Greg Egan on Nostr: nprofile1q…kxprf The simplest definition is that two ellipsoids with the same ...

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The simplest definition is that two ellipsoids with the same centre, and with their axes aligned, are confocal if the squares of their semi-axes all differ by the same amount.

For an ellipse with semi-axes a and b, the distance f from the centre to the foci is given by:

f^2 = a^2 - b^2

so adding the same quantity to both a^2 and b^2 preserves the foci.

So algebraically this generalises that relationship: we add the same quantity to a^2, b^2, c^2 for the ellipsoid.

Geometrically, you can also identify “focal conics” of an ellipsoid that remain the same under this transformation.

https://en.wikipedia.org/wiki/Focal_conics