CHAPTER III: CONICS AND QUADRICS - OCW UPM

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AFFINE <strong>AND</strong> PROJECTIVE GEOMETRY, E. Rosado & S.L. Rueda 4.4.4 Axes of a quadric with proper center Definition. We call axis of a quadric Q to the diameter which is perpendicular to its diametral polar plane. Let Q be a projective quadric with associated matrix A. And let A 00 be the matrix of the conic Q ∩ π ∞ . As Q has a proper center Z, the matrix A 00 is not singular, so its three eigenvalues are not zero λ 1 , λ 2 and λ 3 . Let v 1 , v 2 and v 3 be the eigenvectors associated to λ 1 , λ 2 and λ 3 respectively (we choose the eigenvectors which are orthogonal two by two). The axes of Q are the lines that contain the center, Z, and have as directions the vectors v 1 , v 2 and v 3 , respectively.
AFFINE <strong>AND</strong> PROJECTIVE GEOMETRY, E. Rosado & S.L. Rueda The coordinate system R = {Z, {v 1 , v 2 , v 3 }} gives us a cartesian autoconjugated coordinate system. We can find three situations: 1. The three eigenvalues are different. Then Q has three axes which are orthogonal two by two. 2. An eigenvalue is double, λ 1 = λ 2 , and the other, λ 3 , is simple. Then the dimension of the subspace of eigenvectors associated to the double eigenvalue is dim V 1 = 2 and dim V 3 = 1. Then V 1 is a plane of axes perpendicular to the axis V 3 . In this case the quadric Q is a revolution quadric, whose axis is the one that corresponds to the eigenvalue λ 3 . 3. The three eigenvalues are the same, λ 1 = λ 2 = λ 3 . Then any diameter is the axis and the quadric is a sphere. Definition. We call main planes of a quadric Q to the diametral polar planes of the axes.
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: 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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