Guarding curvilinear art galleries with vertex or point guards

We study a variant of the classical art gallery problem, where an art gallery is modeled by a polygon with curvilinear sides. We focus on piecewise-convex and piecewise-concave polygons, which are polygons whose sides are convex and concave arcs, respectively. It is shown that for monitoring a piecewise-convex polygon with n⩾2 vertices, vertex guards are always sufficient and sometimes necessary. We also present an algorithm for computing at most vertex guards in O(nlogn) time and O(n) space. For the number of point guards that can be stationed at any point in the polygon, our upper bound carries over and we prove a lower bound of . For monitoring a piecewise-concave polygon with n⩾3 vertices, 2n−4 point guards are always sufficient and sometimes necessary, whereas there are piecewise-concave polygons where some points in the interior are hidden from all vertices, hence they cannot be monitored by vertex guards. We conclude with bounds for some special types of curvilinear polygons.