Two remarks on Tau-splines

ABSTRACT

We present a Bézier representation of &tgr;-splines, curvature and torsion-continuous quintics, which were introduced in CAGD by Hagen in 1985 [32]. Explicit formulas are given for the conversion from Bézier representation to &tgr;-spline representation, and vice versa. Thus, by embedding the Bézier representation in a &Bgr;-spline representation of curvature and torsion-continuous quintic spline curves, given in [20], a &Bgr;-spline-Bézier representation of &tgr;-splines results. Second, positivity conditions for the design parameters of the Bézier representation and certain ranges of tension values are derived, which ensure properties like the convex hull and the variation-diminishing property of the &Bgr;-spline-Bézier representation.

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INDEX TERMS

Primary Classification:
  I. Computing Methodologies
  I.3 COMPUTER GRAPHICS
      I.3.5 Computational Geometry and Object Modeling
          Subjects: Curve, surface, solid, and object representations

Additional Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.1 Interpolation
          Subjects: Spline and piecewise polynomial interpolation

General Terms:
Algorithms, Design, Theory