Helical surfaces with a constant ratio of principal curvatures

We determine all helical surfaces in three-dimensional Euclidean space which possess a constant ratio a := K-1/K-2 of principal curvatures (CRPC surfaces), thus providing the first explicit CRPC surfaces beyond the known rotational ones. Our approach is based on the involution of conjugate surface tangents and on well chosen generating profiles such that the characterizing differential equation is sufficiently simple to be solved explicitly. We analyze the resulting surfaces, their behavior at singularities that occur for a > 0, and provide an overview of the possible shapes.

Automated summary

We determine all helical surfaces with a constant ratio of principal curvatures in three-dimensional Euclidean space. Our approach involves the involution of conjugate surface tangents and the use of well chosen generating profiles. The resulting surfaces and their behavior at singularities are analyzed and an overview of the possible shapes is provided.