Polygons with Prescribed Angles in 2D and 3D

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.

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.