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The Space of Spheres

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Problems in Geometry

Part of the book series: Problem Books in Mathematics ((PBM))

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Abstract

We fix an n-dimensional Euclidean space E, and denote by Q ( E ) the space of affine quadratic forms q over E; \(\overrightarrow q \), an element of Q (\(\overrightarrow E \)), will be the symbol of q. A sphere is given by a form q, written as

$$q = k{\left\| {\left. \cdot \right\|} \right.^2} + (\alpha \left| \cdot \right.) + h, where \alpha \in \overrightarrow E and k, h \in R;$$

it is an actual sphere if k ≠ 0 and ‖α2 > 4kh; if k ≠ 0 and ‖α2 = 4kh the image is a single point (sphere of zero radius), and if ‖α2 < 4kh the image is empty, and we say that q represents a sphere “of imaginary radius”. For k = 0, h ≠ 0, we obtain a hyperplane (if α ≠ 0); the case k = 0 and α = 0, h ≠ 0 represents the point at infinity of E.

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© 1984 Marcel Berger

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Berger, M., Pansu, P., Berry, JP., Saint-Raymond, X. (1984). The Space of Spheres. In: Problems in Geometry. Problem Books in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-1836-2_20

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  • DOI: https://doi.org/10.1007/978-1-4757-1836-2_20

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4419-2822-1

  • Online ISBN: 978-1-4757-1836-2

  • eBook Packages: Springer Book Archive

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