mesh2d

Parameters

x: real vector
y: real vector
bdy: integer vector
triEdges: integer matrix

Description

mesh2d computes a triangulation of n points in the plane with coordinates given by vectorsx,y. It returns a matrix triEdges of size [3,nbt] where triEdges(:,i) gives the vertices numbers of triangle #i and nbt is the number of triangles.

When bdy is given as an input parameter this vector defines the boundary and contains the indices of edges belonging to it, grouped by successive connected components. Each component is positively oriented, i.e. successive bdy(i:i+1) segments have the interior of the domain to their left. Hence, for a simply connected domain, the boundary is given counterclockwise, and eventual holes are always given clockwise. Each connected component must be closed and is represented by the vector [i1,..,i_nc] such that i1 == i_nc.

When bdy is given as an output parameter the boundary is computed prior to the triangulation as the convex hull of input points x,y and is returned in bdy with the same convention as above, i.e. counterclockwise sucessive vertices numbers.

Possible error cases are the following:

all nodes are collinear,

some points are identical,

wrong boundary array,

crossed boundary,

wrong orientation of the boundary,

size limitation,

an interior point is too close to the boundary,

an interior point is on the boundary,

Warning

The triangulation computed by mesh2d is not guaranteed to be a Delaunay triangulation of points (x,y).

Examples

function displayTri(X, Y, Tr)
  plot(0,0,rect=[-1 -1 2 2])
  [m, n] = size(Tr);
  xpols = matrix(X(Tr), m, n);
  ypols = matrix(Y(Tr), m, n);
  xpolys(xpols, ypols, color("blue")*ones(n,1));
endfunction

rand("seed", 0);
r1 = 1;
n1 = 20;
u = linspace(2*%pi, 0, n1);
xc1 = r1*cos(u(1:$-1));
yc1 = r1*sin(u(1:$-1));
bdy1 = [1:n1-1, 1];

r2 = 2;
n2 = 40;
v = linspace(0, 2*%pi, n2);
xc2 = r2*cos(v(1:$-1));
yc2 = r2*sin(v(1:$-1));
bdy2 = n1-1+[1:n2-1, 1];

xr = (rand(1, 100, "uniform")-.5)*2*r2;
yr = (rand(1, 100, "uniform")-.5)*2*r2;
r = sqrt(xr.^2+yr.^2);

clf
gcf().position(4)=300

// [t, bdy] = mesh2d(x, y) syntax
subplot(1, 2, 1)
k = find(r <= r2);
[t, bdy] = mesh2d(xr(k), yr(k));
displayTri(xr(k), yr(k), t)
plot(xr(k(bdy)), yr(k(bdy)),"r-o")
xtitle("[triEdges, bdy] = mesh2d(x, y)")
isoview

// t = mesh2d(x, y, bdy) syntax
subplot(1, 2, 2)
k = find((r >=  r1) & (r <=  r2));
x = [xc1 xc2 xr(k)];
y = [yc1 yc2 yr(k)];
t = mesh2d(x, y, [bdy1 bdy2]);
displayTri(x, y, t)
plot(x(bdy1), y(bdy1),"r-o")
plot(x(bdy2), y(bdy2),"r-o")
xtitle("triEdges = mesh2d(x, y, bdy)")
isoview

References

mesh2d was previously part of the metanet ATOMS module.