Is there a way to move the unit circle $e^{i \pi n}$ where $n \in \mathbb{R}$ along the real axis to form a sine wave?
I know I can just replace the $\Re$ part of the result with $n$ but is there a mathematical way of doing that?
Something like $ 2 n \pi + e^{ni/ \pi}$ (below) but an exact sine wave resembling $y=sin(\pi x)$? The wavelength of the graph below is 125.4...
Using complex numbers, a sine wave can be represented as
$$at+ib\sin t$$ where $a$ and $b$ are scaling factors. In terms of the complex exponential,
$$at+\frac b2(e^{it}-e^{-it}).$$
If you absolutely want to "move a circle", which I understand as "consider the trajectory of a point of a rotating circle the center of which moves horizontally" (?), you need to solve
$$x(t)+be^{it}=at+ib\sin t$$ which tells you the motion of the center,
$$x(t)=at-b\cos t.$$
Finally,
$$at-b\cos t+be^{it}$$ which does not seem any better than the very first expression.
Note that the expression
$$at+be^{it}$$ describes a trochoid, a special case of which is a cycloid.