## Abstract

This paper deals with the approximation of the unfolding of a smooth globally developable surface (i.e. "isometric" to a domain of script E sign^{2}) with a triangulation. We prove the following result: let T_{n} be a sequence of globally developable triangulations which tends to a globally developable smooth surface S in the Hausdorff sense. If the normals of T_{n} tend to the normals of S, then the shape of the unfolding of T_{n} tends to the shape of the unfolding of S. We also provide several examples: first, we show globally developable triangulations whose vertices are close to globally developable smooth surfaces; we also build sequences of globally developable triangulations inscribed on a sphere, with a number of vertices and edges tending to infinity. Finally, we also give an example of a triangulation with strictly negative Gauss curvature at any interior point, inscribed in a smooth surface with a strictly positive Gauss curvature. The Gauss curvature of these triangulations becomes positive (at each interior vertex) only by switching some of their edges.

Original language | English (US) |
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Pages (from-to) | 393-418 |

Number of pages | 26 |

Journal | Discrete and Computational Geometry |

Volume | 36 |

Issue number | 3 |

DOIs | |

State | Published - Jan 1 2006 |

## ASJC Scopus subject areas

- Theoretical Computer Science
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics