These curves are arcs of epicycloids. In particular, the one tangent to the vertical line is half of the [cardioid](https://en.wikipedia.org/wiki/Cardioid). If, for a positive integer $k$, you draw a line from $\alpha$ to $k\alpha$ modulo $2\pi$ for all $\alpha$, the envelope will be the complete epicycloid. (For $k$ negative, you get hypocycloids.)

This observation is due to Luigi Cremona.