Polygons with Prescribed Angles in 2D and 3D
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We consider the construction of a polygon P with n vertices whose turning angles at the vertices are given by a sequence A=(α_0,…, 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⊂ℝ^2 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⊂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⊂ℝ^3, and for every realizable sequence the algorithm finds a realization.
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