Something I am particularly interested in is finding a potential way to create an animation illustrating smoothly how points in one 2D space map to another. In particular, I would like to show students how points in Cartesian coordinates become mapped in polar coordinates.
I'd love to be able to recreate this animation here, which illustrates how the curve $y = \sin 6x+2$ becomes transformed to $x = \sin 6y + 2$, and then to the polar equation $r = \sin6\theta + 2$.
I know how to map the points from the second form into the third form mathematically, but what I'd like to be able to do is mathematically derive the result of applying the homeomorphism part-way through, as between the start and end points of the animation. Is it possible to parametrise a "semi-image" under the transformation from Cartesian to polar with some parameter, say $t\in [0,1]$?
I will admit homeomorphisms isn't something I know very much about, so any general ideas would be much appreciated.