Finding the Maximum Subset with Bounded Convex Curvature

We describe an algorithm for solving an important geometric problem arising in computer-aided manufacturing. When machining a pocket in a solid piece of material such as steel using a rough tool in a milling machine, sharp convex corners of the pocket cannot be done properly, but have to be left for finer tools that are more expensive to use. We want to determine a tool path that maximizes the use of the rough tool. Mathematically, this boils down to the following problem. Given a simply-connected set of points $P$ in the plane such that the boundary $\partial P$ is a curvilinear polygon consisting of $n$ line segments and circular arcs of arbitrary radii, compute the maximum subset $Q\subseteq P$ consisting of simply-connected sets where the boundary of each set is a curve with bounded convex curvature. A closed curve has bounded convex curvature if, when traversed in counterclockwise direction, it turns to the left with curvature at most $1$. There is no bound on the curvature where it turns to the right. The difference in the requirement to left- and right-curvature is a natural consequence of different conditions when machining convex and concave areas of the pocket. We devise an algorithm to compute the unique maximum such set $Q$. The algorithm runs in $O(n\log n)$ time and uses $O(n)$ space. For the correctness of our algorithm, we prove a new generalization of the Pestov-Ionin Theorem. This is needed to show that the output $Q$ of our algorithm is indeed maximum in the sense that if $Q'$ is any subset of $P$ with a boundary of bounded convex curvature, then $Q'\subseteq Q$.