Elementary Theory

A circle on S is the intersection of ∑ with a plane Π for which the equation is aξ + bη) + cζ, = a2 + b2 + c2. The plane Π and ∑ intersect iff a2 + b2 + c2 ≤ 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 a2 + b2 + c2 = 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.