Functions with other functions as their $x$-axis

Question

I had this interesting thought today as I was playing with making images with mathematical expressions (vector images with graphing if you like) I wanted to to put a spiky thing (the sine curve) around a circle (to make a sun with its rays). However, I have no idea on how to wrap the sine function around the circle (that is it would look like if the x-axis was arranged into a circle and the sine function was graphed on this bent axis).

As I thought about this more, I even began to wonder how one would "Wrap" any arbitrary function/relation around another one, as it definitely seems possible, but it seems very hard to write down (I can easily draw a sine curve in a circle fashion, but I cannot write it's equation). It seems like, if this is possible, it would produce some very interesting curves. I tried to research but I couldn't even describe it properly, hence why I had to describe it with the x-axis.

Answer 1

For the circle problem: if you use polar coordinates, then wrapping a sinus around the circle means varying the radius with the sin function. So a curve like $$( (R + \sin(\theta)\cdot \cos \theta), (R+\sin\theta)\cdot \sin \theta) $$ might be a solution.

Wrapping a function around a curve might mean translating the point of the curve by a given amount. Now, the direction of the translation matters. In the example above, it was perpendicular to the curve at each point. You may want to look at épicycloïdes and related matters.

Answer 2

Hint.

For a parametrically defined curve $(f_x(t),f_y(t))$, a parallel curve $(F_x[f_x,f_y],F_y[f_x,f_y])$ with distance $a$ is defined as

\begin{align} F_x[f_x,f_y]&=f_x+{\frac {af_y'}{\sqrt {{f_x'}^{2}+{f_y'}^{2}}}} \\ F_y[f_x,f_y]&=f_y-{\frac {af_x'}{\sqrt {{f_x'}^{2}+{f_y'}^{2}}}}. \end{align}

In other words this is a way to wrap a constant function $g(x)=a$ along the curve $(f_x(t),f_y(t))$.