Geometric Deformations There are geometric transformations such as translation, rotation and uniform scaling (Affine transformations). I am interested in knowing whether there is a separate class of transformations that capture deformations for example a square becoming a circle or some other weird shape. I am aware, with a projective transformation a deformation can happen. However, there is no possibility of a square becoming a circle (I doubt).
 A: According to the Riemann mapping theorem, every open topological disk can be mapped conformally and bijectively onto the open unit disk. This includes the square. The class of mappings best suited to describe these transformations are probably the Schwarz-Christoffel mappings, which provide an actual means for computation in the polygonal case. There exist books about these beasts. Looking for suitable images, I found this site about a map of the square by Toby Driscoll. From there I take the final image to illustrate this answer:

A: There are several ways to map a circular disc to a square region and vice versa. For example, given $(u,v)$ in circular coordinates and $(x,y)$ in square coordinates, the following mapping works:

$u = x √( 1 - \frac{1}{2} y^2 )$
$v = y √( 1 - \frac{1}{2} x^2 )$

Its inverse mapping is:

$x =\frac{1}{2}√(2 + u^2 - v^2 + 2u\sqrt2) - \frac{1}{2} √(2 + u^2 - v^2 - 2u\sqrt2)$
$y =\frac{1}{2} √(2 - u^2 + v^2 + 2v\sqrt2) - \frac{1}{2} √(2 - u^2 + v^2 - 2v\sqrt2)$

See http://squircular.blogspot.com/2015/09/mapping-circle-to-square.html for more details on this mapping.
Also, see http://arxiv.org/abs/1509.06344 for the proof/derivation

The mapping is, of course, not unique. See the image below for other mappings:
(including the Schwarz-Christoffel mapping already mentioned previously)



