Abstract: Functions in various kinds of forms, whether to be rational functions or not, to express a plurality of curved surfaces are unified into a set of functions with respect to parameters u and v by an algebraic method or an analytical method. Based on the set of functions, intersections which are necessary for machining of the curved surfaces are calculated, and paths of a cutting tool to machine the curved surfaces are determined.

Abstract: Data about a plurality of curved surfaces which are expressed in various forms are inputted, and based on the inputted data, the curved surfaces are defined by a set of rational functions with respect to parameters u and v. Then, calculation for intersections which are necessary for machining is carried out, and paths of a cutting tool are calculated. In figuring out the solution of the set of rational functions, first, the extremal value and the point of inflection are calculated, and the solution is calculated by an analytically focusing method with the initial value set between the extremal value and the point of inflection. When a plurality of curved surfaces are expressed in the same form, the curved surfaces are unified into a single surface.

Abstract: A plurality of curved surfaces which have distinct characteristics are defined as a unified curved surface. The definition is made by a group of polynomials with respect to parameters u and v in two directions reflecting the characteristics of the curved surfaces. Then, paths of a cutting tool to machine the unified curved surface are generated by using the polynomials. Further, a curved surface is divided by a curve into two regions, and a point is designated in one of the regions. Only with respect to the region which includes the designated point, paths of the cutting tool are generated.

Abstract: A three-dimensional machining method in which paths of the center of a machine tool are calculated from at least one defined curve to machine a surface which is smooth and includes the defined curve. For continuous machining of a first surface and a second surface, the intersection of a tool center path to machine the first surface and the offset surface of the second surface is calculated, and the intersection of a tool center path to machine the second surface and the offset surface of the first surface is calculated. Then, a tool center path to create a connecting surface between the intersections is generated.