Generating a Simple Polygonalizations

Muravitskiy, V.; Tereshchenko, Vasyl

doi:10.1109/iv.2011.88

Cited by 7 publications

(5 citation statements)

References 18 publications

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“…It is perhaps worth noting that, for a related problem on computing the exact arbitrarily-oriented minimum-volume bounding box of a set of n points in R 3 , initially O'Rourke presented an algorithm [17] that runs in O(n 3 ) time (subsequently, improvement in complexity has been attained [18]). Also, for general point sets, as constant factor approximation has proven to be difficult even for a 2D version (minimal area polygon) [7], it is likely that the problems MINVP/MAXVP could end up in similar fate. Two potential applications (4D printing and surface lofting) of volume constrained polyhedronizations have been discussed.…”

Section: Discussionmentioning

confidence: 99%

“…As a consequence, minimal volume polyhedronization (correspondingly maximum volume polyhedronization) of any point set of appreciable size becomes an extremely hard one. The two-dimensional version of the problem (minimum area and maximum area polygons) is already shown to be NP-complete [6] and an infeasibility of a constant factor approximation algorithm is mentioned in [7]. Being a direct extended version of minimal area (maximal area) polygonization, it is most likely that minimum volume (maximum volume) polyhedronization could also end up in a similar fate.…”

Section: Theoretical Challengesmentioning

confidence: 99%

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A Randomized Approach to Volume Constrained Polyhedronization Problem

Peethambaran

Parakkat

Muthuganapathy

2015

Journal of Computing and Information Science in Engineering

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Given a finite set of points in R3, polyhedronization deals with constructing a simple polyhedron such that the vertices of the polyhedron are precisely the given points. In this paper, we present randomized approximation algorithms for minimal volume polyhedronization (MINVP) and maximal volume polyhedronization (MAXVP) of three dimensional point sets in general position. Both, MINVP and MAXVP, problems have been shown to be NP-hard and to the best of our knowledge, no practical algorithms exist to solve these problems. It has been shown that for any point set S in R3, there always exists a tetrahedralizable polyhedronization of S. We exploit this fact to develop a greedy heuristic for MINVP and MAXVP constructions. Further, we present an empirical analysis on the quality of the approximation results of some well defined point sets. The algorithms have been validated by comparing the results with the optimal results generated by an exhaustive searching (brute force) method for MINVP and MAXVP for some well chosen point sets of smaller sizes. Finally, potential applications of minimum and maximum volume polyhedra in 4D printing and surface lofting, respectively, have been discussed.

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Section: Discussionmentioning

confidence: 99%

Section: Theoretical Challengesmentioning

confidence: 99%

A Randomized Approach to Volume Constrained Polyhedronization Problem

Peethambaran

Parakkat

Muthuganapathy

2015

Journal of Computing and Information Science in Engineering

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“…The number of simple polygons constructed by the polar coordinate sorting and dichotomy sorting methods is unique, while the number of simple polygons constructed by the convexhull construction method [7] is non-unique. However, the latter cannot generate a simple polygon using all the given random points as polygon nodes.…”

Section: Comparative Experimentsmentioning

confidence: 99%

“…Currently, several methods are being used to construct a simple polygons from a set of discrete points, such as the polar coordinate sorting [5], dichotomy sorting [6], convex-hull construction [7], and visual region [8] methods. In the dichotomy sorting method [5], two outer points are determined from the maximum and minimum values of x-axis (or y-axis) projection.…”

Section: Introductionmentioning

confidence: 99%

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Generation of simple polygons from ordered points using an iterative insertion algorithm

Zhang

Zhao

2020

PLoS ONE

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To construct a simple polygon from a set of plane points, we propose an iterative inserting ordered points (IIOP) algorithm. Using a given a set of ordered non-collinear points, a simple polygon can be formed and its shape is dependent on the sorting method used. To form such simple polygons with a given set of plane points, the points must first be ordered in one direction (typically, the x-axis is used). The first three points in the set are used to form an initial polygon. Based on the formed polygon, new polygons can be iteratively formed by inserting the first point of from among the remaining set of points, depending on line visibility from that point. This process is carried out until all the points are inserted into the polygon. In this study, we generated 20, 50, and 80 plane points and used the proposed method to construct polygons. Experimental results show that these three polygons are all simple polygons. Through theoretical and experimental verification, we can concluded that when given a set of non-collinear points, a simple polygon can be formed.

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