Polygons with Prescribed Angles in 2D and 3D

TL;DR

This paper studies the problem of constructing polygons with prescribed turning angles in 2D and 3D, providing characterizations, bounds on crossings, and efficient algorithms for realization.

Contribution

It characterizes angle sequences for polygons with minimal crossings in 2D and develops algorithms for realizing sequences in both 2D and 3D.

Findings

Characterization of angle sequences with minimal crossings in 2D.

Efficient algorithms for realizing prescribed angles in 2D and 3D.

Conditions for realizability of angle sequences in 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 \emph{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.