L-system specification of knot-insertion rules for non-uniform B-spline subdivision

Vincent Nivoliers; Cédric Gérot; Victor Ostromoukhov; Neil F. Stewart

hal-00659465, version 1

L-system specification of knot-insertion rules for non-uniform B-spline subdivision

Vincent Nivoliers ¹, Cédric Gérot (, ) ², Victor Ostromoukhov ³, Neil F. Stewart ⁴

Computer Aided Geometric Design 29, 2 (2012) 150-161

Abstract: Subdivision schemes are based on a hierarchy of knot grids in parameter space. A univariate grid hierarchy is regular if all knots are equidistant on each level, and irregular otherwise. We use L-systems to design a wide class of systematically described irregular grid hierarchies. Furthermore, we give sufficient conditions on the L-system which guarantee that the subdivision scheme, based on the non-uniform B-spline of degree d defined on the initial knot grid, is uniformly convergent. If n is the number of symbols in the alphabet of the L-system, this subdivision scheme is defined with a finite set of masks (at most nd+1) which does not depend on the subdivision step. We provide an implementation of such schemes which is available as a worksheet for Sage software.

1: ALICE (INRIA Nancy - Grand Est / LORIA)
INRIA – CNRS : UMR7503 – Université de Lorraine
2: Grenoble Images Parole Signal Automatique (GIPSA-lab)
CNRS : UMR5216 – Université Joseph Fourier - Grenoble I – Université Pierre-Mendès-France - Grenoble II – Université Stendhal - Grenoble III – Institut Polytechnique de Grenoble - Grenoble Institute of Technology
3: Département d'Informatique et de Recherche Opérationnelle [Montreal] (DIRO)
Université de Montréal
4: Laboratoire d'Informatique Graphique de l'Université de Monréal (LIGUM)
Université de Montréal

hal-00659465, version 1
http://hal.archives-ouvertes.fr/hal-00659465
oai:hal.archives-ouvertes.fr:hal-00659465
From: Cédric Gérot <>
Submitted on: Thursday, 12 January 2012 18:04:39
Updated on: Thursday, 25 October 2012 14:28:13

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DOI: 10.1016/j.cagd.2011.11.004

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