I "eyeballed" the problem.
I think I have the solution.
Let be the graph of the parametric equations: .
What is the length of the smallest interval such that the graph
of the equations for all produces the entire graph of ?
At the "start", the graph is at
Here is what I found:
. . and the cycle repeats.
If a given parametric curve is periodic, finding its period is a mystery to me. Here's a problem that I tackled some time ago with its solution; you might try your hand at it (my proof was somewhat long and involved). I wish I knew some general procedure to find the period of such periodic functions.