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I'm a software engineer and look for the answer to a question which I havent found a solution for. So please excuse my bad mathematical terminology in this question.

I want to generate an ellipsoid around $2$ foci points. It should have the following properties:

Every point on the surface of the ellipsoid should have the same combined distance from both focis. Let $S_1$ be the first foci and $S_2$ the second. The distance $r_1$ is the distance from any point on the ellipsoids surface to foci $S_1$ and $r_2$ the distance from any point on the surface to foci $S_2$.

Then $r_1 + r_2$ should be the same for every point on the surface.

In $2D$ I used the simple Pins-and-string construction, which generates such an ellipse.
For $3D$ I don't know how to do it so that i get an ellipsoid with the same properties.
I thought about rotating the ellipse from $2D$ but my research told me if I do that I don't get such an ellipsoid and that I need to use an iterative approach but I didn't find any specific formulas that would help me to do so.

I'm really desperate and hope a mathematician can help me understand the logic behind an ellipsoid.

Thanks!

Daniele Tampieri
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  • FYI: The term-of-art for your ellipsoid is "prolate spheroid", which is in fact the result of rotating a planar ellipse about its major axis. (This is in contrast to an "oblate spheroid", which is the result of rotating a planar ellipse about its minor axis.) – Blue Mar 13 '24 at 19:38
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    Does this answer your question ? https://math.stackexchange.com/questions/4857808/in-three-dimensional-space-when-the-sum-of-the-distances-from-an-unknown-point/4857859#4857859 –  Mar 13 '24 at 23:43
  • The surface you are looking for is just the surface of revolution of the 2D ellipse with your constraint, rotated along the axis through the foci. – 3dguy Mar 15 '24 at 08:59

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