Regulus (geometry)


In three-dimensional space, a regulus R is a set of skew lines, every point of which is on a transversal which intersects an element of R only once, and such that every point on a transversal lies on a line of R
The set of transversals of R forms an opposite regulus S. In ℝ3 the union R ∪ S is the ruled surface of a hyperboloid of one sheet.
Three skew lines determine a regulus:
According to Charlotte Scott, "The regulus supplies extremely simple proofs of the properties of a conic...the theorems of Chasles, Brianchon, and Pascal..."
In a finite geometry PG, a regulus has q + 1 lines. For example, in 1954 William Edge described a pair of reguli of four lines each in PG.
Robert J. T. Bell described how the regulus is generated by a moving straight line. First, the hyperboloid is factored as
Then two systems of lines, parametrized by λ and μ satisfy this equation:
No member of the first set of lines is a member of the second. As λ or μ varies, the hyperboloid is generated. The two sets represent a regulus and its opposite. Using analytic geometry, Bell proves that no two generators in a set intersect, and that any two generators in opposite reguli do intersect and form the plane tangent to the hyperboloid at that point..