Offset surfaces and their usage in the milling Bohum´ ır Bastl Department of Mathematics, University of West Bohemia Universitn´ ı 22, Plzeˇ n, Czech Republic e-mail: bastl@kma.zcu.cz Abstract. The undercut in 3-axis and 5-axis milling causes serious problems because of unexpected damages of the machined surface. Possible solutions for indication of this problem in advance can be based on (general) offset surfaces. This paper presents an algorithm for indication of the undercut problem in the milling for surfaces with rational parametrization for which also a (general) offset surface has rational parametrization (so-called RC surfaces). The method uses algebraic geometry (Gr¨obner bases or Dixon (dialytic) resultant) and computation of a (general) offset surface self-intersection. Keywords : Offset surface, general offset surface, convolution surface, RC surfaces, surface self-intersection. 1 Introduction Milling is a process which is used in mechanical engineering to produce a surface of the desired shape. The path of the milling machine has to be planned on a so-called (general) offset surface which contains all position of a reference point on the milling machine axis during the milling. There are two main types of milling: 3-axis milling (a milling ma- chine is able to do only translational movement, not rotational) and 5-axis milling (a milling machine can do not only translational movement but also rotational movement around two different axes). One of the problems that occurs during milling (in both types, 3-axis and also 5-axis) is the so-called undercut problem. This means that during the movement around the desired surface the milling machine may cause irreversible damage to the already machined part of the desired surface. One of the possible mathematical approaches to solve (or at least to indicate) this problem is based on the study of properties of (general) offset surfaces, especially properties of offset surfaces for the case of 5- axis milling and properties of general offset surfaces for the case of 3-axis milling. Several papers have dealt with the undercut problem in milling, and even with the (general) offset surfaces. Wallner et al. discuss in [8] the self-intersections of offset curves and surfaces mainly from the differen- tial geometry point of view and show how to determine maximum offset distance such that the offset does not neither locally nor globally self- intersect. Glaeser et al. in [4] then focus on finding conditions for col- lision free 3-axis milling of surfaces and also on the selection of cutting tools for a given surface using general offset surfaces and differential ge- ometry. Wallner in [7] studies the connection between singularities and