CHAPTER III: CONICS AND QUADRICS - OCW UPM

AFFINE AND PROJECTIVE GEOMETRY, E. Rosado & S.L. Rueda
2. If sign(λ 1 ) = sign(λ 2 ) ≠ sign(λ 3 ) (+ + − o − − +) we say that A 00 has
signature 1, sig A 00 = 1, and we can encounter the following cases:
a) If sign(d 0 ) ≠ sign(λ 1 ) = sign(λ 2 ) ≠ sign(λ 3 ), then det A > 0 and the
reduced equation of the affine quadric is
1 = x2 1
a + x2 2
2 b − x2 3
2 c 2
where a 2 = −d 0 /λ 1 , b 2 = −d 0 /λ 2 and c 2 = d 0 /λ 3 (as the three of them
are positive) which is the equation of an hyperbolic hyperboloid.
b) If sign(d 0 ) = sign(λ 1 ) = sign(λ 2 ) ≠ sign(λ 3 ), then det A < 0 and the
reduced equation of the quadric is
1 = − x2 1
a − x2 2
2 b + x2 3
2 c 2
where a 2 = d 0 /λ 1 , b 2 = d 0 /λ 2 and c 2 = −d 0 /λ 3 (as the three of them
are positive) which is the equation of an elliptic hyperboloid.