Probing convex polygons with a wedge (Q340529)

For a given geometric probing tool, the authors study the problem of reconstruction of an object with as few probes as possible. The authors introduce a \(w\)-edge probe consisting of two arrays emanating from an apex point forming an angle \(w\) with \(0<w\leq \pi/2\). This probing tool which is inspired by the problem on enclosing triangles [\textit{P. Bose} and \textit{J.-L. De Carufel}, Comput. Geom. 47, No. 1, 90--109 (2014; Zbl 1287.65012)] generalizes earlier work [loc. cit.] and of [\textit{R. Fleischer} and \textit{Y. Wang}, Lect. Notes Comput. Sci. 5878, 255--264 (2009; Zbl 1272.68413)]. The authors present an algorithm that reconstructs a convex \(n\)-gon with all internal angles of size bigger than \(w\) using 2\(n\)-2 \(w\)-probes for \(0<w<\pi/2\). When \(w=\pi/2\) the reconstruction uses 2\(n\)-3 \(w\)-probes. Optimality in both cases is established. It is observed that the probing tool presented here can be manufactured from the tactile whisker-like sensors and does not require complicated software.