Collision-free 3-axis milling and selection of cutting tools

The present paper deals with local and global conditions for collision-free 3-axis milling of sculptured surfaces and the selection of cutting-tools for a given surface. We describe local and global millability results whose proofs have been published in a previous paper. The theoretical background involves general oo-set surfaces. Here an algorithm is presented which after evaluation of the surface curvature yields a diierential inequality for the meridian curve of the cutting-tool, which is fulllled if and only if the cutting-tool is able to mill the entire surface. The choice or even design of the optimal tool then besides this inequality involves further characteristics of the tools, such as its shape and its size. Recently we have studied the problem of locally and globally collision-free milling of sculptured surfaces 9. It turns out that if some conditions on the curvature of the surfaces involved are fulllled, we can show that locally, and in certain cases also globally, no unwanted collision of the cutting-tool with the surface occurs. The present paper deals with the optimal selection (or even design) of cutting tools in order to mill a given surface. We will present algorithms which allow to (i) test whether or not a given cutter is able to mill a given surface, (ii) select a given number of optimal cutters from a given set of available cutting-tools. (iii) calculate the shape of anòptimal' cutting-tool which will be able to mill the surface. The third problem, however hypothetical its engineering applications, is included here because the mathematics which is necessary to solve it will be needed anyway, and also because it is an interesting geometric problem which independently deserves interest. We are going to describe the background and mathematical foundations of the local millability test when restricted to smooth surfaces, which consist of C 2 patches with C 1 join. Non-smooth surfaces will be considered in the next section. We denote the surface by X and the cutter by. X is the boundary of a solid, and we speak of the solid as of the interior of X and will call the ambient space the exterior of X. The cutting-tool is, geometrically, always a convex body of rotational symmetry. The rotation of the cutter around its axis, however important for the mechanical engineering aspect of the problem, can be completely neglected from the geometric point of view. The actual cutter, while rotating, has a …