Siggraph Course

Mesh Parameterization: Theory and Practice

Siggraph Course

Mesh Parameterization: Theory and Practice

Differential Geometry PrimerDifferential Geometry Primer

ParameterizationParameterization

• surface

• parameter domain

• mapping and

Example – Cylindrical CoordinatesExample – Cylindrical Coordinates

•

•

•

•

•

•

•

Example – Orthographic ProjectionExample – Orthographic Projection

•

•

•

•

Example – Stereographic ProjectionExample – Stereographic Projection

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)

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

What is Distortion?What is Distortion?

• parameter point

• surface point

• small disk around

• image of under

• shape of

D

f (D)

LinearizationLinearization

• Jacobian of

• tangent plane at

• Taylor expansion of

• first order approximation of

Infinitesimal Dis(k)tortionInfinitesimal Dis(k)tortion

• small disk around

• image of under

• shape of

– ellipse

– semiaxes and

• behavior in the limit

Linear Map SurgeryLinear Map Surgery

• Singular Value Decomposition (SVD) of

with rotations andand scale factors (singular values)

Notion of DistortionNotion of Distortion

• isometric or length-preserving

• conformal or angle-preserving

• equiareal or area-preserving

• everything defined pointwise on

Example – Cylindrical CoordinatesExample – Cylindrical Coordinates

•

• ⇒ isometric

•

• with

• a ⇒ neither conformal

nor equiareal

Example – Orthographic ProjectionExample – Orthographic Projection

•

• with

• ⇒ conformal

Example – Stereographic ProjectionExample – Stereographic Projection

Computing the Stretch FactorsComputing the Stretch Factors

• first fundamental form

• eigenvalues of

• singular values of

and

Measuring DistortionMeasuring Distortion

• local distortion measure

• has minimum at

– isometric measure

– conformal measure

• overall distortion

Examples – Conformal MeasuresExamples – Conformal Measures

• Conformal energy

• MIPS energy

[Pinkall & Polthier 1993][Lévy et al. 2002]

[Desbrun et al. 2002]

[Hormann & Greiner 2000]

Examples – Isometric MeasuresExamples – Isometric Measures

• Green-Lagrange deformation tensor

• Combined energy

[Maillot et al. 1993]

[Degener et al. 2003]

Examples – Other MeasuresExamples – Other Measures

• Dirichlet energy

• Stretch energies ( , , and symmetric stretch)

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

[Pinkall & Polthier 1993] [Eck et al. 1995]

Piecewise Linear ParameterizationsPiecewise Linear Parameterizations

• piecewise linear atomic maps

• distortion constant per triangle

• overall distortion

Beyond DistortionBeyond Distortion

• surface normal

• surface area

• independent of the particular parameterization

• intrinsic surface properties

CurvatureCurvature

• second fundamental form

• Gaussian curvature

• mean curvature