# point in convex hull?

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Noe alvarado on 23 Mar 2012
how compute if a point point is inside the convex hull ?

Friedrich on 23 Mar 2012
Hi,
I would use convhull and inpolygon:
xx = -1:.05:1; yy = abs(sqrt(xx));
[x,y] = pol2cart(xx,yy);
k = convhull(x,y);
hold on
plot(x(k),y(k),'r-',x,y,'b+')
x_test = rand(10,1);
y_test = rand(10,1);
IN = inpolygon(x_test,y_test,x(k),y(k))
plot(x_test(IN),y_test(IN),'g*',x_test(~IN),y_test(~IN),'y*')
hold off
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Friedrich on 23 Mar 2012
This wont work in higher dimensions. You have to do some smart geometry stuff to get the information you like. you can use convhulln to get the facets of your convex hull and also the corners (x_i, i = 1..m). After that try to get coefficients a_i > 0 so that
sum_i x_i*a_i = v where sum_i a_i = 1 (basically that's the definition of the convex hull). (v is the vector you like to know if its inside or outside of the convex hull). Finding one solution for the system of equations is easy, but the condition that the sum over the coefficients gives 1 is pretty tricky. I don't know if there is an easy way to get solve this.
Seems more like a mathematical question rather than a MATLAB question for me.
If you know an algorithm come back and ask how to implement/use this in MATLAB. But first you need to know how to check this in theory before doing it in MATLAB.

Matt Kindig on 23 Mar 2012
There is a file on the File Exchange called 'inhull'. I have used it for precisely this purpose.
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Noe alvarado on 12 Apr 2012
thanks