Topic
Regular polygon
About: Regular polygon is a research topic. Over the lifetime, 4495 publications have been published within this topic receiving 68009 citations.
Papers published on a yearly basis
Papers
More filters
TL;DR: It is shown by mathematical means that both concepts for solving the point in polygon problem for arbitrary polygons are very closely related, thereby developing a first version of an algorithm for determining the winding number.
Abstract: A detailed discussion of the point in polygon problem for arbitrary polygons is given. Two concepts for solving this problem are known in literature: the even–odd ruleand the winding number, the former leading to ray-crossing, the latter to angle summationalgorithms. First we show by mathematical means that both concepts are very closely related, thereby developing a first version of an algorithm for determining the winding number. Then we examine how to accelerate this algorithm and how to handle special cases. Furthermore we compare these algorithms with those found in literature and discuss the results. 2001 Elsevier Science B.V. All rights reserved.
309 citations
305 citations
301 citations
[...]
01 Jan 2001
TL;DR: A mathematical analysis of the circle is one of the oldest challenges to have faced mathematicians as discussed by the authors, and statements as to how the circumference or the area of a circle can be expressed through other variables are already found in the oldest mathematical documents
Abstract: Mathematical analysis of the circle is one of the oldest challenges to have faced mathematicians Statements as to how the circumference or the area of a circle can be expressed through other variables are already found in the oldest mathematical documents
295 citations
03 Nov 1982
TL;DR: It is proved that it is possible, in O(N) time, to find two vertices a,b in P, such that the segment ab lies entirely inside the polygon P and partitions it into two polygons, each with a weight not exceeding 2C/3.
Abstract: Let P be a simple polygon with N vertices, each being assigned a weight ∈ {0,1}, and let C, the weight of P, be the added weight of all vertices. We prove that it is possible, in O(N) time, to find two vertices a,b in P, such that the segment ab lies entirely inside the polygon P and partitions it into two polygons, each with a weight not exceeding 2C/3. This computation assumes that all the vertices have been sorted along some axis, which can be done in O(Nlog N) time. We use this result to derive a number of efficient divide-and-conquer algorithms for: 1. Triangulating an N-gon in O(Nlog N) time. 2. Decomposing an N-gon into (few) convex pieces in O(Nlog N) time. 3. Given an O(Nlog N) preprocessing, computing the shortest distance between two arbitrary points inside an N-gon (i.e., the internal distance), in O(N) time. 4. Computing the longest internal path in an N-gon in O(N2) time. In all cases, the algorithms achieve significant improvements over previously known methods, either by displaying better performance or by gaining in simplicity. In particular, the best algorithms for Problems 2,3,4, known so far, performed respectively in O(N2), O(N2), and O(N4) time.
294 citations