Loops when drawing constantly changing angles and lines. I start by drawing a line of 1 unit on the $x$ axis. I turn left (from the perspective of an ant on the line) by an angle of $\alpha$ and I draw a second segment of length $u$ from my endpoint of the first segment. I then turn another angle $\alpha$ and then draw a $3^{rd}$ segment of length 1, etc. alternating between segments of length $1,u,1,u,...$ and rotating by a fixed angle each time. I am interested in when does this process returns to the origin and how long does it take for it to return when it does.

The loci are polygons containing start point on the circle as we know the sum of external angles $=360^0$ for a full closed regular polygon. In general it would be infinite sided irregular polygon. The "polygon" can be drawn even on BASIC.
Diameter $D$ of approximate circum-circle so formed $ =\dfrac{1+u}{2 \alpha} $, Full Arc length $=\pi D$ to return to the start point as shown.
It can be seen that when $\alpha$ is small the locus tends to a circle. 
When $n$ is integer, rotation angle at each vertex of polygon $\alpha =2\pi/(2n) $ when repeated forms a regular polygon.
Circles Tgt to X-axis
