Mesh Parameterization: Theory and Practice

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Mesh ParameterizationTheory 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

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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
• ?

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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Piecewise Linear Parameterizations

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

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Linear Methods

• the terms and
  are quadratic in the parameter points
• Dirichlet energy
• Conformal energy
• minimization yields linear problem

Pinkall Polthier 1993Eck et al. 1995
Lévy et al. 2002Desbrun et al. 2002
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Linear 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

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Non-linear Methods

• MIPS energy
• Area-preserving MIPS

Hormann Greiner 2000
Degener et al. 2003
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Non-linear Methods

• Green-Lagrange deformation tensor
• Stretch energies ( , , and symmetric
  stretch)

Maillot et al. 1993
Sander et al. 2001 Sorkine et al. 2002