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.

中文翻译：

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具有指定角度的 2D 和 3D 多边形

我们考虑构建具有 $n$ 个顶点的多边形 $P$，其顶点处的转角由序列 $A=(\alpha_0,\ldots, \alpha_{n-1})$, $\alpha_i\在 (-\pi,\pi)$ 中，对于 $i\in\{0,\ldots, n-1\}$。通过多边形实现 $A$ 的问题可以看作是在顶点处构造具有指定角度的直线图的问题，因此，它是构建\emph{角度图}。在 2D 中，我们表征序列 $A$，其中每个实现 $A$ 的通用多边形 $P\subset\mathbb{R}^2$ 至少有 $c$ 个交叉点，对于每个 $c\in\mathbb{N}$ ，并描述了一种有效的算法，该算法为给定的序列 $A$ 构造一个通用多边形 $P\subset \mathbb{R}^2$，该算法以最少的交叉次数实现 $A$。在 3D 中，