arXiv 1801.021622018-01cs.CG

由凸多边形的$ω$ -云重构凸多边形

Reconstructing a convex polygon from its $ω$-cloud

Elena Arseneva, Prosenjit Bose, Jean-Lou De Carufel, Sander Verdonschot

备注：: 16 pages, 6 figures

摘要：

$ω$ -楔形是包含在从单个点（顶点）发出的两条射线之间的点的闭合集合，并且由角度$ω< π$分开。给定一个凸多边形$P$，我们放置$ω$ -楔形，使得$P$在楔形内，并且两条射线都与$P$相切。$ω$ -楔形的所有这样的放置的顶点位置的集合被称为$P$的$ω$ -云。我们研究重建多边形$P$从其$ω$ -云。以前的工作重建$P$从探头与$ω$ -楔需要的知识之间的相切点$P$和两个射线的$ω$ -楔除了顶点的位置。这里我们考虑仅给出最大$ω$ -云的设置。我们给出了两个条件下，它唯一地定义$P$：（i）当$ω< π$是固定/给定的，或（ii）当什么是已知的是，$ω< π/2$。我们表明，如果这两个条件都不成立，那么$P$可能不是唯一的。我们表明，当唯一性条件成立，多边形$P$可以重建$O(n)$时间与$O(1)$工作空间，除了输入，其中$n$是弧的数量在输入$ω$ -云。

英文摘要：

An $ω$-wedge is the closed set of points contained between two rays that are emanating from a single point (the apex), and are separated by an angle $ω< π$. Given a convex polygon $P$, we place the $ω$-wedge such that $P$ is inside the wedge and both rays are tangent to $P$. The set of apex positions of all such placements of the $ω$-wedge is called the $ω$-cloud of $P$. We investigate reconstructing a polygon $P$ from its $ω$-cloud. Previous work on reconstructing $P$ from probes with the $ω$-wedge required knowledge of the points of tangency between $P$ and the two rays of the $ω$-wedge in addition to the location of the apex. Here we consider the setting where the maximal $ω$-cloud alone is given. We give two conditions under which it uniquely defines $P$: (i) when $ω< π$ is fixed/given, or (ii) when what is known is that $ω< π/2$. We show that if neither of these two conditions hold, then $P$ may not be unique. We show that, when the uniqueness conditions hold, the polygon $P$ can be reconstructed in $O(n)$ time with $O(1)$ working space in addition to the input, where $n$ is the number of arcs in the input $ω$-cloud.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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英文摘要：

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