CHAPTER III: CONICS AND QUADRICS - OCW UPM

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AFFINE <strong>AND</strong> PROJECTIVE GEOMETRY, E. Rosado & S.L. Rueda Theorem. If the point P belongs to the polar plane of a point R, then the point R is in the polar plane of P . This is due to the condition of conjugation f(P, R) = 0 ; it is symmetric in P and R. As we have seen, given a quadric Q, every non sigular point P is assigned a plane (its polar plane) and reciprocally, each plane π is assigned a point (its pole). Definition. We call polarity defined by a quadric Q to the transformation in which each non singular point of Q is assigned to its polar plane. This is, P 3 sign(Q) −→ Planes of P 3 P ↦−→ π P
AFFINE <strong>AND</strong> PROJECTIVE GEOMETRY, E. Rosado & S.L. Rueda Fundamental theorem of polarity The polar planes of the points of a plane π of P 3 , with respect to a regular quadric Q, contain the same point which is precisely the pole of π.
Page 1 and 2: AFFINE AND PROJECTIVE GEOMETRY, E.
Page 9 and 10: 4.1.1 Projective classification AFF
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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
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CHAPTER III: CONICS AND QUADRICS - OCW UPM

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