Scilab function

mesh2d - triangulation of n points in the plane

### Calling Sequence

[nutr,A] = mesh2d(x,y,[front])

### Parameters

• x : real row array
• y : real row array
• front : integer row array
• nutr : integer matrix
• A : sparse 0-1 matrix

### Description

The arrays x and y are the coordinates of n points in the plane. mesh2d returns a matrix nutr(3,nbt) of the numbers of the nodes of the nbt triangles of the triangulation of the points. It returns also a sparse matrix A representing the connections between the nodes (A(i,j)=1 if (i,j) is a side of one of the triangles or i=j). In the case of 3 parameters front is the array defining the boundary: it is the array of the indices of the points located on the boundary . The boundary is defined such that the normal to the boundary is oriented towards outside. The boundary is given by its connected components: a component is the part (i1,i2) such that front(i1)=front(i2) (the external boundary is defined in the counterclockwise way, see the examples below). The error cases are the following: err = 0 if no errors were encountered; err = 3 all nodes are collinear.

If the boundary is given, the other error cases are: err = 2 some points are identical; err = 5 wrong boundary array; err = 6 crossed boundary; err = 7 wrong orientation of the boundary; err = 10 an interior point is on the boundary; err = 8 size limitation; err = 9 crossed boundary; err = 12 some points are identical or size limitation.

### Examples

```// FIRST CASE
theta=0.025*[1:40]*2.*%pi;
x=1+cos(theta);
y=1.+sin(theta);
theta=0.05*[1:20]*2.*%pi;
x1=1.3+0.4*cos(theta);
y1=1.+0.4*sin(theta);
theta=0.1*[1:10]*2.*%pi;
x2=0.5+0.2*cos(theta);
y2=1.+0.2*sin(theta);
x=[x x1 x2];
y=[y y1 y2];
//
nu=mesh2d(x,y);
nbt=size(nu,2);
jj=[nu(1,:)' nu(2,:)';nu(2,:)' nu(3,:)';nu(3,:)' nu(1,:)'];
as=sparse(jj,ones(size(jj,1),1));
ast=tril(as+abs(as'-as));
[jj,v,mn]=spget(ast);
n=size(x,2);
g=make_graph('foo',0,n,jj(:,1)',jj(:,2)');
g('node_x')=300*x;
g('node_y')=300*y;
g('default_node_diam')=10;
show_graph(g)
// SECOND CASE !!! NEEDS x,y FROM FIRST CASE
x3=2.*rand(1:200);
y3=2.*rand(1:200);
wai=((x3-1).*(x3-1)+(y3-1).*(y3-1));
ii=find(wai >= .94);
x3(ii)=[];y3(ii)=[];
wai=((x3-0.5).*(x3-0.5)+(y3-1).*(y3-1));
ii=find(wai <= 0.055);
x3(ii)=[];y3(ii)=[];
wai=((x3-1.3).*(x3-1.3)+(y3-1).*(y3-1));
ii=find(wai <= 0.21);
x3(ii)=[];y3(ii)=[];
xnew=[x x3];ynew=[y y3];
fr1=[[1:40] 1];fr2=[[41:60] 41];fr2=fr2(\$:-1:1);
fr3=[[61:70] 61];fr3=fr3(\$:-1:1);
front=[fr1 fr2 fr3];
//
nu=mesh2d(xnew,ynew,front);
nbt=size(nu,2);
jj=[nu(1,:)' nu(2,:)';nu(2,:)' nu(3,:)';nu(3,:)' nu(1,:)'];
as=sparse(jj,ones(size(jj,1),1));
ast=tril(as+abs(as'-as));
[jj,v,mn]=spget(ast);
n=size(xnew,2);
g=make_graph('foo',0,n,jj(:,1)',jj(:,2)');
g('node_x')=300*xnew;
g('node_y')=300*ynew;
g('default_node_diam')=10;
show_graph(g)
// REGULAR CASE !!! NEEDS PREVIOUS CASES FOR x,y,front
xx=0.1*[1:20];
yy=xx.*.ones(1,20);
zz= ones(1,20).*.xx;
x3=yy;y3=zz;
wai=((x3-1).*(x3-1)+(y3-1).*(y3-1));
ii=find(wai >= .94);
x3(ii)=[];y3(ii)=[];
wai=((x3-0.5).*(x3-0.5)+(y3-1).*(y3-1));
ii=find(wai <= 0.055);
x3(ii)=[];y3(ii)=[];
wai=((x3-1.3).*(x3-1.3)+(y3-1).*(y3-1));
ii=find(wai <= 0.21);
x3(ii)=[];y3(ii)=[];
xnew=[x x3];ynew=[y y3];
nu=mesh2d(xnew,ynew,front);
nbt=size(nu,2);
jj=[nu(1,:)' nu(2,:)';nu(2,:)' nu(3,:)';nu(3,:)' nu(1,:)'];
as=sparse(jj,ones(size(jj,1),1));
ast=tril(as+abs(as'-as));
[jj,v,mn]=spget(ast);
n=size(xnew,2);
g=make_graph('foo',0,n,jj(:,1)',jj(:,2)');
g('node_x')=300*xnew;
g('node_y')=300*ynew;
g('default_node_diam')=3;
show_graph(g)

//An example with a random set of points
function []=test(X,Y)
Tr=mesh2d(X,Y);
plot2d(X,Y,[-1,-2,3]);
[m,n]=size(Tr);
xpols= matrix(X(Tr),m,n);
ypols= matrix(Y(Tr),m,n);
xset("colormap",rand(2*n,3));
xfpolys(xpols,ypols,[n/4:n/4+n-1]);
endfunction
N=1000;xbasc();X=rand(1,N); Y=rand(1,N);
xset("wdim",700,700);
test(X,Y);
xset('default');

```