CHAPTER III: CONICS AND QUADRICS - OCW UPM

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AFFINE <strong>AND</strong> PROJECTIVE GEOMETRY, E. Rosado & S.L. Rueda Table of classification of quadrics with center ⎧ ⎧ { det A > 0 imaginary det A 00 ≠ 0 ⎪⎨ rank(A) = 4 Regular ⎪⎨ ⎪⎩ sig A 00 = 3 Ellipsoids sig A 00 = 1 Hyperboloids det A < 0 real { det A > 0 hyperbolic det A < 0 elliptic ⎪⎩ rank(A) = 3 Cones { sig A00 = 3 Imaginary cone with a real point sig A 00 = 1 Real cone
AFFINE <strong>AND</strong> PROJECTIVE GEOMETRY, E. Rosado & S.L. Rueda 4.5.2 Classification of the quadrics with det A 00 = 0. Because of det A 00 = λ 1 λ 2 λ 3 = 0, we can suppose λ 3 = 0. Hence J = λ 1 λ 2 . In certain coordinate system the matrix of the quadric is ⎛ ⎜ ⎝ b 00 0 0 b 03 0 λ 1 0 0 0 0 λ 2 0 b 03 0 0 0 with det A = −b 2 03λ 1 λ 2 . The reduced equation of the affine quadric is ⎞ ⎟ ⎠ 0 = b 00 + λ 1 x 2 1 + λ 2 x 2 2.
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 and 30: 4.5.1 Classification of quadrics wi
Page 33: AFFINE AND PROJECTIVE GEOMETRY, E.
Page 41: AFFINE AND PROJECTIVE GEOMETRY, E.

CHAPTER III: CONICS AND QUADRICS - OCW UPM

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