Those transformations are non-rigid, trust me 
I want to convert Cartesian coordinates to cylindrical coordinates, so that I could “unwrap” a cylindrical surface, edit it (stretch, push/pull, …) and “wrap” it back to a cylinder.
That means transforming every vertex so that:
x,y,z = [sqrt(x**2,y**2), atan2(y,x), z]
Which is, of many transformations I know, a very non-rigid one 