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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,

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    $\begingroup$ 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. $\endgroup$ – Will Jagy May 19 '12 at 4:51
  • $\begingroup$ Thanks, to be sure then, I'm just interested in $SL(n,\mathbb{R})$? If you write it up I'll check it as an answer, else I'll just up-vote the comment. $\endgroup$ – kηives May 22 '12 at 16:22
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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 $|\omega| = 1.$

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There are a lot of volume preserving maps on $\mathbb{R}^n$. Here are a few simple transformations that preserve volume:

  1. scale by $r$ along one axis and $\frac1r$ along another

  2. rotation preserves distances and therefore volume

  3. translation preserves distances and therefore volume

Combinations of these three transformations are enough to map any sphere to any ellipsoid with the same volume.

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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.

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Any shear is an area/volume-preserving transformation.

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