On the Equation ax—xb = c

Abstract

Beniamino Segre, in his memorial lecture of 1958 [5], [6], inaugurated the study of non-linear geometry in three dimensions over a division ring. In his treatment of sections of quadrics by planes, he is naturally led to consider conics and the problem of tangency. Now in the commutative case the locus of intersection of a quadric and a plane containing a generator is the line-pair consisting of this generator and one from the other family. Such a plane is then the tangent plane of the point of intersection of the two generators. Segre extends this notion to the non-commutative case, where the locus of intersection is not always a line-pair. He joins up the remaining points of intersection in pairs, and calls the points where the lines so formed cut the base generator, the ‘points of contact’ of the plane (π) and the quadric (Q). A line in π is called a ‘tangent’ if it passes through a point of contact, but does not contain any of the points of intersection of Q and π.