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This tour explores 2-D parameterization of 3D surfaces using linear methods.
A review paper for mesh parameterization can be found in:
M.S. Floater and K. Hormann, Surface Parameterization: a Tutorial and Survey in Advances in multiresolution for geometric modelling, p. 157-186, 2005.
See also:
K. Hormann, K. Polthier and A. Sheffer Mesh parameterization: theory and practice, Siggraph Asia Course Notes
addpath('toolbox_signal')
addpath('toolbox_general')
addpath('toolbox_graph')
addpath('solutions/meshdeform_1_parameterization')
The conformal Laplacian uses the cotan weights to obtain an accurate discretization of the Laplace Beltrami Laplacian.
They where first introduces as a linear finite element approximation of the Laplace-Beltrami operator in:
U. Pinkall and K. Polthier, Computing discrete minimal surfaces and their conjugates Experimental Mathematics, 2(1):15-36, 1993.
First load a mesh. The faces are stored in a matrix $F = (f_j)_{j=1}^m \in \RR^{3 \times m}$ of $m$ faces $f_j \in \{1,\ldots,n\}^3$. The position of the vertices are stored in a matrix $X = (x_i)_{i=1}^n \in \RR^{3 \times n}$ of $n$ triplets of points $x_k \in \RR^3$
name = 'nefertiti';
[X,F] = read_mesh(name);
n = size(X,2);
clear options; options.name = name;
In order to perform mesh parameterization, it is important that this mesh has the topology of a disk, i.e. it should have a single B.
First we compute the boundary $B = (i_1,\ldots,i_p)$ of the mesh. By definition, for the edges $(i_k,i_{k+1})$, there is a single adjacent face $ (i_k,i_{k+1},\ell) $.
options.verb = 0;
B = compute_boundary(F, options);
Length of the boundary.
p = length(B);
Display the boundary.
clf; hold on;
plot_mesh(X,F,options);
shading('faceted');
hh = plot3( X(1,B),X(2,B),X(3,B), 'r' );
set_linewidth(hh,4);