Differential Geometry Primer

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					  Mesh Parameterization:
   Theory and Practice

Differential Geometry Primer
Parameterization




• surface
• parameter domain
• mapping          and

Mesh Parameterization: Theory and Practice
Differential Geometry Primer
Example – Cylindrical Coordinates




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Mesh Parameterization: Theory and Practice
Differential Geometry Primer
Example – Orthographic Projection




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•
Mesh Parameterization: Theory and Practice
Differential Geometry Primer
Example – Stereographic Projection




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Mesh Parameterization: Theory and Practice
Differential Geometry Primer
Example – Mappings of the Earth

• usually, surface properties get distorted




orthographic             stereographic           Mercator       Lambert
 ∼ 500 B.C.                ∼ 150 B.C.             1569            1772

                                      conformal              equiareal
                                   (angle-preserving)       (area-preserving)

Mesh Parameterization: Theory and Practice
Differential Geometry Primer
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


Mesh Parameterization: Theory and Practice
Differential Geometry Primer
What is Distortion?

• parameter point
• surface point
• small disk                         around

• image of               under

• shape of

Mesh Parameterization: Theory and Practice
Differential Geometry Primer
Linearization

• Jacobian of


• tangent plane at

• Taylor expansion of

• first order approximation of


Mesh Parameterization: Theory and Practice
Differential Geometry Primer
Infinitesimal Dis(k)tortion

• small disk                                 around
• image of under

• shape of
     – ellipse
     – semiaxes               and
• behavior in the limit


Mesh Parameterization: Theory and Practice
Differential Geometry Primer
Linear Map Surgery

• Singular Value Decomposition (SVD) of



   with rotations              and
   and scale factors (singular values)




Mesh Parameterization: Theory and Practice
Differential Geometry Primer
Notion of Distortion

• isometric or length-preserving


• conformal or angle-preserving


• equiareal or area-preserving


• everything defined pointwise on
Mesh Parameterization: Theory and Practice
Differential Geometry Primer
Example – Cylindrical Coordinates




•

•                                            ⇒ isometric

Mesh Parameterization: Theory and Practice
Differential Geometry Primer
Example – Orthographic Projection




•

•                                    with

•                                   ⇒        neither conformal
                                             nor equiareal
Mesh Parameterization: Theory and Practice
Differential Geometry Primer
Example – Stereographic Projection




•

•                                            with

•                                  ⇒ conformal
Mesh Parameterization: Theory and Practice
Differential Geometry Primer
Computing the Stretch Factors

• first fundamental form



• eigenvalues of


• singular values of
                                       and

Mesh Parameterization: Theory and Practice
Differential Geometry Primer
Measuring Distortion

• local distortion measure


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



Mesh Parameterization: Theory and Practice
Differential Geometry Primer
Piecewise Linear Parameterizations




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

Mesh Parameterization: Theory and Practice
Differential Geometry Primer
Linear Methods

• the terms              and                 are quadratic
  in the parameter points
• Dirichlet energy                            [Pinkall & Polthier 1993]
                                                       [Eck et al. 1995]



• Conformal energy                                  [Lévy et al. 2002]
                                                 [Desbrun et al. 2002]


• minimization yields linear problem

Mesh Parameterization: Theory and Practice
Differential Geometry Primer
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


Mesh Parameterization: Theory and Practice
Differential Geometry Primer
Non-linear Methods

• MIPS energy                                [Hormann & Greiner 2000]




• Area-preserving MIPS                            [Degener et al. 2003]




Mesh Parameterization: Theory and Practice
Differential Geometry Primer
Non-linear Methods

• Green-Lagrange deformation tensor                       [Maillot et al. 1993]




• Stretch energies (                    ,    , and symmetric stretch)



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

Mesh Parameterization: Theory and Practice
Differential Geometry Primer

				
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