Regulus Family The Family of Reguli Determined by a Skew Quadrilateral
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Congruence Skew

Regulus Family Description

A quadrilateral in 3-space such that opposite sides are skew to each other, is called a skew quadrilateral, or skew quad for short. Such a skew quadrilateral determines a tetrahedron, determined by the four corners (or planes) of the quadrilateral. This configuration can be used as the base for building a family of reguli such that through each point of space (except points lying in the four planes of the tetrahedron of the skew quad), exactly one of the reguli passes. Thus, one obtains a foliation of space by these reguli, and a second foliation by their complementary reguli. (Leitschar in german: who knows the correct translation for this?) If the four lines of the skew quad are (a,A, b, B) in their natural order, then the pairs (a,b) resp. (A,B) will belong to each regulus resp. leitschar in the family. To determine a particular regulus of the family, one needs only to specify a third line C which intersects a and b in two points (which are not vertices of the skew quad). Then the set of all lines meeting A, B, and C is the desired regulus, and the lines meeting all lines of this regulus is the desired leitschar. Another characterization of this family of reguli is that their pairwise intersection is the lines A and B, and their union is all lines intersecting both the lines A and B (the hyperbolic line congruence determined by A and B). This application allows the user to control the choice of this third line C in a variety of ways. Since Regulus Family is completely written in Java it should run flawlessly on a variety of platforms.

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