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Breaking a polygon into monotone polygons. A simple polygon may be easily cut into monotone polygons in O(n log n) time. However, since a triangle is a monotone polygon, polygon triangulation is in fact cutting a polygon into monotone ones, and it may be performed for simple polygons in O(n) time with a complex algorithm. [6]
A simple closed polygonal chain in the plane is the boundary of a simple polygon. Often the term "polygon" is used in the meaning of "closed polygonal chain", but in some cases it is important to draw a distinction between a polygonal area and a polygonal chain. A space closed polygonal chain is also known as a skew "polygon".
Chomsky (1959) introduced the Chomsky hierarchy, in which context-sensitive grammars occur as "type 1" grammars; general noncontracting grammars do not occur. [2]Chomsky (1963) calls a noncontracting grammar a "type 1 grammar", and a context-sensitive grammar a "type 2 grammar", and by presenting a conversion from the former into the latter, proves the two weakly equivalent.
A simple polygon is monotone with respect to a line L, if any line orthogonal to L intersects the polygon at most twice. A monotone polygon can be split into two monotone chains. A polygon that is monotone with respect to the y-axis is called y-monotone. A monotone polygon with n vertices can be triangulated in O(n) time. Assuming a given ...
The notions of completely and absolutely monotone function/sequence play an important role in several areas of mathematics. For example, in classical analysis they occur in the proof of the positivity of integrals involving Bessel functions or the positivity of Cesàro means of certain Jacobi series. [ 6 ]
In geometry, a simple polygon is a polygon that does not intersect itself and has no holes. That is, it is a piecewise-linear Jordan curve consisting of finitely many line segments. These polygons include as special cases the convex polygons, star-shaped polygons, and monotone polygons.
And they’re just as mean as he was.” (Kramer gets out of the run-in by accidentally giving them the eight-finger sign—but only because he’s holding garlic powder with the other three fingers.)
The polygonal wraps, weakly simple polygons that use each given point one or more times as a vertex, include all polygonalizations and are connected by local moves. [2] Another more general class of polygons, the surrounding polygons, are simple polygons that have some of the given points as vertices and enclose all of the points. They are ...