On Martin Disteliʹs spatial cycloidal gearing

The modeling of the tooth flanks of gears with skew axes still represents a challenge to geometers. This paper is a contribution along these lines, as pertaining to gears with skew axes and cycloidal teeth. To this end, the authors follow and extend the results reported by Martin Disteli at the turn of the 20th century concerning the general synthesis of gears with skew axes. The main goal is the rendering of the tooth profiles of cycloidal gears with skew axes. This is done by means of the dualization of the geometric relations proposed by Disteli for bevel gears with cycloidal teeth, which leads to ruled surfaces as conjugate tooth flanks such that at any instant the contact points are located on a straight line. All results are proven by means of a consistent use of dual vectors representing directed lines and rigid-body twists.