TY - GEN

T1 - Polygons with Prescribed Angles in 2D and 3D

AU - Efrat, Alon

AU - Fulek, Radoslav

AU - Kobourov, Stephen

AU - Tóth, Csaba D.

N1 - Funding Information:
Research on this paper is supported, in part, by NSF grants CCF-1740858, CCF-1712119, and DMS-1839274. The full version is available at http://arxiv.org/abs/2008. 10192.
Publisher Copyright:
© 2020, Springer Nature Switzerland AG.

PY - 2020

Y1 - 2020

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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U2 - 10.1007/978-3-030-68766-3_11

DO - 10.1007/978-3-030-68766-3_11

M3 - Conference contribution

AN - SCOPUS:85102774426

SN - 9783030687656

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 135

EP - 147

BT - Graph Drawing and Network Visualization - 28th International Symposium, GD 2020, Revised Selected Papers

A2 - Auber, David

A2 - Valtr, Pavel

PB - Springer Science and Business Media Deutschland GmbH

T2 - 28th International Symposium on Graph Drawing and Network Visualization, GD 2020

Y2 - 16 September 2020 through 18 September 2020

ER -