CHAPTER III: CONICS AND QUADRICS - OCW UPM

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this is AFFINE <strong>AND</strong> PROJECTIVE GEOMETRY, E. Rosado & S.L. Rueda ⎧⎪ ⎨ ⎪ ⎩ 2z 0 + z 3 = ρ z 1 + 2z 2 = 0 2z 1 = 0 z 0 + 3z 3 = 0 The center is: Z [(1, 0, 0, −1/3)]. Equation of the asymptotic cone: det A det A 00 x 2 0 − Q = 0 ⇐⇒ 20 12 x2 0 − (x 2 1 + 3x 2 3 + 4x 1 x 2 + 2x 3 x 0 + 2x 2 0) = 0 ⇐⇒ 2x 0 x 3 + 4x 1 x 2 + 1 3 x2 0 + x 2 1 + 3x 2 3 = 0
4.5 Metric invariants of a quadric Q AFFINE <strong>AND</strong> PROJECTIVE GEOMETRY, E. Rosado & S.L. Rueda Let us consider the quadric Q with associated matrix A; this is, Q ≡ X T AX = 0. The following values are euclidean invariants of the quadric: det A Eigenvalues of A 00 : λ 1 , λ 2 , λ 3 or equivalently: det A 00 , tr A 00 = a 11 + a 22 + a 33 , J = ∣ a ∣ 11 a 12 ∣∣∣ + a 12 a 22 ∣ a ∣ 11 a 13 ∣∣∣ + a 13 a 33 ∣ a ∣ 22 a 23 ∣∣∣ a 33 a 23 where A = ⎛ ⎜ ⎝ ⎞ a 00 a 01 a 02 a 03 a 01 a 11 a 12 a 13 ⎟ a 02 a 12 a 22 a 23 a 03 a 13 a 23 a 33 ⎠ and A 00 = ⎛ ⎝ ⎞ a 11 a 12 a 13 a 12 a 22 a 23 ⎠ . a 13 a 23 a 33
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 25: AFFINE AND PROJECTIVE GEOMETRY, E.
Page 29 and 30: 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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