# Functions with other functions as their $x$-axis

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.

• Are you familiar with the methods of vector calculus or linear algebra in general? – Triatticus Aug 24 '17 at 8:54
• Epicycloids and cycloidal gears may interest you. – Cauchy Aug 24 '17 at 8:59

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