TABLE OF CONTENTS The simplest figures in an affine space besides lines and planes are conies which are the intersection curves of planes and right circular cones. Conies were studied by the Greeks, mainly by Menaichmos (about 350 B.C.) and by Apollonios (200 B.C.), who introduced the names ellipse, hyperbola, and parabola. Conies can be conveniently studied using their quadratic equations, and, without additional effort, the analysis of these quadratic equations can be presented for general quadratic surfaces in any dimension, the so-called quadrics. In this chapter affine concepts of quadrics, such as midpoints, singular points, tangents, asymptotes, and polar planes are discussed.
Literature: Berger, Meserve, Samuel
A quadric consists of all points x in an affine space 
satisfying a quadratic equation which can be written as
where C = C t is a symmetric non-zero n n matrix. The quadric described by the equation Q( x) = 0 will also be denoted by Q. The equation can be visualized by blocks:
The intersection of Q with an affine subspace 
of dimension r ? 1 is a quadric again or a hyperplane in 
i.e., a subspace of dimension r - 1. To prove this let 
be represented by
then substitution yields
which is abbreviated by
If C = B t CB ? O, this equation represents a quadric in 
. Note that C is symmetric.
