Geodesic relative curvature of instantaneous conjugation curve on base surface and geometrical differentials of third order

The four independent base geometrical differentials of third order of a surface are introduced. The equations for determining these geometrical differentials for some plain surfaces, which are widely used in gear tooth design and manufacturing, are geometrically deduced. The formula for determining the geodesic relative curvature of the ICC (instantaneous conjugation curve) in terms of parameters of conjugate motion and geometrical differentials up to third order is derived. Also derived are the formulas for determining the geodesic relative curvature of the ICC when the base conjugate surface is one of the plain surfaces. These formulas are instrumental in optimization of the parameters of the gear tooth design, the gear cutting tool design and the machine setting.