## Abstract

Singularities in structured meshes are vertices that have an irregular valency.The integer irregularity in valency is called the singularity index of the vertex of the mesh. Singularities in cross-fields are closely related which are isolated points where the cross-field vectors are defined in its limit neighbourhood but not at the point itself. For a closed surface the genus determines the minimum number of singularities that are required in a structured mesh or in a cross-field on the surface. Adding boundaries and forcing conformity of the mesh or alignment of the cross-field to them also affects the minimum number of singularities required. In this paper a simple formula is derived from Bunin's Continuum Theory for Unstructured Mesh Generation (Bunin, 2008) that specifies the net sum of singularity indices that must occur in a cross-field with even numbers of vectors on a face or surface region with alignment conditions. The formula also applies to mesh singularities in quadrilateral and triangle meshes and the correspondence to 3-D hexahedral meshes is related. Some potential applications are discussed.

Original language | English |
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Pages (from-to) | 11-25 |

Number of pages | 15 |

Journal | CAD Computer Aided Design |

Volume | 105 |

Early online date | 5 Jul 2018 |

DOIs | |

Publication status | Published - 1 Dec 2018 |

## Keywords

- Cross-field
- Singularities
- Structured mesh