B-spline bases and osculating flats: One result of H.-P. Seidel revisited

Marie-Laurence Mazure

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 36, Issue: 6, page 1177-1186
  • ISSN: 0764-583X

Abstract

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Along with the classical requirements on B-splines bases (minimal support, positivity, normalization) we show that it is natural to introduce an additional “end point property". When dealing with multiple knots, this additional property is exactly the appropriate requirement to obtain the poles of nondegenerate splines as intersections of osculating flats at consecutive knots.

How to cite

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Mazure, Marie-Laurence. "B-spline bases and osculating flats: One result of H.-P. Seidel revisited." ESAIM: Mathematical Modelling and Numerical Analysis 36.6 (2010): 1177-1186. <http://eudml.org/doc/194145>.

@article{Mazure2010,
abstract = { Along with the classical requirements on B-splines bases (minimal support, positivity, normalization) we show that it is natural to introduce an additional “end point property". When dealing with multiple knots, this additional property is exactly the appropriate requirement to obtain the poles of nondegenerate splines as intersections of osculating flats at consecutive knots. },
author = {Mazure, Marie-Laurence},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Geometric design; B-spline basis; blossoming; osculating flats.; geometric design; osculating flats},
language = {eng},
month = {3},
number = {6},
pages = {1177-1186},
publisher = {EDP Sciences},
title = {B-spline bases and osculating flats: One result of H.-P. Seidel revisited},
url = {http://eudml.org/doc/194145},
volume = {36},
year = {2010},
}

TY - JOUR
AU - Mazure, Marie-Laurence
TI - B-spline bases and osculating flats: One result of H.-P. Seidel revisited
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 36
IS - 6
SP - 1177
EP - 1186
AB - Along with the classical requirements on B-splines bases (minimal support, positivity, normalization) we show that it is natural to introduce an additional “end point property". When dealing with multiple knots, this additional property is exactly the appropriate requirement to obtain the poles of nondegenerate splines as intersections of osculating flats at consecutive knots.
LA - eng
KW - Geometric design; B-spline basis; blossoming; osculating flats.; geometric design; osculating flats
UR - http://eudml.org/doc/194145
ER -

References

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  1. N. Dyn and C.A. Micchelli, Piecewise polynomial spaces and geometric continuity of curves. Numer. Math.54 (1988) 319-337.  
  2. T.N.T. Goodman, Properties of β-splines. J. Approx. Theory44 (1985) 132-153.  
  3. M.-L. Mazure, Blossoming: a geometrical approach. Constr. Approx.15 (1999) 33-68.  
  4. M.-L. Mazure, Quasi-Chebyshev splines with connexion matrices. Application to variable degree polynomial splines. Comput. Aided Geom. Design18 (2001) 287-298.  
  5. H. Pottmann, The geometry of Tchebycheffian splines. Comput. Aided Geom. Design10 (1993) 181-210.  
  6. L. Ramshaw, Blossoms are polar forms. Comput. Aided Geom. Design6 (1989) 323-358.  
  7. H.-P. Seidel, New algorithms and techniques for computing with geometrically continuous spline curves of arbitrary degree. RAIRO Modél. Math. Anal. Numér.26 (1992) 149-176.  
  8. H.-P. Seidel, Polar forms for geometrically continuous spline curves of arbitrary degree. ACM Trans. Graphics12 (1993) 1-34.  

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