A unified framework for primal/dual quadrilateral subdivision schemes

Denis Zorin, Peter Schröder

Research output: Contribution to journalArticlepeer-review


Quadrilateral subdivision schemes come in primal and dual varieties, splitting faces or respectively vertices. The scheme of Catmull-Clark is an example of the former, while the Doo-Sabin scheme exemplifies the latter. In this paper we consider the construction of an increasing sequence of alternating primal/dual quadrilateral subdivision schemes based on a simple averaging approach. Beginning with a vertex split step we successively construct variants of Doo-Sabin and Catmull-Clark schemes followed by novel schemes generalizing B-splines of bidegree up to nine. We prove the schemes to be C1 at irregular surface points, and analyze the behavior of the schemes as the number of averaging steps increases. We discuss a number of implementation issues common to all quadrilateral schemes. In particular we show how both primal and dual quadrilateral schemes can be implemented in the same code, opening up new possibilities for more flexible geometric modeling applications and p-versions of the Subdivision Element Method. Additionally we describe a simple algorithm for adaptive subdivision of dual schemes.

Original languageEnglish (US)
Pages (from-to)429-454
Number of pages26
JournalComputer Aided Geometric Design
Issue number5
StatePublished - Jun 2001


  • Approximating
  • Averaging
  • B-splines
  • Catmull-Clark
  • Doo-Sabin
  • Dual
  • Primal
  • Quadrilateral
  • Subdivision

ASJC Scopus subject areas

  • Modeling and Simulation
  • Automotive Engineering
  • Aerospace Engineering
  • Computer Graphics and Computer-Aided Design


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