How to Cut Corners and Get Bounded Convex Curvature

Abstract

We describe an algorithm for solving an important geometric problem arising in computer-aided manufacturing. When cutting away a region from a solid piece of material -- such as steel, wood, ceramics, or plastic -- using a rough tool in a milling machine, sharp convex corners of the region cannot be done properly, but have to be left for finer tools that are more expensive to use. We want to determine a toolpath that maximizes the use of the rough tool. In order to formulate the problem in mathematical terms, we introduce the notion of bounded convex curvature. A region of points in the plane $Q$ has \emph{bounded convex curvature} if for any point $x\in\partial Q$, there is a unit disk $U$ and $\varepsilon>0$ such that $x\in \partial U$ and all points in $U$ within distance $\varepsilon$ from $x$ are in $Q$. This translates to saying that as we traverse the boundary $\partial Q$ with the interior of $Q$ on the left side, then $\partial Q$ turns to the left with curvature at most $1$. There is no bound on the curvature where $\partial Q$ turns to the right. Given a region of points $P$ in the plane, we are now interested in computing the maximum subset $Q\subseteq P$ of bounded convex curvature. The difference in the requirement to left- and right-curvature is a natural consequence of different conditions when machining convex and concave areas of $Q$. We devise an algorithm to compute the unique maximum such set $Q$, when the boundary of $P$ consists of $n$ line segments and circular arcs of arbitrary radii. In the general case where $P$ may have holes, the algorithm runs in time $O(n^2)$ and uses $O(n)$ space. If $P$ is simply-connected, we describe a faster $O(n\log n)$ time algorithm.