Singularities in structured meshes and cross-fields

Harold J. Fogg, Liang Sun, Jonathan E. Makem, Cecil G. Armstrong, Trevor T. Robinson

Research output: Contribution to journalArticlepeer-review

14 Citations (Scopus)
125 Downloads (Pure)


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 languageEnglish
Pages (from-to)11-25
Number of pages15
JournalCAD Computer Aided Design
Early online date5 Jul 2018
Publication statusPublished (in print/issue) - 1 Dec 2018


  • Cross-field
  • Singularities
  • Structured mesh


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