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 = (α₀, …, α_n−1), α_i ∈ (−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.