Geometrically continuous piecewise Chebyshevian NU(R)BS

By piecewise Chebyshevian splines we mean splines with pieces taken from different Extended Chebyshev spaces all of the same dimension, and with connection matrices at the knots. Within this very large and crucial class of splines, we are more specifically concerned with those which are good for des...

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Published inBIT Vol. 60; no. 3; pp. 687 - 714
Main Author Mazure, Marie-Laurence
Format Journal Article
LanguageEnglish
Published Dordrecht Springer Netherlands 01.09.2020
Springer Nature B.V
Springer Verlag
Subjects
Chebyshev approximation
Computational Mathematics and Numerical Analysis
Equivalence
Mathematics
Mathematics and Statistics
Numeric Computing
Geometric design
Blossoms
Piecewise Chebyshevian splines
Shape effects
65D17
Rational splines
65D07
NURBS
blossoms
shape effects
geometric design
rational splines
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Abstract By piecewise Chebyshevian splines we mean splines with pieces taken from different Extended Chebyshev spaces all of the same dimension, and with connection matrices at the knots. Within this very large and crucial class of splines, we are more specifically concerned with those which are good for design , in the sense that they possess blossoms, or, equivalently, refinable B-spline bases. In practice, this subclass is known to be characterised by the existence of (infinitely many) piecewise generalised derivatives with respect to which the continuity between consecutive pieces is controlled by identity matrices. Somehow inherent in the previous characterisation, the construction of all associated rational spline spaces creates an equivalence relation between piecewise Chebyshevian spline spaces good for design, among which the famous classical rational splines. We investigate this equivalence relation along with the natural question: Is it or not worthwhile considering the rational framework since it does not enlarge the set of resulting splines? This explains the parentheses inside the acronym NU(R)BS.
AbstractList By piecewise Chebyshevian splines we mean splines with pieces taken from different Extended Chebyshev spaces all of the same dimension, and with connection matrices at the knots. Within this very large and crucial class of splines, we are more specifically concerned with those which are good for design, in the sense that they possess blossoms, or, equivalently, refinable B-spline bases. In practice, this subclass is known to be characterised by the existence of (infinitely many) piecewise generalised derivatives with respect to which the continuity between consecutive pieces is controlled by identity matrices. Somehow inherent in the previous characterisation, the construction of all associated rational spline spaces creates an equivalence relation between piecewise Chebyshevian spline spaces good for design, among which the famous classical rational splines. We investigate this equivalence relation along with the natural question: Is it or not worthwhile considering the rational framework since it does not enlarge the set of resulting splines? This explains the parentheses inside the acronym NU(R)BS.
By piecewise Chebyshevian splines we mean splines with pieces taken from different Extended Chebyshev spaces all of the same dimension, and with connection matrices at the knots. Within this very large and crucial class of splines, we are more specifically concerned with those which are good for design , in the sense that they possess blossoms, or, equivalently, refinable B-spline bases. In practice, this subclass is known to be characterised by the existence of (infinitely many) piecewise generalised derivatives with respect to which the continuity between consecutive pieces is controlled by identity matrices. Somehow inherent in the previous characterisation, the construction of all associated rational spline spaces creates an equivalence relation between piecewise Chebyshevian spline spaces good for design, among which the famous classical rational splines. We investigate this equivalence relation along with the natural question: Is it or not worthwhile considering the rational framework since it does not enlarge the set of resulting splines? This explains the parentheses inside the acronym NU(R)BS.
Author Mazure, Marie-Laurence
Author_xml – sequence: 1
  givenname: Marie-Laurence
  surname: Mazure
  fullname: Mazure, Marie-Laurence
  email: Marie-Laurence.Mazure@univ-grenoble-alpes.fr
  organization: Laboratoire Jean Kuntzmann, UMR5224, Université Grenoble-Alpes
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Issue 3
Keywords Geometric design
Blossoms
Piecewise Chebyshevian splines
Shape effects
65D17
Rational splines
65D07
NURBS
blossoms
shape effects
geometric design
rational splines
Language English
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Snippet By piecewise Chebyshevian splines we mean splines with pieces taken from different Extended Chebyshev spaces all of the same dimension, and with connection...
By piecewise Chebyshevian splines we mean splines with pieces taken from different Extended Chebyshev spaces all of the same dimension, and with connection...
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StartPage 687
SubjectTerms Chebyshev approximation
Computational Mathematics and Numerical Analysis
Equivalence
Mathematics
Mathematics and Statistics
Numeric Computing
Title Geometrically continuous piecewise Chebyshevian NU(R)BS
URI https://link.springer.com/article/10.1007/s10543-019-00795-y
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