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M. R. Jimenez, C. Müller, H. Pottmann:
"Discretizations of Surfaces with Constant Ratio of Principal Curvatures";
Discrete & Computational Geometry, 042019 (2019), 35 pages.

Motivated by applications in architecture, we study surfaces with a constant ratio of principal curvatures. These surfaces are a natural generalization of minimal surfaces, and can be constructed by applying a Christoffel-type transformation to appropriate spherical curvature line parametrizations, both in the smooth setting and in a discretization with principal nets. We link this Christoffel-type transformation to the discrete curvature theory for parallel meshes and characterize nets that admit these transformations. In the case of negative curvature, we also present a discretization of asymptotic nets. This case is suitable for design and computation, and forms the basis for a special type of architectural support structures, which can be built by bending flat rectangular strips of inextensible material, such as sheet metal.

Discrete differential geometry Weingarten surface Christoffel-type transformation Conjugate net Support structure Bending rectangular strips Asymptotic net

http://dx.doi.org/10.1007/s00454-019-00098-7

https://link.springer.com/content/pdf/10.1007%2Fs00454-019-00098-7.pdf