On Ribaucour transformations (Q2781681)

A Ribaucour transformation between two surfaces \(M,\widetilde{M}\subset{\mathbb R}^3\) is a diffeomorphism \(\psi:M\longrightarrow\widetilde{M}\) that preserves curvature lines and is such that for all \(P\in M\) the normal lines \(\nu_P\) and \(\nu_{\widetilde{P}}\) of \(M\) at \(P\) and of \(\widetilde{M}\) at \(\widetilde{P}=\psi(P)\) have a point of intersection \(X=\nu_P\cap\nu_{\widetilde{P}}\) with the same distance \(h(P)=\text{dist}(X,P)=\text{dist}(X,\widetilde{P})\) to \(P\) and \(\widetilde{P}\) respectively. NEWLINENEWLINENEWLINEThe authors recall analytic descriptions and properties of Ribaucour transformations from an earlier paper [\textit{A. V. Corro, W. Ferreira} and \textit{K. Tenenblat}, Mat. Contemp. 17, 137--160 (1999; Zbl 1018.53004)] and describe a special case that provides Ribaucour transformations for minimal surfaces. Proofs concerning this case are postponed to forthcoming papers. However, they provide explicit parametrizations of the families of minimal surfaces obtained by applying these Ribaucour transformations to the Enneper surface and to the catenoid respectively.NEWLINENEWLINEFor the entire collection see [Zbl 0974.00040].