用楔形探测凸多边形
Probing Convex Polygons with a Wedge
摘要：
最大限度地减少探测器的数量是利用探测设备重建几何物体的主要挑战之一。在本文中，我们研究的问题，使用$ω$ -楔形探测工具，以确定准确的形状和方向的凸多边形。一个$ω$ -楔由两条射线从一个点发出称为顶点的楔和两条射线形成一个角度$ω$。要使用$ω$ -wedge进行探测，我们设置探测器顶点必须遵循的方向、线$\overrightarrow L$和两条射线的初始方向。凸多边形$O$的有效$ω$ -探测器包含$O$-楔形内的$O$，其结果包括顶点的坐标、两条射线的方向以及$O$和每条射线之间的最近（到顶点）接触点的坐标。我们提出的算法最小化的探测器的数量，并证明其最优性。特别是，我们展示了如何重建凸$n$ -gon（与所有内角的大小大于$ω$）使用$2n-2$ $ω$ -探针;如果$ω= π/2$，重建使用$2n-3$ $ω$ -探针。我们证明了这两个结果都是最优的。令$N_B$是$O$的内角至多为$ω$的顶点数，（我们证明了$0 \leq N_B \leq 3$）。我们使用$2n-1$（分别为$2n+3$，$2n+5$）$ω$ -探针，确定了具有$N_B=1$（分别为$N_B=2$，$N_B=3$）的一般凸$n$ -gon的形状和方向。我们证明了第一种情况下的最优性。假设算法预先知道$N_B$的值，则可以用$2n+2$探针实现用$N_B=2$或$N_B=3$重建$O$，这是最优的。
英文摘要：
Minimizing the number of probes is one of the main challenges in reconstructing geometric objects with probing devices. In this paper, we investigate the problem of using an $ω$-wedge probing tool to determine the exact shape and orientation of a convex polygon. An $ω$-wedge consists of two rays emanating from a point called the apex of the wedge and the two rays forming an angle $ω$. To probe with an $ω$-wedge, we set the direction that the apex of the probe has to follow, the line $\overrightarrow L$, and the initial orientation of the two rays. A valid $ω$-probe of a convex polygon $O$ contains $O$ within the $ω$-wedge and its outcome consists of the coordinates of the apex, the orientation of both rays and the coordinates of the closest (to the apex) points of contact between $O$ and each of the rays. We present algorithms minimizing the number of probes and prove their optimality. In particular, we show how to reconstruct a convex $n$-gon (with all internal angles of size larger than $ω$) using $2n-2$ $ω$-probes; if $ω= π/2$, the reconstruction uses $2n-3$ $ω$-probes. We show that both results are optimal. Let $N_B$ be the number of vertices of $O$ whose internal angle is at most $ω$, (we show that $0 \leq N_B \leq 3$). We determine the shape and orientation of a general convex $n$-gon with $N_B=1$ (respectively $N_B=2$, $N_B=3$) using $2n-1$ (respectively $2n+3$, $2n+5$) $ω$-probes. We prove optimality for the first case. Assuming the algorithm knows the value of $N_B$ in advance, the reconstruction of $O$ with $N_B=2$ or $N_B=3$ can be achieved with $2n+2$ probes,- which is optimal.
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