"Special issue on Selected papers from the Twenty-eighth International Symposium on Graph Drawing and Network Visualization, GD 2020"
Polygons with Prescribed Angles in 2D and 3D
Vol. 26, no. 3, pp. 363-380, 2022. Regular paper.
Abstract We consider the construction of a polygon $P$ with $n$ vertices whose turning angles at the vertices are given by a sequence $A=(\alpha_0,\ldots, \alpha_{n-1})$, $\alpha_i\in (-\pi,\pi)$, for $i\in\{0,\ldots, n-1\}$. 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 $P\subset \mathbb{R}^2$ realizing $A$ has at least $c$ crossings, for every $c\in \mathbb{N}$, and describe an efficient algorithm that constructs, for a given sequence $A$, a generic polygon $P\subset \mathbb{R}^2$ 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 $P\subset \mathbb{R}^3$, and for every realizable sequence the algorithm finds a realization.

 This work is licensed under the terms of the CC-BY license.
Submitted: October 2020.
Reviewed: April 2022.
Revised: April 2022.
Accepted: May 2022.
Final: May 2022.
Published: June 2022.
Communicated by David Auber and Pavel Valtr
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