Rubio cubics and generalized Cundy–Parry mappings

Abstract

Rubio's mappings between the Thomson and Darboux cubics are generalized for pairs of cubics of the form pα(β 2 − γ 2 )+ qβ(γ 2 − α 2 ) + rγ(α 2 − β 2 ) = 0, where p, r, q, α, β, γ are functions of a triple (a, b, c) of variables or indeterminates. Methods include symbolic substitutions, such as (a, b, c) → (bc, ca, ab). Connections between the generalized Rubio mappings with generalized Cundy-Parry mappings are described.Mathematics Subject Classification (2010). 51N20.