Area of two identical overlapping ellipses

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;

Answer 1

[John D'Errico]

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.

[laoliu102]

I see! Thank you so much!

Thinking about your answer geometrically, b=sqrt(2) can be viewed as if the unit circle forms a cylinder and the cylinder is cut 45 degrees. The cross section at 45 degrees is the ellipse, with b=sqrt(2), a=1=r_circle, X_e=x_c*a, Y_e=Y_c*b.

If the ellipse offset is 0.2, then the circle offset is 0.2/sqrt(2), which is why you said the 0.2/b is important. Then I analytically solve for the overlapping area (A_c) of the unit circles with this 0.2/b offset, and the ellipse overlapping area is just sqrt(2)*A_c.

[Walter Roberson]

I think the solution would need a little modification if the second ellipse were also rotated instead of just displaced.

[John D'Errico]

I saw no reason to worry about that case, because the ellipses were explicitly simple ones.

[laoliu102]

The ellipses are not rotated. The second ellipse only moves a distance in the y direction on the plane where the ellipse sits.

[John D'Errico]

Exactly. Were one of the ellipses rotated, it gets a bit messy. But this case is a simple one.