Generalized Rational B-Spline Bases via Homographic Parametrization and Applications

A new class of rational parametrization has been developed and was used to generate a new family of rational B-splines,  αGiki=0k, which depend on a parameter α∈−∞,0∪1,+∞. This family of functions satisfies, among other properties, positivity and the partition of unity. For a given degree k, it constitutes a proper basis for the approximation of continuous functions. However, we lose the classical optimal regularity associated with knot multiplicity, which is recovered in the asymptotic case when α⟶∞. The associated B-spline curves satisfy the traditional properties, particularly the convex hull property, and exhibit a form of “conjugated symmetry†with respect to α. The case of open knot vectors without interior knots leads to a new family of rational Bézier curves, which will be the subject of a separate, in-depth analysis. Consequently, a comparative analysis with the classical NURBS framework has been developed in order to position our construction precisely within the landscape of existing rational methods. We have also proved, by induction on the degree, that the basis functions are differentiable with respect to α on the admissible domain. Numerical experiments visually and quantitatively confirm the theoretical results established in this work.