TABLE OF CONTENTS Many properties of quadrics do not depend on the constan t terms of their equations. These properties reveal interesting relations among quadrics that differ only in the constant terms of their equations. Such a family of quadrics is called a homothetic pencil, and the family's definition does not depend on a specific affine coordinate system. In particular, homothetic pencils are useful in analyzing intersections of quadrics with pencils of parallel lines and planes.
Literature: Berger, Coxeter, Samuel
The family of quadrics represented by
where C and c are fixed, but c varies is called a homothetic pencil. Since Q( x, c) is linear in c, every point x lies on exactly one member of the pencil. This property has two immediate consequences. The quadrics of such a pencil are pairwise disjoint, and the pencil covers the entire space. Moreover, because of Remark 1 in Section 13.1, a homothetic pencil is determined by any one of its members.
It follows from Chapter 13 that the definition of a homothetic pencil does not depend on a particular affine system. Moreover, all quadrics of a homothetic pencil have the same midpoints, if there are any; the same diametric plane with respect to some arbitrary direction, and therefore the same pairs of conjugate directions; and if there are any, the same asymptotic and the same axial directions.
Furthermore, the polar planes of some fixed pole x with respect to...
