NURBS or not NURBS
Résumé
In this talk the expression NURBS is meant with the general meaning of Geometrically
Continuous Piecewise Quasi-Chebyshevian NURBS, that is, rational B-splines built from
the largest class $\mathcal C$ of spline spaces which can be used for design. A spline space in $\mathcal C$ has
different Quasi Extended Chebyshev spaces (QEC-spaces) as section-spaces, and
we allow connection matrices at the knots. Moreover, as usual for design, we must require
the presence of blossoms. We recently achieved a recursive constructive characterisation
of the class $\mathcal C$. The important part of this characterisation consists in proving
that a spline space in $\mathcal C$ can automatically be based on infinitely many possible Piecewise
QEC-spaces.
Interpreted in an appropriate way, the first step of this construction can be viewed as the
construction of all rational spline spaces based on a spline space in $\mathcal C$. This guarantees that
any such rational spline space belongs in turn to the class $\mathcal C$ . It thus possesses blossoms as
well as B-spline bases (NURBS in the sense explained earlier).
The classical NURBS are thus examples of parametrically continuous splines in the
class $\mathcal C$. Compared to their polynomial counterparts, one major interest of introducing them
was the shape effects permitted by the parameters defining them. Now, a natural question
arises: is it worthwhile building NURBS when the class $\mathcal C$ already provides us with such
a great variety of shape parameters (coming either from the section-spaces or from the
connection matrices) and of exactly represented curves, and when this does not increase the
class $\mathcal C$ ?