CHAPTER III: CONICS AND QUADRICS - OCW UPM

AFFINE AND PROJECTIVE GEOMETRY, E. Rosado & S.L. Rueda
4.4.2 Relative position of the quadric and the plane at infinity
Let π ∞ ≡ x 0 = 0 be the equation of the plane at infinity and let us consider
the projective quadric Q determined by a quadratic form ω and with
associated matrix ⎛
⎞
a 00 a 01 a 02 a 03
a
A = 01 a 11 a 12 a 13
⎜
⎟
⎝ a 02 a 12 a 22 a 23 ⎠ .
a 03 a 13 a 23 a 33
We have:
this is,
Q ∩ π ∞ = {X ∈ π ∞ | ω(X) = 0} = { (0, x 1 , x 2 , x 3 ) | X t AX = 0 }
Q ∩ π ∞ ≡ a 11 x 2 1 + a 22 x 2 2 + a 33 x 2 3 + 2a 12 x 1 x 2 + 2a 13 x 1 x 3 + 2a 23 x 2 x 3 = 0,