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
it is an actual sphere if k ≠ 0 and ‖α‖2 > 4kh; if k ≠ 0 and ‖α‖2 = 4kh the image is a single point (sphere of zero radius), and if ‖α‖2 < 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
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