On the symmetry of finite sums of exponentials II

In this note we extend our study of the rich geometry of the graph of a curve defined as the weighted sum of two exponentials. Let ${\gamma }_{a,b}^s:[0,1]\to \mathbb{C}$ be defined as $${\gamma }_{a,b}^s(t)=(1-s)\exp (2\pi iat)+(1+s)\exp (2\pi ibt)$$ in which $1\le a<b$ are two positive integers and $s\in [-1,1]$. In the first part we determined the symmetry groups of the graphs of ${\gamma }_{a,b}:={\gamma }_{a,b}^0$. The main aim of this note is to study the continuous transition of the graph of the curve when $s$ changes from $-1$ to $1$. As a main result we determine the winding numbers $\mathrm{w}\mathrm{i}\mathrm{n}\mathrm{d}({\gamma }_{a,b}^s,0)$ for $s\in [-1,1]\setminus \{0\}$ as well as the set of cusp points of each such curve. This sheds further light on our initial symmetry result and provides more non-trivial albeit easy-to-state examples of advanced concepts of geometry and topology.