On Helical Surfaces with a Constant Ratio of Principal Curvatures

We determine all helical surfaces in three-dimensional Euclidean space which possess a constant ratio $a:=κ_1/κ_2$ of principal curvatures (CRPC surfaces), thus providing the first explicit CRPC surfaces beyond the known rotational ones. A key ingredient in the successful determination of these surfaces is the proper choice of generating profiles. We employ the contours for parallel projection orthogonal to the helical axis. This has the advantage that the CRPC property can be nicely expressed with the help of the involution of conjugate surface tangents. The arising ordinary differential equation has an explicit parametric solution, which forms the basis for a further study and classification of the possible shapes and the singularities arising for $a>0$.