NURBS or not NURBS?

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