Abstract
The theory of quadrics in spaces of several different types is presented. The chapter begins with quadrics in projective spaces. The principle of projective duality is extended to nonsingular quadrics and is illustrated with Pascal’s and Brianchon’s theorems. The isotropic subspaces of maximum possible dimension on a nonsingular quadric in complex projective spaces are investigated. Then quadrics in real projective spaces and real affine spaces are considered. The projective, affine, and metric classifications of quadrics are obtained. Finally, quadrics in the real plane are considered in greater detail.
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Notes
- 1.
A clarification of this term, that is, an explanation of what this has to do with a cone, will be given somewhat later.
- 2.
Here we move away somewhat from the intuition of elementary geometry, where by a side we mean not the entire line passing through two points, but only the segment connecting them. This extended notion of a side is necessary if we wish to include the case of an arbitrary field \({\mathbb{K}}\), for instance, \({\mathbb{K}}= {\mathbb{C}}\).
- 3.
Such a proof can be found, for example, in the book Algebraic Curves, by Robert Walker (Springer, 1978).
- 4.
The reader can find them, for example, in the book Methods of Algebraic Geometry, by Hodge and Pedoe (Cambridge University Press, 1994).
- 5.
The proof of this fact is due to the Franco-Belgian mathematician Germinal Pierre Dandelin. It can be found, for example, in A.P. Veselov and E.V. Troitsky, Lectures in Analytic Geometry (in Russian); B.N. Delone and D.A. Raikov, Analytic Geometry (in Russian); P. Dandelin, Mémoire sur l’hyperboloïde de révolution, et sur les hexagones de Pascal et de M. Brianchon; D. Hilbert and S. Cohn-Vossen, Geometry and the Imagination.
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© 2012 Springer-Verlag Berlin Heidelberg
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Shafarevich, I.R., Remizov, A.O. (2012). Quadrics. In: Linear Algebra and Geometry. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-30994-6_11
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DOI: https://doi.org/10.1007/978-3-642-30994-6_11
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-30993-9
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