Geometrically Controlled 4-Point Interpolatory Schemes

  • Martin Marinov
  • Nira Dyn
  • David Levin
Part of the Mathematics and Visualization book series (MATHVISUAL)


We present several non-linear 4-point interpolatory schemes, derived from the “classical” linear 4-point scheme. These new schemes have variable tension parameter instead of the fixed tension parameter in the linear 4-point scheme. The tension parameter is adapted locally according to the geometry of the control polygon within the 4-point stencil. This allows the schemes to remain local and in the same time to achieve two important shape-preserving properties - artifact elimination and convexity-preservation. The proposed schemes are robust and have special features such as “double-knot” edges corresponding to continuity without geometrical smoothness and inflection edges support for convexity-preservation. A convergence proof is given and experimental smoothness analysis is done in detail, which indicates that the limit curves are C1.


Subdivision Scheme Limit Curve Straight Edge Tension Parameter Control Polygon 
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  1. 1.
    Aspert, N., Ebrahimi, T., Vandergheynst, P.: Non-linear subdivision using local spherical coordinates. Computer Aided Geometric Design, 20, 165–187 (2003).MathSciNetCrossRefGoogle Scholar
  2. 2.
    Dyn, N., Gregory, J., Levin., D.: A 4-point interpolatory subdivision scheme for curve design. Computer Aided Geometric Design, 4, 257–268 (1987).MathSciNetCrossRefGoogle Scholar
  3. 3.
    Dyn, N., Kuijt, F., Levin, D., van Damme, R. M. J.: Convexity preservation of the four-point interpolatory subdivision scheme. Computer Aided Geometric Design, 16, 789–792 (1999).MathSciNetCrossRefGoogle Scholar
  4. 4.
    Dyn, N., Levin, D., Liu, D.: Interpolatory convexity-preserving subdivision schemes for curves and surfaces. Computer Aided Geometric Design, 24, 211–216 (1992).Google Scholar
  5. 5.
    Kuijt, F.: Convexity Preserving Interpolation — Stationary Nonlinear Subdivision and Splines. PhD thesis, University of Twenty, Faculty of Mathematical Sciences, (1998).Google Scholar
  6. 6.
    Levin, D.: Using Laurent polynomial representation for the analysis of the nonuniform binary subdivision schemes. Advances in Computational Mathematics, 11, 41–54 (1999).zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Martin Marinov
    • 1
  • Nira Dyn
    • 2
  • David Levin
    • 2
  1. 1.Computer Graphics GroupRWTH AachenGermany
  2. 2.School of Mathematical SciencesTel-Aviv UniversityIsrael

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