Guarding curvilinear art galleries with vertex

10 One of the earliest and most well known problems in computational geometry is 11 the so-called art gallery problem. The goal is to compute the minimum number of 12 guards needed to cover the interior of any polygon with n vertices; the guards are 13 to be placed on the vertices of the polygon. We consider the problem of guarding an 14 art gallery which is modeled as a polygon with curvilinear walls. Our main focus is 15 on polygons the edges of which are convex or concave arcs (but not both), named 16 piecewise-convex and piecewise-concave polygons, respectively. We prove that, in 17 the case of piecewise-convex polygons, if we only allow vertex guards, ⌊ 7 ⌋ − 1 18 guards are sometimes necessary, and ⌊ 3 ⌋ guards are always sufficient. Moreover, 19 an O(n log n) time and O(n) space algorithm is described that produces a vertex 20 guard set of size at most ⌊ 3 ⌋. When we allow point guards the afore-mentioned 21 lower bound drops to ⌊n2 ⌋. In the case of piecewise-concave polygons, we show that 22 2n−4 point guards are always sufficient and sometimes necessary, whereas it might 23 not be possible to guard such polygons by vertex guards. We conclude with bounds 24 for other types of curvilinear polygons. 25