Problems in Real and Complex Analysis pp 371-380 | Cite as

# Elementary Theory

Chapter

## Abstract

A circle on S is the intersection of ∑ with a plane Π for which the equation is representing a circle in ℂ or, if

*a*ξ +*b*η) +*c*ζ, =*a*^{2}+*b*^{2}+*c*^{2}. The plane Π and ∑ intersect iff*a*^{2}+*b*^{2}+*c*^{2}≤ 1. The equation ξ^{2}+ η^{2}+ ζ^{2}= 1 and the formulae for the coordinates of \(\Theta \left( {\xi ,\eta ,\zeta \mathop = \limits^{{\text{def}}} (x,y)} \right)\) lead to the equation$$
\left( {a^2 + b^2 + c^2 - c} \right)\left( {x^2 + y^2 } \right) - 2ax - 2by + a^2 + b^2 + c^2 + c = 0
$$

*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.## Keywords

Elementary Theory Double Sequence Cauchy Integral Formula Rectifiable Curve Circle Passing
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Copyright information

© Springer Science+Business Media New York 1992