## Abstract

Motivated by applications in architectural geometry, we study and compute surfaces with a constant ratio of principal curvatures (CRPC surfaces) based on their characteristic parameterizations. For negative Gaussian curvature K, these parameterizations are asymptotic. For positive K they are conjugate and symmetric with respect to the principal curvature directions. CRPC surfaces are described by characteristic parameterizations whose parameter lines form a constant angle. We use them to derive characteristic parameterizations of rotational CRPC surfaces in a simple geometric way. Pairs of such surfaces with principal curvature ratio κ_{1}/κ_{2}=±a can be seen as equilibrium shapes and reciprocal force diagrams of each other. We then introduce discrete CRPC surfaces, expressed via discrete isogonal characteristic nets, and show how to efficiently compute them through numerical optimization. In particular, we derive discrete helical and spiral CRPC surfaces. We provide various ways how these and other special types of CRPC surfaces can serve as a basis for computational design of more general CRPC surfaces. Our computational tools may also serve as an experimental basis for mathematical studies of the largely unexplored class of CRPC surfaces.

Original language | English (US) |
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Article number | 102074 |

Journal | Computer Aided Geometric Design |

Volume | 93 |

DOIs | |

State | Published - Feb 2022 |

### Bibliographical note

Publisher Copyright:© 2022 Elsevier B.V.

## Keywords

- Architectural geometry
- Characteristic parameterization
- Constant ratio of principal curvatures
- Discrete differential geometry
- Principal symmetric net
- Weingarten surface

## ASJC Scopus subject areas

- Modeling and Simulation
- Automotive Engineering
- Aerospace Engineering
- Computer Graphics and Computer-Aided Design