The representation of the general three-system of screws by a sphere

In his classic Treatise on the Theory of Screws, R.S. Ball devotes Chapter V on “The Representation of the Cylindroid by a Circle”. He offers a remarkably simple geometric model of the general two-system of screws as a circle in a plane with coordinates the pitch and the elevation of the screw axis. The present work generalizes the model to three dimensions. The general three-system is represented by a sphere through the origin. The highest- and lowest-pitch screws meet it at two antipodes, defining the pitch axis. A line from the origin, parallel to each screw, intersects the sphere at an image point. Its projection on the pitch axis measures the screwʹs pitch; a translation along a directed segment, equal to the pointʹs velocity under unit clockwise rotation about the pitch axis, positions it on the screwʹs axis.