Area Preserving Transformations from a Circle to an Ellipse I am looking for the class/set of transformations that map a circle to an ellipse while preserving the area inside.  Generally, maps of n-spheres to n-ellipsoids that preserve volume, if such a class/set exists.
Thank you,
 A: If it takes all spheres to ellipsoids and preserves $n$-volume in the Euclidean space $\mathbb R^n,$ it is just linear with determinant $\pm 1.$ Plus a translation, either before or after. So the basic object is, indeed, $SL_n(\mathbb R).$   
If you relax either restriction you immediately get many more mappings. In particular, Moebius transformations on $\mathbb C \cup \{ \infty \}$ take any circle to a circle or line, any line to a circle or line. But area is not preserved unless you have $f(z) = \omega z + \beta,$ with $|\omega| = 1.$ 
A: There are a lot of volume preserving maps on $\mathbb{R}^n$. Here are a few simple transformations that preserve volume:


*

*scale by $r$ along one axis and $\frac1r$ along another

*rotation preserves distances and therefore volume

*translation preserves distances and therefore volume
Combinations of these three transformations are enough to map any sphere to any ellipsoid with the same volume.
A: The folks in the animation business are always inventing new (and not-so-new) ways to do "volume-preserving deformations". Their idea is that preserving the volume of the object will make the deformation more "natural". It makes some sense -- when you flex your muscles, their overall volume doesn't change (I suppose). Anyway, if you google the term "volume-preserving deformation" you will find piles of stuff. 
Generally, these deformations are only approximately volume-preserving, though. Good enough for animation, but maybe not good enough for your purposes. Only you can decide.
A: Any shear is an area/volume-preserving transformation.
