Biharmonic coordinates

Ofir Weber, Roi Poranne, Craig Gotsman

Research output: Contribution to journalArticlepeer-review


Barycentric coordinates are an established mathematical tool in computer graphics and geometry processing, providing a convenient way of interpolating scalar or vector data from the boundary of a planar domain to its interior. Many different recipes for barycentric coordinates exist, some offering the convenience of a closed-form expression, some providing other desirable properties at the expense of longer computation times. For example, harmonic coordinates, which are solutions to the Laplace equation, provide a long list of desirable properties (making them suitable for a wide range of applications), but lack a closed-form expression. We derive a new type of barycentric coordinates based on solutions to the biharmonic equation. These coordinates can be considered a natural generalization of harmonic coordinates, with the additional ability to interpolate boundary derivative data. We provide an efficient and accurate way to numerically compute the biharmonic coordinates and demonstrate their advantages over existing schemes. We show that biharmonic coordinates are especially appealing for (but not limited to) 2D shape and image deformation and have clear advantages over existing deformation methods.

Original languageEnglish
Pages (from-to)2409-2422
Number of pages14
JournalComputer Graphics Forum
Issue number8
StatePublished - Dec 2012
Externally publishedYes


  • barycentric coordinates
  • boundary element method
  • image deformation

ASJC Scopus subject areas

  • Computer Graphics and Computer-Aided Design


Dive into the research topics of 'Biharmonic coordinates'. Together they form a unique fingerprint.

Cite this