The title describes what I'm looking for:
Is there a (canonical) way of mapping an ellipse (interior and boundary) to an ellipse with different eccentricity that maps focal points to focal points?
(Obviously, an affine linear map won't do it and I think it can't be a conformal mapping either.)
Put in other words: Is there among all (let's say differentiable) maps that map a given ellipse (interior and boundary) to a given other ellipse and preserving the focal point one canonical (e.g. based on geometric reasons or extremal principles, e.g. mean square of deplacement of the points is minimal or the like).
P.S. Note that you won't be able to map first to a circle then to the other ellipse, because mapping to the circle won't be injective (both focal points map to the center.
Best Answer
Hint:
From two foci, we can define two families of confocal conics, ellipses and hyperbolas. Furthermore, they are orthogonal.
Every point in the plane can be referred to by $(u,v)$ coordinates associated to this network of curves. Then any transformation of the form $(u,v)\to(\phi(u),v)$ achieves the desired effect. Obviously, the $v$-curves map to themselves.
See https://en.wikipedia.org/wiki/Confocal_conic_sections.