Coordinates of a point on regular polygon perimeter

Question

Having an angle $\alpha$ between the $y$ axis and a line intersecting the origin, natural number $n$ being the number of sides of regular polygon, radius $R$ and assuming the bottom side of the polygon is parallel to the $x$ axis, how can I calculate coordinates of the point $I$?

Angle $\alpha$ being $15°$ in the picture is just an example.

Answer 1

With this from wikipedia

The circumradius R from the center of a regular polygon to one of the vertices is related to the side length s or to the apothem a by

$$ { R={\frac {s}{2\sin \left({\frac {\pi }{n}}\right)}}={\frac {a}{\cos \left({\frac {\pi }{n}}\right)}}} $$

you can find the length $a$ from the origin to the side of your polygon. That gives you the $y$ coordinate of $I$. For the $x$ coordinate use $\tan \alpha$.

Note that if $n$ is odd the "bottom edge" of the polygon won't be horizontal if one vertex lies on the $x$ axis.