Suppose I have the map $$ (x,y) \rightarrow(y\cos(x),y\sin(x)), \text{valid for } (x,y)\in\left[-\pi,\pi\right)\times\mathbb{R}^+ $$
This is going from Cartesian to polar coordinates with the assumption that the input is the identified semi-infinite strip. It maps lines of constant $x$ into radial lines, and it maps lines of constant $y$ into concentric circles. Lines at right angles in the input are at right angles in the output. I'm not 100% sure but I think this transformation generally preserves all local angles, not just right angles, and is thus conformal.
I am looking for a family of mappings $(x,y)\rightarrow T(x,y,\alpha)$ with a parameter $\alpha\in[0,1]$ that in some sense smoothly, continuously interpolates between these two maps in a sensible way. The conditions I want of this family of transformations are:
- $T(x,y,0) = (x,y)$
- $T(x,y,1) = (y\cos(x),y\sin(x))$
- Smooth lines in the input don't become non-smooth or cross over themselves as $\alpha$ goes from 0 to 1.
- Lines that are parallel in the input don't cross over each other as $\alpha$ goes from 0 to 1. They stay locally parallel.
- I think I want the map $(x,y)\rightarrow T(x,y,\alpha)$ to always have the angle preserving quality noted above, i.e., at least right angles in the input are right angles in the output. If the original map is truly conformal, then I want the intermediate maps to be conformal as well.
The application here is to animate a plot in polar coordinates being smoothly "unwrapped" into Cartesian coordinates. I think the constraints above do what I want, but feel free to relax / generalize as needed to get to the idea of implementing the animation. I have a vague notion that continuous deformations, conformal mappings, level sets of some higher dimensional function, maybe gradients of a scalar potential, and Jacobians are involved but can't find the right terms to search and can't quite formulate this myself. I've tried the "linear" deformation of $\alpha T_2+(1-\alpha)T_1$ but this definitely doesn't have the nice properties of the intermediate transforms not having kinks and not having the x-y grid lines cross.
What is the general method to derive a family of mappings that interpolates between conformal maps?
Does the answer have a closed form in my specific case?
If not, can the solution be written as some definite integrals and solved numerically?