Specific values of paramaters for which curve is closed

Question

In my study of curves, I encountered this family of parametrized curves in $ \mathbb{R}^2 $

$ \cosh(y)=-A\cos(x)+B $ for real parameters A and B such that $ 0 < |A| < 1 $

My problem is to find when the curve is closed and when it is open as I got no real idea where to start even or how to simplify the expression. Help appreciated.

Answer 1

Note that the right side is periodic in $x$, with period $2\pi$, and varies between $B - |A|$ and $B + |A|$. The left side, on the other hand, takes any value $\ge 1$. There are three main cases (I won't bother with the cases of equality):

1. $B + |A| < 1$: no real solutions.
2. $B - |A| < 1 < B + |A|$: infinitely many closed curves.

3. $1 < B -|A|$: two curves defined for all $x$.