Transformations and singularities of polarized curves

TL;DR

This paper investigates the limiting behavior of Darboux and Calapso transforms of polarized curves in conformal spheres near poles of first and second order, revealing convergence patterns and limit structures.

Contribution

It provides a detailed analysis of how Darboux and Calapso transforms behave near singularities, extending understanding of isothermic surface transformations with singular umbilics.

Findings

Darboux transforms converge to the original curve near first-order poles.

Calapso transforms also converge near first-order poles.

Near second-order poles, Calapso transforms can approach a point or circle depending on parameters.

Abstract

We study the limiting behaviour of Darboux and Calapso transforms of polarized curves in the conformal n-dimensional sphere, when the polarization has a pole of first or second order at some point. We prove that for a pole of first order, as the singularity is approached all Darboux transforms converge to the original curve and all Calapso transforms converge. For a pole of second order, a generic Darboux transform converges to the original curve while a Calapso transform has a limit point or a limit circle, depending on the value of the transformation parameter. In particular, our results apply to Darboux and Calapso transforms of isothermic surfaces when a singular umbilic with index 1/2 or 1 is approached along a curvature line.