Abstract In this paper we study helix surfaces whose unit normals make a constant angle with a fixed direction. As an application of the singularity theory, we classify the generic singularities of helix surfaces, which are cuspidaledges and swallowtails. These singularities are deeply related to the order of contact between the generating curves of helix surfaces and the affine tangent bundles over unit spheres. We also show that the unit normals and rulings of helix surfaces are spherical Legendrian dual to each other. Moreover, we investigate the singularities of helix surfaces from the viewpoint of opening map-germ.

中文翻译：

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欧几里得 3 空间中螺旋面的奇异性

摘要 在本文中，我们研究了单位法线与固定方向成恒定角度的螺旋曲面。作为奇点理论的一个应用，我们对螺旋面的一般奇点进行了分类，即尖角和燕尾。这些奇点与螺旋面的母曲线和单位球体上的仿射切丛之间的接触顺序密切相关。我们还表明，螺旋曲面的单位法线和线数是彼此对偶的球面勒让德函数。此外，我们从打开图谱的角度研究螺旋表面的奇异性。