The numerical control (NC) community lacks a tradition of publication and the concomitant striving for excellence. Practitioners, generally from an engineering background, have developed algorithms that operate satisfactorily under a large range of circumstances. A major requirement for computer-aided manufacturing (CAM) programs that drive milling machines is the ability to remove metal from bounded areas (possibly containing islands). This task is variously known as “pocket machining,” “pocket milling,” or “area clearance.”

Two major methods of pocket milling have been developed: boundary parallel machining and axis parallel machining. In the first method, a series of passes are made parallel to the boundary, starting from some point inside the boundary and gradually moving outwards until the boundary is reached. In the second method, all the cutting moves are either parallel to an axis or are moves from one of these paraxial passes to the next. Axis parallel machining is normally generalized to allow the parallel passes to lie in any direction.

This book develops theory and algorithms for pocket machining with the rigor that is now becoming commonplace in computational geometry literature. The book is aimed both at computational geometers who have an academic interest in the pocket machining problem and at CAM system developers who wish to implement algorithms with formally demonstrable capabilities.

The book uses the term “contour parallel” and “direction parallel” for the two methods of pocket machining. Approximately two thirds of the book is devoted to contour parallel milling, the harder problem, and one third addresses direction parallel milling.

The basis of the method for producing a contour parallel cutter path comes from the work of Persson [1], which is acknowledged. Many readers may be familiar with the Voronoi diagram of a set of points in the plane: the division of the plane into disjoint regions such that all points within one region are closer to one of the given points than to any other. What is not so well known is the extension to include other bounded geometric elements as well as points. This book considers boundaries composed of straight line segments and circular arcs meeting at points (the great majority of pockets encountered in practice do have such boundaries, and more complex curves can be approximated). The Voronoi diagram of a boundary divides the interior of the boundary into regions such that points within a region are closer to one boundary element than to any other element. Given the Voronoi diagram, the generation of each offset cutter path becomes relatively straightforward. The concepts of Voronoi diagrams and pockets are developed completely formally, and several theorems are proven that form the basis of the algorithms. Held presents the algorithms for generating the Voronoi diagram and producing the contour parallel cutter path (with the necessary extensions to handle islands and bottlenecks).

The method for generating a direction parallel cutter path is split into two parts. First all the endpoints of the straight line passes are found, and then the points are linked together to form the cutter path. To determine all the endpoints, a sweep line algorithm is used. Initially all local maxima and minima in the direction at right angles to the passes are found. The sweep line algorithm then generates all the additional pass endpoints; the interaction of the sweep line with the pocket is updated only at the maximum and minimum points. The second stage is to construct a tour of the pass endpoints and maxima and minima that joins the correct pass endpoints to produce the direction parallel passes; moves along the contour, where possible, from one pass to the next; and only lifts the cutter where absolutely necessary.

The book is a revised version of the author’s Ph.D. thesis. Its purpose is to present pocket machining algorithms for the two main approaches--boundary parallel and axis parallel machining--with a sound theoretical basis. It is not intended to be a textbook. By drawing on the established tradition of publication in computational geometry, it succeeds in presenting algorithms for CAM applications in a formal mathematical manner.

The book is compact. It does contain overview sections, but the reader is quickly taken into areas that are definitely not for the dilettante. The book suffers slightly from stilted English, but as one moves from the largely textual to the largely mathematical and algorithmic sections this problem almost disappears. The mathematical concepts are presented formally.

This book is the result of combining algorithmic ideas developed within the NC community with the form of rigorous presentation now expected in computational geometry. The formalism engendered by this ancestry may put some readers off, but even for them, the algorithms are presented in a form that makes implementation reasonably straightforward. The reference list is extensive but reflects the theoretical aspects of the book. The book has no index, but this omission is not unusual in a text of this form.

Held has taken some poorly defined notions from the CAM practitioners and elevated them to a position of sound theoretical standing. This foundation provides many opportunities for development, both academic and practical.