Isometric Mappings (Length-Preserving)

An allowable mapping from $S$S to $S^{*}$S∗ is isometric if and only if the coefficients of the first fundamental forms are the same, i.e.,

$I = I^{*}$I=I∗

Isometric surfaces have the same Gaussian curvature at corresponding pairs of points. (In differential geometry, the Gaussian Curvature or Gauss Curvature $k$k of a surface at a point is the product of the principal curvatures $k_1$k1​ and $k_2$k2​, at the given point;

$k = k_{1}k_{2}$k=k1​k2​

The two principal curvatures at a given point of a surface are the eigenvalues of the shape operator at the point. They measure how the surface bends by different amounts in different direction at that point.)

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Conformal Mappings (Angle-Preserving)

An allowable mapping from $S$S to $S^{*}$S∗ is conformal or angle-preserving if and only if the coefficients of the first fundamental forms are proportional, i.e,

$I = \eta(x,y)I^{*},$I=η(x,y)I∗,

for some scalar $\eta \neq 0$η̸​=0.

Consider for the case of mappings from a planar region $S$S to the plane. Such a mapping can be viewed as a function of a complex variable, $w = f(z)$w=f(z). Locally, a conformal map is simply any function f which is analytic in a neighborhood of a point $z$z and such that $f^{\prime}(z) \neq 0$f′(z)̸​=0. A conformal mapping $f$f thus satisfies the Cauchy-Riemann equations, which, with $z = x+iy$z=x+iy and $w = u+iv$w=u+iv, are

$\begin{aligned} \frac{\partial(u)}{\partial(x)} = \frac{\partial(v)}{\partial(y)}, \\ \frac{\partial(u)}{\partial(y)} = -\frac{\partial(v)}{\partial(x)} \end{aligned}$∂(x)∂(u)​=∂(y)∂(v)​,∂(y)∂(u)​=−∂(x)∂(v)​​

Discrete Conformal Mappings

There are several approaches that maximize the conformality of the piecewise linear mapping without demanding the mesh boundary to be mapped onto a fixed shape, like the Discrete Harmonic Mappings below. Instead, these methods allow the parameter values of the boundary points to be included into the optimization problem and the shape of the parameter domain is determined by the method.

• Most Isometric Parameterizations
• Angle-based Flattening
• Linear Methods

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Harmonic Mappings

When the Laplace equations of the above $w = f(z)$w=f(z) equal zero, the mapping is called Harmonic Mapping.

$\triangle(u) = 0, \triangle(v) = 0,$△(u)=0,△(v)=0,

where

$\triangle = \frac{\partial}{\partial(x^2)} + \frac{\partial}{\partial(y^2)}.$△=∂(x2)∂​+∂(y2)∂​.

If $f : S \rightarrow R^2$f:S→R2 is harmonic and maps the boundary of $S$S homemorphically into the boundary of $S^{*}$S∗ (some convex regions belong to $R^2$R2), the $f$f is one-to-one.

However, the harmonic maps are not in general conformal and do not preserve angles. And the inverse of a harmonic mapping is not necessarily harmonic.

Discrete Harmonic Mappings

Common to almost all surface parameterization methods is to approximate the underlying smooth surface $S$S by a piecewise surface $S_T$ST​, in the form of a triangular mesh, i.e. the union of a set of $T = {T_1, T_2, …, T_M}$T=T1​,T2​,…,TM​ of triangles $T_i$Ti​, such that the triangles intersect only at common vertices or edges.

1. Fix the boundary mapping for the vertices lying on the boundary

2. The Boundary Mapping

• Riemann Mapping Theorem
• Chord Length Parameterization

3. Find the piecewise linear mapping for the interior vertices

• Finite Element Method
• Convex Combination Maps
• Mean Value Coordinates

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Equiareal Mappings (Area-Preserving)

An allowable mapping from $S$S to $S^{*}$S∗ is equiareal if and only if the discriminants (判别式) of the first fundamental forms are equal, i.e,

$g = g^{*}$g=g∗

Every isometric mapping is conformal and equiareal, and every conformal and equiareal mapping is isometric, i.e,

Isometric <—> conformal + equiareal

Discrete Equiareal Mappings

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Planar Mappings

Please refer to the paper ”Surface Parameterization: a Tutorial and Survey” by Michael S. Floater and Kai Hormann.