A new look at the Ball–Disteli diagram and its relevance to spatial gearing

The subject of this paper is a diagram due to Sir Robert Stawell Ball and Martin Disteli. The diagram, illustrating the relations among the velocities of a pair of gears with skew axes and the geometric parameters of the relative layout of the axes, is intended for the analysis and design of the gear train. A brief historical account on the contributions of both Ball and Disteli is included, which shows that, very likely, these two geometers found the diagram independently. While this diagram appears to be an invaluable tool for the design of gears with skew axes, its significance has remained virtually unknown to the community of gear-geometry researchers and designers. One aim of this paper is to provide a succinct derivation of the diagram, which we do with the aid of dual algebra. The significance of the geometric and kinematic relations summarized in the diagram are thus made more transparent to the reader.