In this paper, a new system for designing drills with a full parametric cycle of connection of the main parameters of the rear surface of the drill, based on the identified functional relationships between the parameters of the original tool surface and the kinematics of the movement of the drill tooth. For the first time, a mechanism for determining the shape of the back surface of a three-tooth drill with the possibility of regrinding by various methods has been established. This work will allow not only to develop design support for tool production, but also to create a system for the automatic production of three-tooth drills and writing control programs for CNC machines. The work is the first to propose an analytical mathematical model for solving the solve problem of profiling the helical groove of a twist drill and cutting edges, which is the main stage in the design of a new class of drills with a toroidal cutting surface. For the first time, a study of the field of change in the rake angle and the normal rake angle in the region of the generating surface was carried out. As a result, it was found that in the f range from [-1.2; 1] the normal rake angle γN for almost the entire length of the section takes values in the range [20°; 32°], with a displacement of the cutting edge f by a distance of 2.8 mm, it will be possible to obtain the geometry of the helical surface on the cutting surface in a constant range from 32° with a tolerance range of 2°.