Computing the Tool Path of an Externally Monotone Polygon in Linear Time

A Numerically-Controlled (NC) machine typically consists of a worktable and a spindle (or cutter) with several axes of freedom for positioning the tool. In this paper, we restrict our attention to machines having only translational freedom. We focus on 2D milling, which can only cut out planar objects, and 2 1 2D milling where only two of the axes are continuous-path controlled and the third axis is point-to-point or straight-line controlled. More than 80% of all mechanical parts to be machined can be cut by applying 2D or 2 2D for path control [11]. We study contour-parallel milling where a pocket is machined by having a cutter following paths that are equidistant offset curves from the boundary of the object. Although there are many types of cutters, the most common is the ball-end cutter. Such a cutter removes a disc whose radius is the radius of the ball. We focus on the following basic problem: given an object, modelled by a simple polygon on n vertices, and the radius of the ball-end cutter, compute the boundary or outer tool path for the cutter and the complete tool path for the cutter. In Held [11], it is shown how to compute both these tool paths by using the medial axis (see [2]). The medial axis can be computed fairly easily in O(n log n) time. In the specific case of a simple polygon, it can even be computed in O(n) time [6]. However, the O(n) time algorithm is of theoretical interest only since it is quite complex and also uses Chazelle’s [4] linear–time triangulation algorithm which in itself is extremely complex. The challenge is to simply and efficiently compute both the outer and complete tool path of a simple polygon in O(n) time. In this paper, we demonstrate a simple and efficient method to compute the outer and complete tool path of a fairly general class of polygons called externally monotone. A simple polygon is externally monotone if for every point inside a pocket, there is a path to the lid of the pocket that is monotone in the direction normal to the lid. To date, this is the most general class of polygons for which a simple and efficient linear time algorithm to compute the tool path is known. In Section 2, we review notation and preliminaries related to the results. In Section 3, we show how to com-