Abstract
A circle on S is the intersection of ∑ with a plane Π for which the equation is aξ + bη) + cζ, = a 2 + b 2 + c 2. The plane Π and ∑ intersect iff a 2 + b 2 + c 2 ≤ 1. The equation ξ2 + η2 + ζ2 = 1 and the formulae for the coordinates of \(\Theta \left( {\xi ,\eta ,\zeta \mathop = \limits^{{\text{def}}} (x,y)} \right)\) lead to the equation
representing a circle in ℂ or, if a 2 + b 2 + c 2 = c, a straight line in ℂ. The latter circumstances imply that Π passes through (0,0,1). The reasoning is reversible and leads from a circle in Π to a circle on Σ\ {(0, 0, 1)} or from a straight line in ℂ to a circle passing through (0, 0, 1) on Σ.
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© 1992 Springer Science+Business Media New York
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Gelbaum, B.R. (1992). Elementary Theory. In: Problems in Real and Complex Analysis. Problem Books in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-0925-6_22
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DOI: https://doi.org/10.1007/978-1-4612-0925-6_22
Publisher Name: Springer, New York, NY
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