Self-Inversive Cubic Curves

Abstract

Let γ denote an irreducible nonsingular cubic curve which inverts onto itself with respect to a circle ω with center X. Depending on the type of γ, we show that γ inverts onto itself via a second circle orthogonal to ω or that γ inverts onto itself via two additional circles υ, η with ω,υ,η mutually orthogonal. To accomplish this an algebra (γ,δ) with a ternary operation δ is defined on the points of γ by setting δ(a,b,c) equal to the fourth point, counting multiplicities, on circle (a,b,c) and on γ. If * is the binary operation defined on the points of γ by setting a*b equal to the third point, counting multiplicities, on the line [a,b] and on γ, we show that δ(a,b,c) = X*((X*a)*(b*c)). This equation is used extensively to determine automorphisms of (γ,δ) and to discuss subalgebras.