CHAPTER III: CONICS AND QUADRICS - OCW UPM

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AFFINE <strong>AND</strong> PROJECTIVE GEOMETRY, E. Rosado & S.L. Rueda 4.3.1 Tangent variety to a quadric Definition. The tangent variety to a quadric Q in a point P ∈ P 3 , is the set of points X ∈ P 3 such that the line that joins P and X is tangent to the quadric Q; this is, Observations. T P Q = {X ∈ P 3 | line XP is tangent to Q} = {X ∈ P 3 | ∆ = f(P, X) 2 − ω(P )ω(X) = 0} = {X ∈ P 3 | f(P, X) 2 = ω(P )ω(X)}. 1. T P Q is a degenerate quadric which has P as singular point. 2. If P ∈ Q is a regular point, then T P Q = {X ∈ P 3 | f(P, X) 2 = 0} = {X ∈ P 3 | P t AX = 0} is a plane, called the tangent plane to Q in P . In fact, it is the polar plane of the point P ; this is, T P Q = π p .
3. If P ∈ Q is a singular point, then T P Q = P 3 . AFFINE <strong>AND</strong> PROJECTIVE GEOMETRY, E. Rosado & S.L. Rueda 4.4 Affine classification and notable elements of quadrics Let A 3 = P(R 4 ) be the projectivized affine space, with coordinate system R = {O, B}. And let ω be a quadratic form with associated matrix A. Let Q = {X ∈ P 3 (R 4 ) | ω(X) = 0 be a projective quadric with affine quadric Q = Q ∩ A 3 = {X ∈ A 3 | ω( ˜X) = 0}, where ˜X = (1, x 1 , x 2 , x 3 ).
Page 1 and 2: AFFINE AND PROJECTIVE GEOMETRY, E.
Page 9 and 10: 4.1.1 Projective classification AFF
Page 15: with discriminant AFFINE AND PROJEC
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 41: AFFINE AND PROJECTIVE GEOMETRY, E.

CHAPTER III: CONICS AND QUADRICS - OCW UPM

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