On parametric smoothness of generalised B-spline curves

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In a Riemannian manifold, generalised B-spline curves are piecewise C curves defined by a generalisation of the classical Cox-de Boor algorithm, in which line segments are replaced by minimal geodesics. Their applications include rigid body motion planning and computer graphics. We prove that, like classical B-spline curves, they are C1 at knots of multiplicity at most m - 1, where m is the degree. We then compute the difference between their left and right (covariant) accelerations at knots of multiplicity at most m - 2. Unlike classical B-spline curves, generalised B-spline curves are not in general C2 at such knots.

Original languageEnglish
Pages (from-to)655-668
Number of pages14
JournalComputer Aided Geometric Design
Issue number8
Publication statusPublished - Nov 2006
Externally publishedYes


  • B-spline
  • Cox-de Boor algorithm
  • Riemannian manifold
  • Spline

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