Projective Geometry (Volume 2)

This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1918 edition. Excerpt: ...the last theorem implies Corollary 1. Any direct projective collineation of a quadric is a product of four harmonic homologies whose centers are polar to their respective planes of fixed points. Corollary 2. Any nondirect projective collineation of a quadric is a product of an odd number of harmonic homologies whose centers are polar to tlieir respecti ve planes of fixed points. Proof. If a projective collineation T interchanges the two reguli, and A is a harmonic homology of the sort described in the statement of the corollary, then TA=A is a projective collineation leaving each regulus invariant. By Cor. 1, A is a product of an even number of harmonic homologies of the required sort, and hence T = AA is a product of an odd number. 103. Real quadrics. The isomorphism between the real inversion group and the projective collineation group of the real quadric (or sphere) (21) may now be studied more in detail. Since a circular transformation leaving three given points of the inversion plane Tt invariant is the identity or an inversion (Theorem 21), and since a collineation of S3 leaving three points of the quadric (21) hi variant is the identity or a harmonic homology whose center is polar to its plane of fixed points, it follows that inversions in Tt correspond to homologies of S, . Hence the direct circular transformations of Tt correspond to the direct collineations of S3 transforming (21) into itself. An involution in Tt is a product of two inversions whose invariant circles intersect and are perpendicular. To say that the invariant circles intersect and are perpendicular is to say that they intersect in such a way that one of the circles is transformed into itself by the inversion with respect to the other. Now suppose that Oa and ptt..