Engineers and designers use a wide variety of curve and surface formulations to describe products. The process of producing the physical shape of these products has remained essentially unchanged for many years. Traditionally, the process of finish surface machining has been error prone and inefficient due in large part to the mathematical basis used to control the positioning, orientation and movement of cutting tools in five-axis machining centers. This paper presents swept silhouette curvature matching algorithms for positioning and orienting a cutter such that tool and surface curvatures match. Formulations are given for both flat and filleted end mill cutters. The benefits of curvature matching are: reduction of local machining errors, reduction or elimination of grinding of the finished machined surface, and the improvement of machine tool efficiency. Examples are given that compare curvature matching to traditional machining methods. The paper concludes by discussing current research into a priori gouge detection methods based on intersection contact between the cutting tool and the design surface or the lower tolerance-bound offset surface to the design surface. An a priori gouge detection algorithm is necessary for the development of optimal tool motion and the reduction of time spent in tool path editing and verification. Techniques involving collinear normals, Bézier clipping, triangulation, normal intersection and swept volumes are suggested as techniques for examining the positional and translational tool gouge problem.