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.
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