CHAPTER III: CONICS AND QUADRICS - OCW UPM

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4.4.5 Asymptotic cones AFFINE <strong>AND</strong> PROJECTIVE GEOMETRY, E. Rosado & S.L. Rueda Definition. We call asymptotes of a quadric Q to the tangents of a conic in its improper points. Let Q be a projective quadric with proper center Z. Definition. The tangent variety to the quadric Q from the center Z [(z 0 , z 1 , z 2 , z 3 )] is a cone that is called asymptotic cone. The equation of the asymptotic cone is the following one: f(Z, X) 2 − ω(Z)ω(X) = 0 ⇐⇒ (Z t AX)(Z t AX) − (Z t AZ)(X t AX) = 0 equivalently ⇐⇒ x 2 0 − z 0 (X t AX) = 0 ⇐⇒ x 2 0 − det A 00 det A (Xt AX) = 0 det A det A 00 x 2 0 − Q = 0. The quadrics of ellyptic type have an imaginary asymptotic cone and the quadrics of hyperbolic type have a real asymptotic cone.
AFFINE <strong>AND</strong> PROJECTIVE GEOMETRY, E. Rosado & S.L. Rueda The generatrixs of the cone (lines of the cone) are the diameters tangent to the quadric. We call asymptotic plane to the polar planes of the points of the improper conic of Q (Q ∩ π ∞ = C) (if there exists any). Example 1. Let us consider the quadric Q ≡ x 2 1 + 3x 2 3 + 4x 1 x 2 + 2x 3 + 2 = 0. The matrix of Q is: ⎛ ⎞ 2 0 0 1 0 1 2 0 A = ⎜ ⎟ ⎝ 0 2 0 0 ⎠ 1 0 0 3 The determinant of A is det A = −20, quadric with proper center: ⎛ ⎞ 2 0 0 1 ( ) 0 1 2 0 z0 z 1 z 2 z 3 ⎜ ⎟ ⎝ 0 2 0 0 ⎠ = ρ ( 1 0 0 0 ) , 1 0 0 3
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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
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Page 27 and 28: 4.5 Metric invariants of a quadric
Page 29 and 30: 4.5.1 Classification of quadrics wi
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CHAPTER III: CONICS AND QUADRICS - OCW UPM

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