de Boor-Fix dual functionals and algorithms for Tchebycheffian B-spline curves

The de Boor-Fix dual functionals are a potent tool for deriving results about piecewise polynomial B-spline curves. In this paper we extend these functionals to Tchebycheffian B-spline curves and then use them to derive fundamental algorithms that are natural generalizations of algorithms for piecewise polynomial B-spline algorithms. Then, as a further example of the utility of this approach, we introduce “geometrically continuous Tchebycheffian spline curves,” and show that a further generalization works for them as well.