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Point in polygon queries with respect to a monotone polygon may be answered in logarithmic time after linear time preprocessing (to find the leftmost and rightmost vertices). [1] A monotone polygon may be easily triangulated in linear time. [4] For a given set of points in the plane, a bitonic tour is a
A polygonal chain is called monotone if there is a straight line L such that every line perpendicular to L intersects the chain at most once. Every nontrivial monotone polygonal chain is open. In comparison, a monotone polygon is a polygon (a closed chain) that can be partitioned into exactly two monotone chains. [2]
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 ]
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 ...
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. [ 1 ] [ 2 ] [ 3 ] This concept first arose in calculus , and was later generalized to the more abstract setting of order theory .
A monotone polygon with respect to a line L is a polygon with the property that every line orthogonal to L intersects the polygon in a single interval. When the input is a monotone polygon, its straight skeleton can be constructed in time O(n log 2 n). [13]
A monotone planar subdivision with some monotone chains highlighted. A (vertical) monotone chain is a path such that the y-coordinate never increases along the path. A simple polygon is (vertical) monotone if it is formed by two monotone chains, with the first and last vertices in common. It is possible to add some edges to a planar subdivision ...
Monotone class theorem, in measure theory; Monotone convergence theorem, in mathematics; Monotone polygon, a property of a geometric object; Monotonic function, a property of a mathematical function; Monotonicity of entailment, a property of some logical systems; Monotonically increasing, a property of number sequence