Geometric Contribution 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 Σ 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 Σ. Second, we want to avoid that while milling in one part of X, Σ 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 Σ. Third, in practice a cuttingtool 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 Σ. If we allow larger scallop heights, the path finding also requires the study of local and global properties.