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Mesh Parameterization:Theory and Practice
Mesh Parameterization:Theory and Practice
Differential Geometry PrimerDifferential Geometry Primer
Mesh Parameterization: Theory and PracticeDifferential Geometry Primer
• surface• parameter domain• mapping and
ParameterizationParameterization
Mesh Parameterization: Theory and PracticeDifferential Geometry Primer
Example – Cylindrical CoordinatesExample – Cylindrical Coordinates
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•
•
Mesh Parameterization: Theory and PracticeDifferential Geometry Primer
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•
•
•
Example – Orthographic ProjectionExample – Orthographic Projection
Mesh Parameterization: Theory and PracticeDifferential Geometry Primer
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•
Example – Stereographic ProjectionExample – Stereographic Projection
Mesh Parameterization: Theory and PracticeDifferential Geometry Primer
Example – Mappings of the EarthExample – Mappings of the Earth
• usually, surface properties get distorted
orthographic ∼ 500 B.C.
stereographic ∼ 150 B.C.
Mercator1569
Lambert1772
conformal(angle-preserving)
equiareal(area-preserving)
Mesh Parameterization: Theory and PracticeDifferential Geometry Primer
Distortion is (almost) InevitableDistortion 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
Mesh Parameterization: Theory and PracticeDifferential Geometry Primer
What is Distortion?What is Distortion?
• parameter point• surface point• small disk around
• image of under
• shape of
Mesh Parameterization: Theory and PracticeDifferential Geometry Primer
LinearizationLinearization
• Jacobian of
• tangent plane at
• Taylor expansion of
• first order approximation of
Mesh Parameterization: Theory and PracticeDifferential Geometry Primer
Infinitesimal Dis(k)tortionInfinitesimal Dis(k)tortion
• small disk around• image of under
• shape of– ellipse – semiaxes and
• behavior in the limit
Mesh Parameterization: Theory and PracticeDifferential Geometry Primer
Linear Map SurgeryLinear Map Surgery
• Singular Value Decomposition (SVD) of
with rotations andand scale factors (singular values)
Mesh Parameterization: Theory and PracticeDifferential Geometry Primer
Notion of DistortionNotion of Distortion
• isometric or length-preserving
• conformal or angle-preserving
• equiareal or area-preserving
• everything defined pointwise on
Mesh Parameterization: Theory and PracticeDifferential Geometry Primer
Example – Cylindrical CoordinatesExample – Cylindrical Coordinates
•
• ⇒isometric
Mesh Parameterization: Theory and PracticeDifferential Geometry Primer
•
• with
• ⇒
Example – Orthographic ProjectionExample – Orthographic Projection
neither conformalnor equiareal
Mesh Parameterization: Theory and PracticeDifferential Geometry Primer
•
• with
• ⇒ conformal
Example – Stereographic ProjectionExample – Stereographic Projection
Mesh Parameterization: Theory and PracticeDifferential Geometry Primer
Computing the Stretch FactorsComputing the Stretch Factors
• first fundamental form
• eigenvalues of
• singular values of and
Mesh Parameterization: Theory and PracticeDifferential Geometry Primer
Measuring DistortionMeasuring Distortion
• local distortion measure
• has minimum at– isometric measure– conformal measure
• overall distortion
Mesh Parameterization: Theory and PracticeDifferential Geometry Primer
Piecewise Linear ParameterizationsPiecewise Linear Parameterizations
• piecewise linear atomic maps • distortion constant per triangle
• overall distortion
Mesh Parameterization: Theory and PracticeDifferential Geometry Primer
Linear MethodsLinear Methods
• the terms and are quadratic in the parameter points
• Dirichlet energy
• Conformal energy
• minimization yields linear problem
[Pinkall & Polthier 1993][Eck et al. 1995]
[Lévy et al. 2002][Desbrun et al. 2002]
Mesh Parameterization: Theory and PracticeDifferential Geometry Primer
Linear MethodsLinear 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
Mesh Parameterization: Theory and PracticeDifferential Geometry Primer
Non-linear MethodsNon-linear Methods
• MIPS energy
• Area-preserving MIPS [Degener et al. 2003]
[Hormann & Greiner 2000]