 RegularStarPolygon - Maple Help

geometry

 RegularStarPolygon
 define a regular star polygon Calling Sequence RegularStarPolygon(p, n, cen, rad ) Parameters

 p - the name of the regular star polygon n - positive rational number > 2 cen - point which is the center of the n-gon rad - number which is the radius of the circumscribed circle of the n-gon Description

 • 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 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:

 form(p) returns the form of the geometric object (i.e., RegularStarPolygon2d if p is a regular polygon). DefinedAs(p) returns a list of vertices of p. sides(p) returns the side of p. center(p) returns the center of p. radius(p) returns the radius of the circum-circle of p. InteriorAngle(p) returns the interior angle of p. ExteriorAngle(p) returns the exterior angle of p. perimeter(p) returns the perimeter of p. area(p) returns the area of p. detail(p) returns a detailed description of the given regular polygon p.

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

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