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Surfaces of constant principal-curvatures ratio in isotropic geometry

Khusrav Yorov, Mikhail Skopenkov, Helmut Pottmann

TL;DR

This work develops a comprehensive classification of CRPC (constant ratio of principal curvatures) surfaces in Euclidean and simply isotropic geometry, covering rotational, channel, ruled, helical, and translational families under regularity assumptions. It unifies diverse methods—differential geometry, line geometry, Lie sphere geometry, ODEs, and algebraic geometry—to derive explicit models, such as isotropic rotational CRPC surfaces $z=(x^2+y^2)^{(1+a)/2}$ or $z=(x^2+y^2)^{(1+a)/(2a)}$ (and $z=\log(x^2+y^2)$ for $a=-1$), parabolic rotational CRPC surfaces $z=x^2+ay^2$, and various translational and dual-translational families with explicit parameterizations. The paper establishes duality results showing CRPC properties are preserved under isotropic metric duality and provides complete classifications in several isotropic and Euclidean contexts, including hyperbolic paraboloid, helicoid, and spiral ruled surfaces, as well as helically and translationally generated CRPC surfaces with exact forms. The findings have implications for geometric modeling and architectural design via CRPC structures and reveal deep connections between curvature constraints, symmetry groups, and duality in non-Euclidean geometries. The work also outlines open problems, notably translational surfaces with two spatial generators and extensions to other Cayley--Klein geometries, inviting further exploration of CRPC surfaces and their singularities.

Abstract

We study surfaces with a constant ratio of principal curvatures in Euclidean and simply isotropic geometries and characterize rotational, channel, ruled, helical, and translational surfaces of this kind under some technical restrictions (the latter two cases only in isotropic geometry). We use the interlacing of various methods of differential geometry, including line geometry and Lie sphere geometry, ordinary differential equations, and elementary algebraic geometry.

Surfaces of constant principal-curvatures ratio in isotropic geometry

TL;DR

This work develops a comprehensive classification of CRPC (constant ratio of principal curvatures) surfaces in Euclidean and simply isotropic geometry, covering rotational, channel, ruled, helical, and translational families under regularity assumptions. It unifies diverse methods—differential geometry, line geometry, Lie sphere geometry, ODEs, and algebraic geometry—to derive explicit models, such as isotropic rotational CRPC surfaces or (and for ), parabolic rotational CRPC surfaces , and various translational and dual-translational families with explicit parameterizations. The paper establishes duality results showing CRPC properties are preserved under isotropic metric duality and provides complete classifications in several isotropic and Euclidean contexts, including hyperbolic paraboloid, helicoid, and spiral ruled surfaces, as well as helically and translationally generated CRPC surfaces with exact forms. The findings have implications for geometric modeling and architectural design via CRPC structures and reveal deep connections between curvature constraints, symmetry groups, and duality in non-Euclidean geometries. The work also outlines open problems, notably translational surfaces with two spatial generators and extensions to other Cayley--Klein geometries, inviting further exploration of CRPC surfaces and their singularities.

Abstract

We study surfaces with a constant ratio of principal curvatures in Euclidean and simply isotropic geometries and characterize rotational, channel, ruled, helical, and translational surfaces of this kind under some technical restrictions (the latter two cases only in isotropic geometry). We use the interlacing of various methods of differential geometry, including line geometry and Lie sphere geometry, ordinary differential equations, and elementary algebraic geometry.
Paper Structure (26 sections, 29 theorems, 74 equations, 7 figures)

This paper contains 26 sections, 29 theorems, 74 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Previous work
  3. Preliminaries
  4. Admissible surfaces and isotropic curvatures
  5. Isotropic spheres and circles
  6. Rotational surfaces
  7. Isotropic rotational CRPC surfaces
  8. Euclidean rotational CRPC surfaces
  9. Parabolic rotational CRPC surfaces
  10. Channel surfaces
  11. Euclidean channel CRPC surfaces
  12. Surfaces with both isotropic principal curvatures constant
  13. Isotropic channel CRPC surfaces
  14. Ruled surfaces
  15. Euclidean ruled CRPC surfaces.
  16. ...and 11 more sections

Key Result

Theorem 2

(See sachs, strubecker:1942) Let an admissible surface $\Phi$ have isotropic normal curvature $\kappa_n\ne 0$ at a point $p \in \Phi$ along a surface tangent line $T$. Then the osculating isotropic circles of all curves on $\Phi$ that are tangent to $T$ at $p$ lie on the parabolic isotropic sphere o

Figures (7)

  • Figure 1: An asymptotic gridshell (top) schling:2018. Straight lamellas (middle) follow asymptotic curves of the reference surface and intersect under a right angle. This simplifies the manufacturing process as all steel joints (bottom) are identical but forces the reference surface to be a Euclidean minimal surface. If the intersection angle is constant and not right, then the joints are still identical, and we get more general surfaces: ones with a constant ratio of principal curvatures.
  • Figure 2: Rotational isotropic CRPC surfaces (from the left to the right): the first of two surfaces \ref{['eq-rotational']} for $a>0$, $0>a>-1$, and $a<-1$ respectively; surface \ref{['eq-logarithmoid']}.
  • Figure 3: The notation in the proofs of Lemmas \ref{['l-osculating-Dupin-cyclide']}, \ref{['l-isotropic-circle-center']}, and \ref{['l-conoid']} (from the left to the right).
  • Figure 4: Ruled isotropic CRPC surfaces (from the left to the right): a hyperbolic paraboloid, helicoid \ref{['eq-helicoid']}, and spiral ruled surface \ref{['eq-spiral']}.
  • Figure 5: Helical isotropic CRPC surfaces (from the left to the right): surface \ref{['helicalane-1']} for $a>0$; its outer part; its inner part; surface \ref{['helicalane-1']} for $a<0$; surface \ref{['helical-1']}. The singular curve $\tan^2(u) = a$ of the leftmost surface is depicted in red; it splits the surface into the parts $\tan^2(u) > a$ and $\tan^2(u) < a$ shown separately.
  • ...and 2 more figures

Theorems & Definitions (55)

  • Example 1
  • Theorem 2
  • Proposition 3
  • Proposition 4
  • Theorem 5
  • Lemma 6
  • proof
  • proof : Proof of Theorem \ref{['th-Euclidean-Weingarten']}
  • Lemma 7
  • proof
  • ...and 45 more