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.