# Differential Geometry Primer by MikeJenny

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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

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•                                            ⇒ isometric

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

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•                                    with

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

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•                                            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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