The geometry of Tchebycheffian splines

This paper shows that many properties of Bézier and B-spline curves hold for a much wider class of curves. Using a "normal curve" associated with an extended Tchebycheff space, we derive a Bézier representation of Tchebycheffian spline curve segments. These are affine or projective images of segments of the normal curve. A generalization of the blossoming method allows us to study Tchebycheffian B-spline curves and their segments in a simple geometric way. The basic algorithms such as knot insertion and construction of the Bézier points are described. Whereas the generation of tensor product surfaces is straightforward, some preliminary studies indicate that a similarly natural generalization of Bézier triangles does not exist.