CHAPTER III: CONICS AND QUADRICS - OCW UPM
CHAPTER III: CONICS AND QUADRICS - OCW UPM
CHAPTER III: CONICS AND QUADRICS - OCW UPM
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AFFINE <strong>AND</strong> PROJECTIVE GEOMETRY, E. Rosado & S.L. Rueda<br />
The coordinate system R = {Z, {v 1 , v 2 , v 3 }} gives us a cartesian autoconjugated<br />
coordinate system.<br />
We can find three situations:<br />
1. The three eigenvalues are different. Then Q has three axes which are<br />
orthogonal two by two.<br />
2. An eigenvalue is double, λ 1 = λ 2 , and the other, λ 3 , is simple. Then the<br />
dimension of the subspace of eigenvectors associated to the double<br />
eigenvalue is dim V 1 = 2 and dim V 3 = 1. Then V 1 is a plane of axes<br />
perpendicular to the axis V 3 . In this case the quadric Q is a revolution<br />
quadric, whose axis is the one that corresponds to the eigenvalue λ 3 .<br />
3. The three eigenvalues are the same, λ 1 = λ 2 = λ 3 . Then any diameter<br />
is the axis and the quadric is a sphere.<br />
Definition. We call main planes of a quadric Q to the diametral polar planes<br />
of the axes.
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