DOUBLE LINES

A path curve transformation (or space collineation) has an invariant tetrahedron with 6 invariant or double lines at least two of which must be real (see Path Curves). While this is quite easy to show algebraically, it is no trivial matter to derive a method of construction for the real double lines which is purely synthetic. For those who enjoy advanced pure projective geometry the proof and method is outlined below, and the full nine-page proof may be downloaded. If A1 A2 A3 are three successive corresponding points of a space collineation C then the bundles of planes in A1 A2 and A3 are collinear i.e. projectively related by the collineation. The triples of corresponding planes of the bundles meet in points describing a cubic surface C (see e.g. Semple and Kneebone "Algebraic Projective Geometry"). C possesses in general six special lines each of which contains three corresponding planes, and which are thus double-lines of C. C intersects an arbitrary plane p in a plane cubic C1 i.e. the points in p where triples of corresponding planes meet all lie on C1. Each double-line of C must lie in all three planes of such a triple, so it must intersect C1. If we consider the plane cubic C’ in which C intersects the plane at infinity, generally a plane triple meeting in one of the points of C’ is such that its planes meet in pairs in three parallel lines, and these will coincide for the double-lines. We select one line of each of these triples generated by the planes in A2 and A3 to intersect p in a second plane cubic C2, which will generally intersect C1 in 9 points. Six of these are significant and the lines through them are the double-lines of C e.g. if the two nodes coincide then so do 4 points of intersection leaving 5 others, giving 6 actual points. In these cases the three parallel lines of the plane triples must coincide as the triples also meet in p, so those six points give the double-lines of C. Since two plane cubics must meet in at least one real point we see that there is at least one real double-line of C. The double lines of C form the invariant tetrahedron of the collineation.

The reason for using plane cubics is that they are guaranteed to meet in at least one real point, unlike conics! From these ideas a method of construction (in principle, this is all in 3 dimensions) can be derived to find three of the double lines without having to construct anything more complicated than a conic. The other three require the additional construction of a plane cubic. 