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Communication Dans Un Congrès Année : 2015

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 $\mathcal C$ ?
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Dates et versions

hal-01162969 , version 1 (11-06-2015)

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  • HAL Id : hal-01162969 , version 1

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Marie-Laurence Mazure. NURBS or not NURBS. CGTA 2015: Conference on Geometry: Theory and Applications, Univ. Linz, Austria, Jun 2015, Kefermarkt, Austria. ⟨hal-01162969⟩
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