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In geometry, a star-shaped polygon is a polygonal region in the plane that is a star domain, that is, a polygon that contains a point from which the entire polygon boundary is visible. Formally, a polygon P is star-shaped if there exists a point z such that for each point p of P the segment ¯ lies entirely within P. [1]
A regular star polygon is a self-intersecting, equilateral, and equiangular polygon. A regular star polygon is denoted by its Schläfli symbol { p / q }, where p (the number of vertices) and q (the density ) are relatively prime (they share no factors) and where q ≥ 2.
A star domain (equivalently, a star-convex or star-shaped set) is not necessarily convex in the ordinary sense. An annulus is not a star domain.. In geometry, a set in the Euclidean space is called a star domain (or star-convex set, star-shaped set [1] or radially convex set) if there exists an such that for all , the line segment from to lies in .
All convex polygons are star-shaped. Self-intersecting: the boundary of the polygon crosses itself. The term complex is sometimes used in contrast to simple, but this usage risks confusion with the idea of a complex polygon as one which exists in the complex Hilbert plane consisting of two complex dimensions. Star polygon: a polygon which self ...
As well as star-shaped polygonalizations, every non-collinear set of points has a polygonalization that is a monotone polygon. This means that, with respect to some straight line (which may be taken as the x {\displaystyle x} -axis) every perpendicular line to the reference line intersects the polygon in a single interval, or not at all.
For the same choice of a pivot point for the sorting algorithm, connecting all of the other points in their sorted order around this point rather than performing the remaining steps of the Graham scan produces a star-shaped polygon, a polygonalization of the input. [4]
This is the smallest star polygon that can be drawn in two forms, as irreducible fractions. The two heptagrams are sometimes called the heptagram (for {7/2}) and the great heptagram (for {7/3}). The previous one, the regular hexagram {6/2}, is a compound of two triangles. The smallest star polygon is the {5/2} pentagram.
In geometry, a polygon is traditionally a plane figure that is bounded by a finite chain of straight line segments closing in a loop to form a closed chain. These segments are called its edges or sides, and the points where two of the edges meet are the polygon's vertices (singular: vertex) or corners.