Residue designs modulo a given number on the circle

The distribution of the total length of chords of the unit circle joining the vertices of a regular $q$-gon with numbers $k$ and $ak^{2} \ (\operatorname{mod} q)$, $k=1,2,\dots, q$, is studied in the case where $q$ is a prime and $a$ ranges over the complete residual system modulo $q$.