### Abstract

In the present paper, we propose a new discrete surface theory on 3-valent embedded graphs in the 3-dimensional Euclidean space which are not necessarily “discretization” or “approximation” of smooth surfaces. The Gauss curvature and the mean curvature of discrete surfaces are defined which satisfy properties corresponding to the classical surface theory. We also discuss the convergence of a family of subdivided discrete surfaces of a given 3-valent discrete surface by using the Goldberg–Coxeter construction. Although discrete surfaces in general have no corresponding smooth surfaces, we may find some in the limit.

Original language | English |
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Pages (from-to) | 24-54 |

Number of pages | 31 |

Journal | Computer Aided Geometric Design |

Volume | 58 |

DOIs | |

Publication status | Published - 2017 Nov |

### Keywords

- Discrete curvature
- Discrete minimal surface
- Discrete surfaces theory

### ASJC Scopus subject areas

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

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## Cite this

Kotani, M., Naito, H., & Omori, T. (2017). A discrete surface theory.

*Computer Aided Geometric Design*,*58*, 24-54. https://doi.org/10.1016/j.cagd.2017.09.002