies ( , , and symmetric
stretch)
Maillot et al. 1993
Sander et al. 2001 Sorkine et al. 2002