Say that we have a circle in the $xy$ plane with radius $r$ and center $(C_x,C_y)$ and a point given by $(p_x,p_y)$. I was able to deduce that this point can be brought onto the surface of the circle by making its coordinates depend on the parameter $t$ where $t\ge0$. The coordinates of the point with $t$ dependence are rewritten as:
$(p_x + sign(C_x - p_x)(-\frac{1}{(t + \frac{1}{\sqrt L})^2}+L) , p_y + sign(C_y - p_y)sign(\tan(a-\pi))\tan(a-\pi)(-\frac{1}{(t + \frac{1}{\sqrt L})^2}+L))$
where $a = \arctan(\frac{C_y - p_y}{C_x - p_x})$ and $L = (\sqrt{(C_x - p_x)^2+(C_y - p_y)^2} - r)\cos(a)$.
Now this is all fine and interesting enough, but I reasoned that since any given graph of a function $f(x)$ is just a set of points, perhaps it is possible to perform the operations described above on every point of $f(x)$ and deform it into all or part of the circle. I would just like to know if such a thing is even possible, since I found that most of my attempts to do so were in vain. If such a thing is possible I would like to know what path in general to take in order to arrive at the result, as I feel like a full blown explanation of how to get there would ruin the journey.