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 $$\frac{1}{2}$$12 or 1 is approached along a curvature line.

中文翻译：

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极化曲线的变换和奇异性

我们研究了当极化在某个点具有一阶或二阶极点时，共形 n 维球体中极化曲线的 Darboux 和 Calapso 变换的极限行为。我们证明，对于一阶极点，随着接近奇点，所有 Darboux 变换收敛到原始曲线，所有 Calapso 变换收敛。对于二阶极点，通用 Darboux 变换收敛到原始曲线，而 Calapso 变换具有极限点或极限圆，具体取决于变换参数的值。特别是，当沿着曲率线接近指数为 $$\frac{1}{2}$$12 或 1 的奇异脐带时，我们的结果适用于等温面的 Darboux 和 Calapso 变换。