TY - JOUR

T1 - Polygons with prescribed angles in 2D and 3D

AU - Efrat, Alon

AU - Fulek, Radoslav

AU - Kobourov, Stephen

AU - Tóth, Csaba D.

N1 - Publisher Copyright:
Copyright © 2020, The Authors. All rights reserved.
Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2020/8/24

Y1 - 2020/8/24

N2 - We consider the construction of a polygon P with n vertices whose turning angles at the vertices are given by a sequence A = (α0, . . ., αn−1), αi ∈ (−π, π), for i ∈ {0, . . ., n− 1}. The problem of realizing A by a polygon can be seen as that of constructing a straight-line drawing of a graph with prescribed angles at vertices, and hence, it is a special case of the well studied problem of constructing an angle graph. In 2D, we characterize sequences A for which every generic polygon P ⊂ R2 realizing A has at least c crossings, for every c ∈ N, and describe an efficient algorithm that constructs, for a given sequence A, a generic polygon P ⊂ R2 that realizes A with the minimum number of crossings. In 3D, we describe an efficient algorithm that tests whether a given sequence A can be realized by a (not necessarily generic) polygon P ⊂ R3, and for every realizable sequence the algorithm finds a realization.

AB - We consider the construction of a polygon P with n vertices whose turning angles at the vertices are given by a sequence A = (α0, . . ., αn−1), αi ∈ (−π, π), for i ∈ {0, . . ., n− 1}. The problem of realizing A by a polygon can be seen as that of constructing a straight-line drawing of a graph with prescribed angles at vertices, and hence, it is a special case of the well studied problem of constructing an angle graph. In 2D, we characterize sequences A for which every generic polygon P ⊂ R2 realizing A has at least c crossings, for every c ∈ N, and describe an efficient algorithm that constructs, for a given sequence A, a generic polygon P ⊂ R2 that realizes A with the minimum number of crossings. In 3D, we describe an efficient algorithm that tests whether a given sequence A can be realized by a (not necessarily generic) polygon P ⊂ R3, and for every realizable sequence the algorithm finds a realization.

KW - Angle graph

KW - Crossing number

KW - Polygon

KW - Spherical polygon

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M3 - Article

AN - SCOPUS:85095555740

JO - Nuclear Physics A

JF - Nuclear Physics A

SN - 0375-9474

ER -