On the geometry of the general three-system of screws

The general three-system of screws exhibits a rich geometric structure in which the constant-pitch screws are arranged on a one-parameter family of concentric quadrics, generically hyperboloids. Hunt’s sextic is the algebraic surface on which the family equation, viewed as a cubic in the pitch, has a repeated root; it separates the region where three distinct real quadrics of the family pass through a point from the region where only one does. This paper provides an analytical study of two aspects of this geometry that have received limited treatment in the classical literature. First, the real intersections of two and three quadrics of the family are characterized analytically. Necessary and sufficient conditions on their pitches for real intersections are established, and explicit representations of the corresponding intersection curves are derived. Second, building on a prior description by Hunt and McAree, the paper parametrizes Hunt’s sextic in terms of the repeated root of the pitch cubic, enabling an analytical description of its constant-pitch curves and singular locus. The resulting analysis extends and refines the classical geometric description of the general three-system.