Polygons with Prescribed Angles in 2D and 3D

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.