Direct Relations Between Curvatures of Surfaces Being in Continuous Tangency

Abstract The motion of two solids provided with surfaces Σ1 and Σ2 being in continuous tangency either along a line or at a point at every instant is considered. The authors propose an approach for determining direct relations between: (i) the principal curvatures and directions of Σ1 and Σ2, (ii) the normal curvatures of Σ1 and Σ2 along two mutually perpendicular directions chosen in the tangent plane. The performed diagonalization of the curvature matrix enables to determine the directions and magnitude for the extreme relative normal curvatures of Σ1 and Σ2. The proposed approach enables: (i) to simplify substantially the determination of curvatures when one of the surfaces is represented in threeparametric form but with related parameters, (ii) to simplify the computerized simulation of contact of gear tooth surfaces, and (iii) to simplify the determination of curvatures of a surface generated by a tool with the given surface. The approach is illustrated with the example of generation of a worm (screw) by a cone surface of the tool.