THE GEOMETRY OF TCHEBYCHEFFIAN SPLINES

This paper shows that many properties of Bezier and B-spline curves ho ld for a much wider class of curves. Using a ''normal curve'' associat ed with an extended Tchebycheff space, we derive a Bezier representati on of Tchebycheffian spline curve segments. These are affine or projec tive images of segments of the normal curve. A generalization of the b lossoming method allows us to study Tchebycheffian B-spline curves and their segments in a simple geometric way. The basic algorithms such a s knot insertion and construction of the Bezier points are described. Whereas the generation of tensor product surfaces is straightforward, some preliminary studies indicate that a similarly natural generalizat ion of Bezier triangles does not exist.