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Helical surfaces with a constant ratio of principal curvatures

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Abstract

We determine all helical surfaces in three-dimensional Euclidean space which possess a constant ratio \(a:=\kappa _1/\kappa _2\) 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.

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Notes

  1. We always assume that the constant C is large enough s.t. \(I_C\) is nonempty.

  2. This new parameterization considerably reduces the calculation for derivatives in the light of ODE (9) when \(t=0\).

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Acknowledgements

The authors gratefully acknowledge the support by KAUST baseline funding.

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Authors and Affiliations

  1. KAUST Visual Computing Center, King Abdullah University of Science and Technology, Thuwal, 23955-6900, Saudi Arabia

    Yang Liu, Olimjoni Pirahmad, Hui Wang, Dominik L. Michels & Helmut Pottmann

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  1. Yang Liu
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  2. Olimjoni Pirahmad
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  3. Hui Wang
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  4. Dominik L. Michels
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  5. Helmut Pottmann
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Correspondence to Helmut Pottmann.

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Liu, Y., Pirahmad, O., Wang, H. et al. Helical surfaces with a constant ratio of principal curvatures. Beitr Algebra Geom 64, 1087–1105 (2023). https://doi.org/10.1007/s13366-022-00670-y

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