CHAPTER III: CONICS AND QUADRICS - OCW UPM

AFFINE AND PROJECTIVE GEOMETRY, E. Rosado & S.L. Rueda
4.1 Singular points and projective classification
Let Q be a projective quadric determined by a quadratic form ω : R 4 −→ R,
with polar form f : R 4 × R 4 −→ R and associated matrix A with respect to
certain coordinate system.
Definitions.
We say that two points A, B ∈ P 3 are conjugated with respect to Q if
f(A, B) = 0.
We say that a point P ∈ P 3 is an autoconjugated point with respect to
Q if ω(P ) = f(P, P ) = 0.
We say that a point P ∈ P 3 is a singular point of Q if it is conjugated
with every point of P 3 ; this is, f(P, X) = 0 for every point X ∈ P 3 . This
is, if
f(P, X) = P T AX = 0, ∀X ∈ P 3 ,
or equivalently,
P T A = 0.