Simple formulas for quasiconformal plane deformations

Yaron Lipman, Vladimir G. Kim, Thomas A. Funkhouser

Research output: Contribution to journalArticlepeer-review

19 Scopus citations


We introduce a simple formula for 4-point planar warping that produces provably good 2D deformations. In contrast to previous work, the new deformations minimize the maximum conformal distortion and spread the distortion equally across the domain. We derive closed-form formulas for computing the 4-point interpolant and analyze its properties. We further explore applications to 2D shape deformations by building local deformation operators that use thin-plate splines to further deform the 4-point interpolant to satisfy certain boundary conditions. Although this modification no longer has any theoretical guarantees, we demonstrate that, practically, these local operators can be used to create compound deformations with fewer control points and smaller worst-case distortions in comparisons to the state-of-the-art.

Original languageEnglish (US)
Article number124
JournalACM Transactions on Graphics
Issue number5
StatePublished - Aug 2012

All Science Journal Classification (ASJC) codes

  • Computer Graphics and Computer-Aided Design


  • Conformal
  • Deformation
  • Planar warping
  • Quasiconformal


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