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Siggraph Course
Mesh Parameterization: Theory and Practice
Siggraph Course
Mesh Parameterization: Theory and Practice
Differential Geometry PrimerDifferential 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)
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
•
• with
• a ⇒ neither conformal
nor equiareal
Example – Orthographic ProjectionExample – Orthographic 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