Locally injective parametrization with arbitrary fixed boundaries

Ofir Weber, Denis Zorin

Research output: Contribution to journalConference articlepeer-review

68 Scopus citations


We present an algorithm for mapping a triangle mesh, which is homeomorphic to a disk, to a planar domain with arbitrary fixed boundaries. The algorithm is guaranteed to produce a globally bi-jective map when the boundary is fixed to a shape that does not self-intersect. Obtaining a one-to-one map is of paramount importance for many graphics applications such as texture mapping. However, for other applications, such as quadrangulation, remesh-ing, and planar deformations, global bijectively may be unnecessarily constraining and requires significant increase on map distortion. For that reason, our algorithm allows the fixed boundary to intersect itself, and is guaranteed to produce a map that is injective locally (if such a map exists). We also extend the basic ideas of the algorithm to support the computation of discrete approximation for extremal quasiconformal maps. The algorithm is conceptually simple and fast. We demonstrate the superior robustness of our algorithm in various settings and configurations in which state-of-the-art algorithms fail to produce injective maps.

Original languageEnglish
Article number75
JournalACM Transactions on Graphics
Issue number4
StatePublished - 2014
Event41st International Conference and Exhibition on Computer Graphics and Interactive Techniques, ACM SIGGRAPH 2014 - Vancouver, BC, Canada
Duration: 10 Aug 201414 Aug 2014

Bibliographical note

Funding Information:
This research was supported by grant number 2012264 from the United States-Israel Binational Science Foundation (BSF). We gratefully acknowledge the support of NVIDIA Corporation with the donation of the Quadro k6000 GPU used for this research.


  • Bijective mapping
  • Conformal mapping
  • Deformation
  • Parametrization
  • Quadrangulation
  • Quasiconformal
  • Remeshing


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