Constructive Approximation

, Volume 7, Issue 1, pp 257–279

Symmetric recursive algorithms for surfaces: B-patches and the de boor algorithm for polynomials over triangles

  • Hans-Peter Seidel

DOI: 10.1007/BF01888157

Cite this article as:
Seidel, HP. Constr. Approx (1991) 7: 257. doi:10.1007/BF01888157


Using the concept of a symmetric recursive algorithm, we construct a new patch representation for bivariate polynomials: the B-patch. B-patches share many properties with B-spline segments: they are characterized by their control points and by a three-parameter family of knots. If the knots in each family coincide, we obtain the Bézier representation of a bivariate polynomial over a triangle. Therefore B-patches are a generalization of Bézier patches. B-patches have a de Boor-like evaluation algorithm, and, as in the case of B-spline curves, the control points of a B-patch can be expressed by simply inserting a sequence of knots into the corresponding polar form. In particular, this implies linear independence of the blending functions. B-patches can be joined smoothly and they have an algorithm for knot insertion that is completely similar to Boehm's algorithm for curves.

AMS classification


Key words and phrases

Bézier patchBlossomde Boor algorithmB-patchB-splinede Casteljau algorithmControl pointKnot insertionKnot netPolar formSymmetric algorithmTriangular patch

Copyright information

© Springer-Verlag New York Inc 1991

Authors and Affiliations

  • Hans-Peter Seidel
    • 1
  1. 1.Wilhelm-Schickard-Institut für InformatikUniversität TübingenTübingenGermany