CHAPTER III: CONICS AND QUADRICS - OCW UPM

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We can distinguish two cases: AFFINE <strong>AND</strong> PROJECTIVE GEOMETRY, E. Rosado & S.L. Rueda 1. the eigenvalues of A 00 have the same sign 2. two of the eigenvalues of A 00 have the same sign and the other the opposite sign.
AFFINE <strong>AND</strong> PROJECTIVE GEOMETRY, E. Rosado & S.L. Rueda 1. If sign(λ 1 ) = sign(λ 2 ) = sign(λ 3 ) (+ + + o − − −), we say that A 00 has signature 3, sig A 00 = 3, 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 imaginary ellipsoid. b) 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 ellipsoid, and if besides a 2 = b 2 = c 2 we obtain a sphere.
Page 1 and 2: AFFINE AND PROJECTIVE GEOMETRY, E.
Page 9 and 10: 4.1.1 Projective classification AFF
Page 15 and 16: with discriminant AFFINE AND PROJEC
Page 17 and 18: 3. If P ∈ Q is a singular point,
Page 21 and 22: 4.4.3 Diameters of a quadric AFFINE
Page 27 and 28: 4.5 Metric invariants of a quadric
Page 29: 4.5.1 Classification of quadrics wi
Page 41: AFFINE AND PROJECTIVE GEOMETRY, E.

CHAPTER III: CONICS AND QUADRICS - OCW UPM

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