Which cross fields can be quadrangulated? Global Parameterization from Prescribed Holonomy Signatures

Hanxiao Shen, Leyi Zhu, Ryan Capouellez, Daniele Panozzo, Marcel Campen, Denis Zorin

Research output: Contribution to journalArticlepeer-review


We describe a method for the generation of seamless surface parametrizations with guaranteed local injectivity and full control over holonomy. Previous methods guarantee only one of the two. Local injectivity is required to enable these parametrizations' use in applications such as surface quadrangulation and spline construction. Holonomy control is crucial to enable guidance or prescription of the parametrization's isocurves based on directional information, in particular from cross-fields or feature curves, and more generally to constrain the parametrization topologically. To this end we investigate the relation between cross-field topology and seamless parametrization topology. Leveraging previous results on locally injective parametrization and combining them with insights on this relation in terms of holonomy, we propose an algorithm that meets these requirements. A key component relies on the insight that arbitrary surface cut graphs, as required for global parametrization, can be homeomorphically modified to assume almost any set of turning numbers with respect to a given target cross-field.

Original languageEnglish (US)
Article number3530187
JournalACM Transactions on Graphics
Issue number4
StatePublished - Jul 22 2022


  • Conformal map
  • Cross field
  • Holonomy
  • Quad mesh
  • Seamless parametrization
  • Turning number

ASJC Scopus subject areas

  • Computer Graphics and Computer-Aided Design


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