Method for transforming one curve around another?

Question

I'm working with a complex problem involving waveforms. Essentially I want to bend a given waveform around a circle.

At it's most basic, I want to take one curve on a linear graph and map it onto a circular graph.

I'm wondering if anyone knows what the mathematical formula is for the transformation around the circle if I know what the original curve is?

To help explain the problem I've drawn this diagram. My starting point is the top graph and my desired end point is the bottom one.

I might be thinking about the problem the wrong way, so any guidance would be appreciated. :)

Answer 1

A simple example like

$ y= A \sin ^{2} 2 \pi \,x/ \lambda + b $

And you want to wind it around a circle radius $r_1$

Angle at circle center $ \theta = x/r_1 $ then

$ r = r_1+ A \sin ^{2} 2 \pi r_1 \theta / \lambda + b $

So you basically substitute $ x = \theta r_1$ and add to the radius to be bent.

Next new $(X,Y):$

$ X = r \cos \theta, Y = r \sin \theta ..$