Geometric Model for Generation of Contour- Parallel Lines’ Family for Cutting Tool’s Path Automated Computation

Abstract

In this paper has been proposed a geometric model for forming problem of contour-parallel lines (equidistant lines) for a flat contour with an island, and has been obtained the problem’s analytical solution, which is relevant for computer-aided design of cutting tools processing pocket surfaces on CNC machines. The proposed geometric model is based on cyclograph mapping of space on a plane. Beyond the analytical solution the geometric model differs from the known algebraic models and their solutions for considered forming problem also by the fact that it allows obtain a more complete and evident representation on the relationship and interaction for all its geometric components at the stages of 3D computer visualization. A 3D geometric model based on a cyclograph mapping of space has been proposed for obtaining the families of equidistant lines for connected and multiply connected regions with closed contours taken as a basis for pocket surfaces modeling. An algorithm for the analytical solution of the problem related to equidistant families generation is getting from the geometric model. All stages of the analytical solution are accompanied by a figurative representation of geometric objects and their relations in the geometric model’s virtual electronic space. The proposed in this paper algorithm for the case of a doubly connected polygonal region can be used as a basis for generation of equidistant families for multiply connected polygonal regions. The presence of the analytical solution for the problem related to equidistant families generation simplifies greatly the automated calculation of the tool path and preparation of control programs for pocket surfaces manufacturing on CNC machines. Have been presented an example and algorithm providing support for working capacity of the proposed geometric model for considered forming problem.