Geometric conditions for tangent continuity of interpolatory planar subdivision curves

Nira Dyn, Kai Hormann*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


Curve subdivision is a technique for generating smooth curves from initial control polygons by repeated refinement. The most common subdivision schemes are based on linear refinement rules, which are applied separately to each coordinate of the control points, and the analysis of these schemes is well understood. Since the resulting limit curves are not sufficiently sensitive to the geometry of the control polygons, there is a need for geometric subdivision schemes. Such schemes take the geometry of the control polygons into account by using non-linear refinement rules and are known to generate limit curves with less artefacts. Yet, only few tools exist for their analysis, because the non-linear setting is more complicated. In this paper, we derive sufficient conditions for a convergent interpolatory planar subdivision scheme to produce tangent continuous limit curves. These conditions as well as the proofs are purely geometric and do not rely on any parameterization.

Original languageEnglish
Pages (from-to)332-347
Number of pages16
JournalComputer Aided Geometric Design
Issue number6
StatePublished - Aug 2012


  • Convergence
  • Non-linear interpolatory subdivision
  • Tangent continuity


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