Epicycloid: How do I get the characteristic equation given a picture and specific points I am simulating a process and the resulting line has the shape of an epicyloid. the epicycloid- like shape of the line For my next steps I need an equation to approach this shape. I got the paramatic equation of an epicycloid: x = (r+R) _cos(t)-a_ cos((1+R/r)*t); y = (r+R) _sin(t)-a_ sin((1+R/r)*t); And I got the specific points. So i got the size of x and y at given t. My consideration was to take 3 specific points in order to solve 3 equations with 3 unknown variables and define R, r and a in this way. But I can't find a way to solve the equations. Do one of you got an idea how to get the unknown Parameters ?

You are talking about prolate epicycloid. If you take function $f(t) = x^2 + y^2$:

$$ f(t) = a^2 +(R+r)^2 - 2a(R+r)\cos\left(\frac Rrt\right) = A - B\cos \omega t $$

You can find the period of the function, maximal and minimal value of the function this will give you all the parameters.

In other words, you find the angle $\theta$ between to successive points closest to origin, you find the minimal and maximal distances to origin $D$ and $d$:

$$ R/r = 2\pi/\theta,\qquad D = a+R+r,\qquad d = R+r-a,\\\ a = \frac{D+d}2,\qquad R= \frac{D-d}{2+\theta/\pi},\qquad r=\frac{D-d}{4\pi/\theta+2} $$