3rd order double B-splint surfaces and the 3rd order contact in NC machining

Abstract

In order to increase machining efficiency in machining sculptured surfaces, the contact principle of differential geometry is applied in the case of 5-axis NC machining. Optimization of contact condition between tool and surface was investigated in this paper. The degree of contact between the line of intersection and between the cutter and the surface is determined analytically. As compared to previous studies, the theory of this paper goes beyond second order parameters to third order, thereby areconciling the primary surfaces of analytical geometry with computer-generated surface conventions of computational geometry. It explains definite procedure of calculation, and the equations mentioned are easy to solve. The presented underlying idea is quite simple: first, finding a surface of third order which have arbitrary coefficients. Secondly finding the best posture of a circle in order that it and the chosen surface will conform most closely at a locally chosen origin and cartesian base frame. Finally, the desired surface is carefully approximated by a 3-order surface at every point of machining. The results of this procedure are used to find the best cutter posture at every machining position. Thus it may be seen that the equations are easy to solve and the concept is clear.