The Wikipedia article for Diffeomorphism has the below picture of an example diffeomorphism of a square onto itself, but it does not seem to specify the specific mapping used to create it. Obviously, I don't need to know the exact mapping used to make the example image, but what would be a similar diffeomorphism that would map an arbitrary rectangle in $\mathbb{R}^2$ centered at the x,y origin onto itself and would produce a somewhat similar style of grid distortion? What would the inverse then be?
The application I'm trying implement is pretty much exactly what the image is: distorting a rectangular grid in different ways such that it can be inverted. I'm just trying to find a concrete example as a starting point as I'm am still rather unfamiliar with this area of mathematics. So pardon any misunderstandings I may be having.
Best Answer
The following figure was drawn with Mathematica. I have used a rotation of the plane by a variable angle $\theta$. This $\theta$ is ${\pi\over4}$ at the origin and decreases to $0$ when we come the boundary of the square. In fact $$\theta(x,y)={\pi\over4}\,\cos x\>\cos y\ .$$