Comments on Helical Developables

All normal lines of a helical developable Φ have the same distance to its axis a and make the same angle with a . This implies that on coaxial one-sheet hyperboloids of revolution or skew ruled helical surfaces with generators included in the normal-congruence of Φ any orthogonal trajectory of the rulings is located on a translate of Φ . Let line n be the normal of the helical developable Φ at C . When Φ performs a helical motion or rotation about its axis a with constant velocity, then the point C of intersection with the fixed normal line n runs along n with constant velocity, too. The same holds for circular involutes in the plane. This property offers the most simple way to grasp why helical developables serve as tooth flanks for skew gearing according to Jack Phillips (2003) and why the gear ratio is independent from the relative position of their axes. The generators of two conjugate tooth flanks 1 Φ and 2 Φ passing through the instantaneous point C of contact remain parallel to their respective initial position. The corresponding osculating cones of the tooth flanks have axes fixed in the gear box and parallel to the related helical axis.