# Elementary Theory

• Bernard R. Gelbaum
Part of the Problem Books in Mathematics book series (PBM)

## 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
$$\left( {a^2 + b^2 + c^2 - c} \right)\left( {x^2 + y^2 } \right) - 2ax - 2by + a^2 + b^2 + c^2 + c = 0$$
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.

## 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.