CHAPTER III: CONICS AND QUADRICS - OCW UPM

AFFINE AND PROJECTIVE GEOMETRY, E. Rosado & S.L. Rueda
The following identities are satisfied:
det A 00 = λ 1 λ 2 λ 3
J = λ 1 λ 2 + λ 1 λ 3 + λ 2 λ 3
tr A 00 = λ 1 + λ 2 + λ 3
The charasteristic equation of A 00 is:
|A 00 − λI 3 | = −λ 3 + tr A 00 λ 2 − Jλ + det A 00 = 0.
Therefore, λ 1 , λ 2 and λ 3 are the roots of the equation |A 00 − λI 3 | = 0.
If det A 00 ≠ 0, then the conic Q ∩ π ∞ is regular and Q has a center.
If det A 00 = 0, then the conic Q ∩ π ∞ is not regular. It is a quadric of parabolloid
type, it may not have a center, have a line of centers or even have a
plane of centers.