MATLAB: Area of two identical overlapping ellipses

intersecting area

Hi, I have two identical overlapping ellipses (one is moved by 0.2 in the y direction). How can I calculate the overlapping area?
I know how to analytically solve the overlapping area of two identical circles, but the analytical solution for two identical overlapping ellipses is very hard to find…Maybe Matlab can do it numerically or by comparing pixels?
My two ellipses are generated using the following code:
a = 1;
b = sqrt(2);
x0=0;
y0=0;
t=-pi:0.01:pi;
x=x0+a*cos(t);
y=y0+b*sin(t);
m1=fill(x,y,'b')
m1.FaceAlpha=0.2;
y01=y0+0.2;
y1=y01+b*sin(t);
hold on
m2=fill(x,y1,'r')
m2.FaceAlpha=0.2;

Best Answer

Pretty simple actually as just a transformation of variables.
You claim to have the analytical solution for a pair of circles, with a simple y offset.
Solve the problem for a pair of circles with unit radius, where one circle is offset in y by a delta of 0.2/b. (.2/b is important here.) Get the area as A_c. (Thus the area of intersection for two circles of radius 1.)
Now, if you implicitly transform the variables such that
X_e = x_c*a
Y_e = Y_c*b
The area of the ellipse intersection will be a*b*A_c.
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