On parametric smoothness of generalised B-spline curves

Authors:
Tomasz Popiel
Affiliations:
School of Mathematics and Statistics, The University of Western Australia, Crawley, Western Australia, Australia
Venue:
Computer Aided Geometric Design
Year:
2006

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Abstract

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.