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The intersection of two convex polygons is a convex polygon. A convex polygon may be triangulated in linear time through a fan triangulation, consisting in adding diagonals from one vertex to all other vertices. Helly's theorem: For every collection of at least three convex polygons: if all intersections of all but one polygon are nonempty ...
If every internal angle of a simple polygon is less than a straight angle (π radians or 180°), then the polygon is called convex. In contrast, an external angle (also called a turning angle or exterior angle) is an angle formed by one side of a simple polygon and a line extended from an adjacent side. [1]: pp. 261–264
All convex polygons are simple. Concave: Non-convex and simple. There is at least one interior angle greater than 180°. Star-shaped: the whole interior is visible from at least one point, without crossing any edge. The polygon must be simple, and may be convex or concave. All convex polygons are star-shaped. Self-intersecting: the boundary of ...
A non-convex regular polygon is a regular star polygon. The most common example is the pentagram, which has the same vertices as a pentagon, but connects alternating vertices. For an n-sided star polygon, the Schläfli symbol is modified to indicate the density or "starriness" m of the polygon, as {n/m}.
A p-gonal regular polygon is represented by Schläfli symbol {p}. Many sources only consider convex polygons, but star polygons, like the pentagram, when considered, can also be regular. They use the same vertices as the convex forms, but connect in an alternate connectivity which passes around the circle more than once to be completed.
One might characterise the Greek definition as follows: A regular polygon is a planar figure with all edges equal and all corners equal. A regular polyhedron is a solid (convex) figure with all faces being congruent regular polygons, the same number arranged all alike around each vertex.
In the case of genus one, a fundamental convex polygon is sought for the action by translation of Λ = Z a ⊕ Z b on R 2 = C where a and b are linearly independent over R. (After performing a real linear transformation on R 2, it can be assumed if necessary that Λ = Z 2 = Z + Z i; for a genus one Riemann surface it can be taken to have the form Λ = Z 2 = Z + Z ω, with Im ω > 0.)
A related theorem states that every simple polygon that is not a convex polygon has a mouth, a vertex whose two neighbors are the endpoints of a line segment that is otherwise entirely exterior to the polygon. The polygons that have exactly two ears and one mouth are called anthropomorphic polygons. [16]