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A spheroid is bisected into two spheroidal caps by a plane, such that the shape of the area of the plane inside the spheroid is elliptical. The alignment of the plane is defined by two angles theta1 and theta2, which are the angles made between the plane and the normal planes to the spheroid at the two points at either end of the major axis of the ellipse. I know the volume V of the spheroidal cap. However I want to find the length of the major axis of the ellipse. So I need an equation for the length of this major axis in terms of V, theta1 and theta2.

How can I derive such an equation please? I can't embed an image as I'm a newbie but this might hopefully illustrate the scenario a little more clearly, it shows a cross-section through the middle of the spheroid, with the length a being the major axis of the ellipse.

enter image description here

Grateful for any help.

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Are the dimensions of the spheroid known? If so, the problem looks overconstrained to me. If not, you don't have enough data. – Henning Makholm Aug 15 '13 at 8:49
No the dimensions of the spheroid are not known. Are you sure there is not enough data? What else is needed? Given the volume and the two angles, I can only picture one possible spheroid that can be constructed. – user90356 Aug 15 '13 at 10:49
Suppose the two angles given are both $\pi/2$. Then your cap is half the spheroid, and your question is just "what is the diameter of a spheroid with such-and-such volume?". That cannot be answered without knowing at least its oblateness. – Henning Makholm Aug 16 '13 at 8:44
I see your point - and I think I did not define the problem correctly. In the case I am interested in, if both angles are pi/2 then the shape becomes a spherical cap rather than a spheroidal cap. In fact if both angles become equal, whatever their value, the shape becomes a spherical cap. I'm not sure how to define this shape - is it a special case of a spheroid? Does this extra symmetry make the problem solvable? – user90356 Aug 16 '13 at 14:32

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