Agnes Scott College

Ruth G. Wood

The Collineations of Space with Transforms a Non-Degenerate Quadric Surface into Itself
The Annals of Mathematics, 2nd Ser., Vol. 2, No. 1/4 (1900-1901), 161-171


The object of this paper is to discuss the ∞6 collineations of space which transform a non-degenerate quadric surface into itself. These fall into two classes. The collineations of the first kind leave the two systems of generators of the quadric invariant, while the collineations of the second kind interchange the two systems. When the quadric surface degenerates into the imaginary circle at infinity, these collineations of the first and second kinds become respectively the displacements and the symmetry transformations of euclidean geometry. We shall then call a collineation of the first kind a non-euclidean displacement, and a collineation of the second kind a non-euclidean symmetry transformation. These transformations for euclidean geometry have been discussed by Mr. Gale in a paper entitled Wiener's Theory of Displacements, with Application to the Proof of Four Theorems of Chasles. The methods here employed are similar to those used by Mr. Gale, while the theorems obtained correspond in non-euclidean space to the four theorems of Chasles proved in his paper.

This correspondence is readily seen if we choose the quadric surface for the absolute. Two lines are then said to be conjugate polars with respect to the absolute when the polar planes of all the points of one line pass through the other. Two planes are said to be conjugate or perpendicular when the pole of one lies in the other. Two intersecting lines are said to be perpendicular when each intersects the conjugate of the other, and a line is said to be perpendicular to a plane when it passes through the pole of the plane.

The fundamental operation in this discussion is the involutory transformation known as a skew reflection on conjugate polars of the quadric. By this operation a point P is transformed into a point P' such that the line PP' is divided harmonically by the two polars, which are called the directrices of the skew reflection. When the quadric degenerates into the imaginary circle at infinity the skew reflection becomes a reflection in a line, the transformation used by Mr. Gale in his discussion. We shall then call a skew reflection on conjugate polars of the quadric a non-euclidean line reflection.

We shall first show that any non-euclidean displacement may be resolved into the product of two non-euclidean line reflections. This affords a simple method of resolving non-euclidean displacements and symmetry transformations so that some of their fundamental properties are at once evident. From these resolutions theorems concerning the displacements of straight lines and plane figures in non-euclidean space are readily deduced, which, as before mentioned, become well known theorems of Chasles when the quadric degenerates into the imaginary circle at infinity.