How to integrate over a triangle in MATLAB? - matlab

Question

let's consider a two dimensional fonction f(x,y)
and tree points A,B,C with ABC a triangle
and i want to integrate the function f over the triangle ABC,
is there a way to do that in matlab?
thank you.

Answer 1

You can create a new function h(x,y), that returns f(x,y) if (x,y) is inside the polygon, and 0 otherwise.
For instance:
A = [0, 0];
B = [0, 5];
C = [5, 0];
triangleX = [A(1) B(1) C(1)];
triangleY = [A(2) B(2) C(2)];
f = #(x,y) (1);
h = #(x,y) (inpolygon(x, y, triangleX, triangleY) .* f(x,y));
q = integral2(h, min(triangleX), max(triangleX), min(triangleY), max(triangleY)
'Method', 'iterated')
Outputs (which may be close enough for you):
q =
12.500070877352647
And another function:
f = #(x,y) (x .* y);
q = integral2(#foo, min(triangleX), max(triangleX), min(triangleY), max(triangleY),
'Method', 'iterated')
q =
26.042038561947592
Notice that the integral2 documentation states:
When integrating over nonrectangular regions, the best performance and
accuracy occurs when ymin, ymax, (or both) are function handles. Avoid
setting integrand function values to zero to integrate over a
nonrectangular region. If you must do this, specify 'iterated' method.
So it'll be better if instead of using the above solution, you write two functions that given an x coordinate, give you the minimal and maximal y coordinates of the polygon (triangle).

Answer 2

I found the correct answer,
thanks to this https://math.stackexchange.com/questions/954409/double-integral-over-an-arbitrary-triangle
function r = intm2(f, t)
% f: function
% t: three points of a triangle
% r: integration of f over t
a = t(1,:);
b = t(2,:);
c = t(3,:);
jg = abs((b(1)-a(1))*(c(2)-a(2))-(c(1)-a(1))*(b(2)-a(2)));
ftilda = #(u,v) f(a(1)+u*(b(1)-a(1))+v*(c(1)-a(1)), a(2)+u*(b(2)-a(2))+v*(c(2)- a(2)));
fy = #(x) 1-x;
r = jg * integral2(ftilda, 0,1, 0,fy);
end