# how to calculate the intersection area of two ellipses

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amal Mbarki on 18 Feb 2016
Commented: Image Analyst on 2 Dec 2018
hello, how can i calculte the intersection area of two ellipses, each ellipse is characterized by(x,y,a,b,w).I thought of solving the system of equations of the two ellipses ,then calculate the area bounded by the points found as solution. would you please help me ?
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Image Analyst on 2 Dec 2018
The only solution I've seen is a numerical one, not an analytical one.

Kevin Claytor on 18 Feb 2016
Here is an arxiv paper on ellipse intersection algorithms.

Image Analyst on 18 Feb 2016
To create an ellipse numerically in 2-D is very easy - just see the FAQ. http://matlab.wikia.com/wiki/FAQ#How_do_I_create_an_ellipse.3F To rotate it, just multiply the coordinates by the rotation matrix [cos(theta), -sin(theta); cos(theta), sin(theta)]. Then just AND the two ellipse images
intersectionArea = ellipsoid1 & ellipsoid2;
pixelArea = sum(intersectionArea (:)); % Compute the area in pixels.
amal Mbarki on 19 Feb 2016
In fact each ellipse is characterized by (x,y)its center coordinate, it big and small axes(a,b) and it's orientation w. I’ve read a code which calculate the intersection area of two circles analytically .the equation is: when (d(i,j)> abs(ri-rj)) & (d(i,j)<(ri+rj), d is the distance between the centers of the two objects) then the intersection area is M(i,j) = f(xi,yi,ri,xj,yj,rj) = ri^2*arctan2(yk,xk)+rj^2*arctan2(yk,d(i,j)-xd(i,j)*yk where xk = (ri^2-rj^2+d(i,j)^2)/(2*d(i,j)) and yk = sqrt(ri^2-xk^2). So I thought why not i calculate the intersection area of two ellipse with the same way.But i don't know how to extract the analytical formula?