A new class of rational parametrization has been developed and it was used to generate a new family of rational k functions B-splines which depends on an index α ∈ ]−∞ , 0[ ∪ ]1 , +∞[. This family of functions verifies, among other things, the properties of positivity, of partition of the unit and, for a given degree k, constitutes a true basis approximation of continuous functions. We loose, however, the regularity classical optimal linked to the multiplicity of nodes, which we recover in the asymptotic case, when α → ∞. The associated B-splines curves verify the traditional properties particularly that of a convex hull and we see a certain “conjugated symmetry” related to α. The case of open knot vectors without an inner node leads to a new family of rational Bezier curves that will be separately, object of in-depth analysis.