# mesh2d

Triangulation of n points in the plane

### Calling Sequence

triEdges = mesh2d(x, y) [triEdges, bdy] = mesh2d(x, y) triEdges = mesh2d(x, y, bdy)

### 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 vectors`x,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:

### 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 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)-.5)*2*r2; yr = (rand(1, 100)-.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.

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