Say we have the curve $$\vec r(t)=(at-a\sin t,a+a\cos t)$$ for $a$ a positive constant, how do we find it's explicit equation? I've tried plotting it on desmos and I got this (choosing $a=1$):
Which if you find the points on the extremes and on the minima and got $(0,2a),(\pi a,0),(2\pi a,2a)$ which has all the characteristics of a periodic equation but I don't know how to find it, is it even possible?
(Note: by equation I mean the form $f(x,y)=0$)