# How to formulate the convex hull which is a regular polygon on the complex plane

Suppose that I have a convex regular polygon with $$k$$ vertices on the complex plane, and the first vertex lies on the positive real axis. Is there a neat way to formulate the convex hull with the shape of regular polygon and make it acceptable for CVX solver?
I could take the intersection of the half-spaces of each side of the polygon, but it seems to be a little bit complicated, when $$k$$ grows large.

• Do you have a convex polygon or you just have its vertices? Are you vertices equidistant from (0, 0)? Oct 17, 2023 at 12:54
• Thanks for the comment. I have a convex polygon. Suppose the radius is $r$, and the first vertex will be at $(r,0)$. Right now I am using the intersection of halfplanes of each side of the polygon to represent the convex hull. As the $K$ normally is not that large, maybe it's fine that I continue to use it. Oct 17, 2023 at 13:52
• I don't understand, if you already have a convex polygon why do you need to compute its convex hull? Oct 17, 2023 at 14:31
• Also, since you have the number of vertices and all lie at a distance $r$ from the origin and know the position for the first one, you can determine analytically all the positions and connectivities. Oct 17, 2023 at 14:44
• Maybe I didn't demonstrate my problem quite well. Yes, I can use the halfplanes of each side to express all the points that are in the interior and on the boundary of this polygon. I am just wondering whether there is a simpler way to formulate it. Sorry for not state my problem clearly. Oct 17, 2023 at 23:52

Suppose you know that your boundary points are at a distance $$r$$ from the origin in the plane. In that case, you can filter them according to their distance from $$(0, 0)$$ and join them according to the angle with the horizontal — that you can compute using the function $$\operatorname{atan2}$$.