Geometric contributions to 3-axis milling of sculptured surfaces

When we are trying to shape a surface X by 3-axis milling, we encounter a list of problems: First we have to decide if locally the milling tool Sigma is able to move along the surface such that its envelope during the motion is the given surface. This is a question involving the curvatures of X and Sigma. Second, we want to avoid that while milling in one part of X, Sigma intersects another, already finished, part of the surface. This is a problem which involves global shape properties of the surface and can be successfully attacked by considering the general offset surface of X with respect to Sigma. Third, in practice a cutting-tool is not able to perform a 2-dimensional motion along a surface. It has to trace out a finite number of piecewise smooth paths such that the resulting surface does not differ from X too much. This question again involves, in the limit case of very small error tolerance, only the curvatures of X and Sigma. If we allow larger scallop heights, the path finding also requires the study of local and global properties.