Volume-Preserving free-form solids

Ari Rappoport*, Alia Sheffer, Michel Bercovier

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

45 Scopus citations


Some important trends in geometric modeling are the reliance on solid models rather than surface-based models and the enhancement of the expressive power of models, by using free-form objects in addition to the usual geometric primitives and by incorporating physical principles. An additional trend is the emphasis on interactive performance. In this paper we integrate all of these requirements in a single geometric primitive by endowing the tri-variate tensor product free-form solid with several important physical properties, including volume and internal deformation energy. Volume preservation is of benefit in several application areas of geometric modeling, including computer animation, industrial design and mechanical engineering. However, previous physics-based methods, which usually have used some forms of energy, have neglected the issue of volume (or area) preservation. We present a novel method for modeling an object composed of several tensor-product solids while preserving the desired volume of each primitive and ensuring high-order continuity constraints between the primitives. The method utilizes the Uzawa algorithm for non-linear optimization, with objective functions based on deformation energy or least squares. We show how the algorithm can be used in an interactive environment by relaxing exactness requirements while the user interactively manipulates free-form solid primitives. On current workstations, the algorithm runs in real-time for tri-quadratic volumes and close to real-time for tri-cubic volumes.

Original languageAmerican English
Pages (from-to)19-27
Number of pages9
JournalIEEE Transactions on Visualization and Computer Graphics
Issue number1
StatePublished - 1996


  • Continuity constraints
  • Energy constraints
  • Free-form deformations (ffd)
  • Free-form solids
  • Physics-based modeling
  • Uzawa's algorithm
  • Volume preservation


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