Let S be a rotation through angle $\frac{2\mathrm{\pi }}{n}$, and let A_0 be any point not on the axis of S. Then the points $A_i={A_0}^{S^i},i=...,-2,-1,0,1,2,...$ are the vertices of a regular polygon n whose sides are the segments A_0A_1, A_1A_2, ...

When n is an integer (greater than 2) this definition is equivalent to that given for regular polyhedra. But the polygon can be closed without n being integral; it is merely necessary that the period of S to be finite, i.e., that n be rational. We still stipulate that $2<n$ since a positive rotation through an angle greater than Pi is the same as a negative rotation through an angle less than Pi.

To access the information relating to a regular star polygon p, use the following function calls:

The command with(geometry,RegularStarPolygon) allows the use of the abbreviated form of this command.

$\mathrm{with}\left(\mathrm{geometry}\right)\&colon;$

$\mathrm{RegularStarPolygon}\left(\mathrm{gon}\&comma;\frac{5}{2}\&comma;\mathrm{point}\left(o\&comma;0\&comma;0\right)\&comma;1\right)$

$\mathrm{gon}$

$\mathrm{detail}\left(\mathrm{gon}\right)$

$\begin{array}{ll}\text{name of the object}& \mathrm{gon}\\ \text{form of the object}& \mathrm{RegularStarPolygon2d}\\ \text{the side of the polygon}& \frac{2}{\csc \left(\frac{2\mathrm{\pi }}{5}\right)}\\ \text{the center of the polygon}& \left[0\&comma;0\right]\\ \text{the radius of the circum-circle}& 1\\ \text{the interior angle}& \frac{\mathrm{\pi }}{5}\\ \text{the exterior angle}& \frac{4\mathrm{\pi }}{5}\\ \text{the perimeter}& \frac{10}{\csc \left(\frac{2\mathrm{\pi }}{5}\right)}\\ \text{the area}& \frac{5\sin \left(\frac{2\mathrm{\pi }}{5}\right)\cos \left(\frac{2\mathrm{\pi }}{5}\right)}{2}\\ \text{the vertices of the polygon}& \left[\left[1\&comma;0\right]\&comma;\left[-\cos \left(\frac{\mathrm{\pi }}{5}\right)\&comma;\sin \left(\frac{\mathrm{\pi }}{5}\right)\right]\&comma;\left[\cos \left(\frac{2\mathrm{\pi }}{5}\right)\&comma;-\sin \left(\frac{2\mathrm{\pi }}{5}\right)\right]\&comma;\left[\cos \left(\frac{2\mathrm{\pi }}{5}\right)\&comma;\sin \left(\frac{2\mathrm{\pi }}{5}\right)\right]\&comma;\left[-\cos \left(\frac{\mathrm{\pi }}{5}\right)\&comma;-\sin \left(\frac{\mathrm{\pi }}{5}\right)\right]\right]\end{array}$