Siggraph Course Mesh Parameterization: Theory and Practice

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Siggraph CourseMesh Parameterization Theory
and Practice

• Differential Geometry Primer

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Parameterization

• surface
• parameter domain
• mapping and

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Example Cylindrical Coordinates

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Example Orthographic Projection

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Example Stereographic Projection

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Example 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)
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Distortion 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

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What is Distortion?

• parameter point
• surface point
• small disk around
• image of under
• shape of

D
f (D)
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Linearization

• Jacobian of
• tangent plane at
• Taylor expansion of
• first order approximation of

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Infinitesimal Dis(k)tortion

• small disk around
• image of under
• shape of
• ellipse
• semiaxes and
• behavior in the limit

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Linear Map Surgery

• Singular Value Decomposition (SVD) of
• with rotations and
• and scale factors (singular values)

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Notion of Distortion

• isometric or length-preserving
• conformal or angle-preserving
• equiareal or area-preserving
• everything defined pointwise on

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Example Cylindrical Coordinates

• ? isometric

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Example Orthographic Projection

• with
• a ? neither conformal
  nor equiareal

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Example Stereographic Projection

• with
• ? conformal

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Computing the Stretch Factors

• first fundamental form
• eigenvalues of
• singular values of
• and

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Measuring Distortion

• local distortion measure
• has minimum at
• isometric measure
• conformal measure
• overall distortion

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Examples Conformal Measures

• Conformal energy
• MIPS energy

Pinkall Polthier 1993Lévy et al.
2002Desbrun et al. 2002
Hormann Greiner 2000
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Examples Isometric Measures

• Green-Lagrange deformation tensor
• Combined energy

Maillot et al. 1993
Degener et al. 2003
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Examples Other Measures

• Dirichlet energy
• Stretch energies ( , , and symmetric
  stretch)

Pinkall Polthier 1993 Eck et al. 1995
Sander et al. 2001 Sorkine et al. 2002
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Piecewise Linear Parameterizations

• piecewise linear atomic maps
• distortion constant per triangle
• overall distortion

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Beyond Distortion

• surface normal
• surface area
• independent of the particular parameterization
• intrinsic surface properties

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Curvature

• second fundamental form
• Gaussian curvature
• mean curvature