On Beloch's curve that appears when solving real cubic with origami

Abstract

The Justin--Huzita--Hatori Axiom 6 of origami related to so-called neusis construction assures the solution of real cubic equations shown by Beloch in 1936. We investigate a certain real cubic curve $F(x,y)=0$, say, Beloch's curve that appears in the algorithm and prove that its shape is determined by the signature of the Hessian $\mathcal{H}_F=-4(4p+q^2)$ at its uniquely existing singular point $P(p,q)$. This viewpoint would shed new light on the relationship between Axioms 5 and 6.