An interest problem arises to determine the surfaces in the Euclidean three space, which admit at least one nontrivial isometry that preserves the principal curvatures. This leads to a class of surface known as a Bonnet surface. The intention of this study is to examine a Bonnet ruled surface in dual space and to calculate the dual geodesic trihedron of the dual curve associated with the Bonnet ruled surface and derivative equations of this trihedron by the dual geodesic curvature. Also, we find that the dual curvature, the dual torsion for the dual curves associated with the Bonnet ruled surface which are different from any dual curves. Moreover, some examples are obtained about the Bonnet ruled surface.

中文翻译：

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与 Bonnet 直纹曲面相关的双曲线

确定欧几里得三空间中的曲面时出现了一个有趣的问题，该曲面允许至少一个保留主曲率的非平凡等距。这导致了一类称为阀帽曲面的曲面。本研究的目的是研究对偶空间中的 Bonnet 直纹面，并计算与 Bonnet 直纹面相关的对偶曲线的对偶测地线三面体以及该三面体通过对偶测地曲率的导数方程。此外，我们发现与 Bonnet 直纹面相关的对偶曲线的对偶曲率、对偶扭转不同于任何对偶曲线。此外，还获得了一些关于 Bonnet 直纹面的例子。