# Basics About 2D Triangulation¶

$\newcommand{\dotp}[2]{\langle #1, #2 \rangle}$ $\newcommand{\enscond}[2]{\lbrace #1, #2 \rbrace}$ $\newcommand{\pd}[2]{ \frac{ \partial #1}{\partial #2} }$ $\newcommand{\umin}[1]{\underset{#1}{\min}\;}$ $\newcommand{\umax}[1]{\underset{#1}{\max}\;}$ $\newcommand{\umin}[1]{\underset{#1}{\min}\;}$ $\newcommand{\uargmin}[1]{\underset{#1}{argmin}\;}$ $\newcommand{\norm}[1]{\|#1\|}$ $\newcommand{\abs}[1]{\left|#1\right|}$ $\newcommand{\choice}[1]{ \left\{ \begin{array}{l} #1 \end{array} \right. }$ $\newcommand{\pa}[1]{\left(#1\right)}$ $\newcommand{\diag}[1]{{diag}\left( #1 \right)}$ $\newcommand{\qandq}{\quad\text{and}\quad}$ $\newcommand{\qwhereq}{\quad\text{where}\quad}$ $\newcommand{\qifq}{ \quad \text{if} \quad }$ $\newcommand{\qarrq}{ \quad \Longrightarrow \quad }$ $\newcommand{\ZZ}{\mathbb{Z}}$ $\newcommand{\CC}{\mathbb{C}}$ $\newcommand{\RR}{\mathbb{R}}$ $\newcommand{\EE}{\mathbb{E}}$ $\newcommand{\Zz}{\mathcal{Z}}$ $\newcommand{\Ww}{\mathcal{W}}$ $\newcommand{\Vv}{\mathcal{V}}$ $\newcommand{\Nn}{\mathcal{N}}$ $\newcommand{\NN}{\mathcal{N}}$ $\newcommand{\Hh}{\mathcal{H}}$ $\newcommand{\Bb}{\mathcal{B}}$ $\newcommand{\Ee}{\mathcal{E}}$ $\newcommand{\Cc}{\mathcal{C}}$ $\newcommand{\Gg}{\mathcal{G}}$ $\newcommand{\Ss}{\mathcal{S}}$ $\newcommand{\Pp}{\mathcal{P}}$ $\newcommand{\Ff}{\mathcal{F}}$ $\newcommand{\Xx}{\mathcal{X}}$ $\newcommand{\Mm}{\mathcal{M}}$ $\newcommand{\Ii}{\mathcal{I}}$ $\newcommand{\Dd}{\mathcal{D}}$ $\newcommand{\Ll}{\mathcal{L}}$ $\newcommand{\Tt}{\mathcal{T}}$ $\newcommand{\si}{\sigma}$ $\newcommand{\al}{\alpha}$ $\newcommand{\la}{\lambda}$ $\newcommand{\ga}{\gamma}$ $\newcommand{\Ga}{\Gamma}$ $\newcommand{\La}{\Lambda}$ $\newcommand{\si}{\sigma}$ $\newcommand{\Si}{\Sigma}$ $\newcommand{\be}{\beta}$ $\newcommand{\de}{\delta}$ $\newcommand{\De}{\Delta}$ $\newcommand{\phi}{\varphi}$ $\newcommand{\th}{\theta}$ $\newcommand{\om}{\omega}$ $\newcommand{\Om}{\Omega}$

This tour explores some basics about 2D triangulated mesh (loading, display, manipulations).

In [2]:
addpath('toolbox_signal')


## Planar Triangulation¶

A planar triangulation is a collection of |n| 2D points, whose coordinates are stored in a |(2,n)| matrix |vertex|, and a topological collection of triangle, stored in a |(m,2)| matrix |faces|.

Number of points.

In [3]:
n = 200;


Compute randomized points in a square.

In [4]:
vertex = 2*rand(2,n)-1;


A simple way to build a triangulation of the convex hull of the points is to compute the Delaunay triangulation of the points.

In [5]:
faces = delaunay(vertex(1,:),vertex(2,:))';


One can display the triangulation.

In [6]:
clf;
subplot(1,2,1);
hh = plot(vertex(1,:),vertex(2,:), 'k.');
axis('equal'); axis('off');
set(hh,'MarkerSize',10);
title('Points');
subplot(1,2,2);
plot_mesh(vertex,faces);
title('Triangulation');


## Point Modification¶

It is possible to modify the position of the points like a particles system. The dynamics is govered by the connectivity to enfoce an even distribution. During the modification of the positions, the connectivity is updated.

Fix some points on a disk.

In [7]:
m = 20;
t = linspace(0,2*pi,m+1); t(end) = [];
vertexF = [cos(t);sin(t)];
vertex(:,1:m) = vertexF;
faces = delaunay(vertex(1,:),vertex(2,:))';


Initialize the positions.

In [8]:
vertex1 = vertex;


Compute the delaunay triangulation.

In [9]:
faces1 = delaunay(vertex1(1,:),vertex1(2,:))';


Compute the list of edges.

In [10]:
E = [faces([1 2],:) faces([2 3],:) faces([3 1],:)];
p = size(E,2);


We build the adjacency matrix of the triangulation.

In [11]:
A = sparse( E(1,:), E(2,:), ones(p,1) );


Normalize the adjacency matrix to obtain a smoothing operator.

In [12]:
d = 1./sum(A);
iD = spdiags(d(:), 0, n,n);
W = iD * A;


Apply the filtering.

In [13]:
vertex1 = vertex1*W';


Set of the position of fixed points.

In [14]:
vertex1(:,1:m) = vertexF;


Display the positions before / after.

In [15]:
clf;
subplot(1,2,1);
plot_mesh(vertex,faces);
title('Before filering');
subplot(1,2,2);
plot_mesh(vertex1,faces1);
title('After filtering');


Exercise 1

Compute several steps of iterative filterings, while ensuring the positions of the fixed points.

In [16]:
exo1()

In [17]:
%% Insert your code here.