Provably good planar mappings

Roi Poranne, Yaron Lipman

Research output: Contribution to journalArticlepeer-review


The problem of planar mapping and deformation is central in computer graphics. This paper presents a framework for adapting general, smooth, function bases for building provably good planar mappings. The term "good" in this context means the map has no fold-overs (injective), is smooth, and has low isometric or conformal distortion. Existing methods that use mesh-based schemes are able to achieve injectivity and/or control distortion, but fail to create smooth mappings, unless they use a prohibitively large number of elements, which slows them down. Meshless methods are usually smooth by construction, yet they are not able to avoid fold-overs and/or control distortion. Our approach constrains the linear deformation spaces induced by popular smooth basis functions, such as B-Splines, Gaussian and Thin-Plate Splines, at a set of collocation points, using specially tailored convex constraints that prevent fold-overs and high distortion at these points. Our analysis then provides the required density of collocation points and/or constraint type, which guarantees that the map is injective and meets the distortion constraints over the entire domain of interest. We demonstrate that our method is interactive at reasonably complicated settings and compares favorably to other state-of-the-art mesh and meshless planar deformation methods.

Original languageEnglish
Article number76
JournalACM Transactions on Graphics
Issue number4
StatePublished - 2014
Externally publishedYes
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 study was part of the Llyn Brianne Acid Waters Project funded by the National Rivers Authority, Department of Environment, and Welsh Office. Staff from the NRA and the Catchment Research Group at UC Cardiff collected and analysed samples. Catherine Shaw of the Institute of Hydrology helped retrieve the data. The author is also grateful to the Carnegie Trust for the Universities of Scotland for financial support, and to two anonymous referees for their helpful comments.


  • Bijective mappings
  • Bounded isometric distortion
  • Conformal distortion
  • Meshless deformation

ASJC Scopus subject areas

  • Computer Graphics and Computer-Aided Design


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