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How can you estimate the distortion introduced by a parametrization?
Given a parametrization
F(u,v) : \Omega \to \mathbb{R}^3
consider the Jacobian of
at some point (u_0, v_0):
\nabla F(u_0, v_0) = \begin{pmatrix} \frac{\partial F(u_0, v_0)}{\partial u} & \frac{\partial F(u_0, v_0)}{\partial v} \end{pmatrix} \in \mathbb{R}^{3 \times 2}
and its Singular Value Decomposition
and 
are orthonormal, i.e. rotation matrices. 
is a diagonal matrix and corresponds to an axis-aligned scaling. 
thus can be interpreted as a concatenation of a rotation, a scaling and again a rotation. The distortion at 
can thus be inferred from 
and \gamma:
F(u,v) : \Omega \to \mathbb{R}^3
consider the Jacobian of
at some point (u_0, v_0):\nabla F(u_0, v_0) = \begin{pmatrix} \frac{\partial F(u_0, v_0)}{\partial u} & \frac{\partial F(u_0, v_0)}{\partial v} \end{pmatrix} \in \mathbb{R}^{3 \times 2}
and its Singular Value Decomposition
and are orthonormal, i.e. rotation matrices. is a diagonal matrix and corresponds to an axis-aligned scaling. thus can be interpreted as a concatenation of a rotation, a scaling and again a rotation. The distortion at can thus be inferred from and \gamma:- Conformal / angle preserving:
- Area preserving:
- Isometry:
Karteninfo:
Autor: janisborn
Oberthema: Informatik
Thema: Computergrafik
Schule / Uni: RWTH Aachen
Ort: Aachen
Veröffentlicht: 18.05.2022