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

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• ⇒isometric

Mesh Parameterization: Theory and PracticeDifferential Geometry Primer

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• with

• ⇒

Example – Orthographic ProjectionExample – Orthographic Projection

neither conformalnor equiareal

Mesh Parameterization: Theory and PracticeDifferential Geometry Primer

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• 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]

Mesh Parameterization: Theory and PracticeDifferential Geometry Primer

Non-linear MethodsNon-linear Methods

• Green-Lagrange deformation tensor

• Stretch energies ( , , and symmetric stretch)

[Sander et al. 2001][Sorkine et al. 2002]

[Maillot et al. 1993]