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

Hull 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Springer Science+Business Media New York 1992

Authors and Affiliations

  • Bernard R. Gelbaum
    • 1
  1. 1.Department of MathematicsState University of New York (SUNY)BuffaloUSA

Personalised recommendations