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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Noninterpolatory Hermite subdivision schemes
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by Bin Han, Thomas P.-Y. Yu and Yonggang Xue PDF
Math. Comp. 74 (2005), 1345-1367 Request permission

Abstract:

Bivariate interpolatory Hermite subdivision schemes have recently been applied to build free-form subdivision surfaces. It is well known to geometric modelling practitioners that interpolatory schemes typically lead to “unfair" surfaces—surfaces with unwanted wiggles or undulations—and noninterpolatory (a.k.a. approximating in the CAGD community) schemes are much preferred in geometric modelling applications. In this article, we introduce, analyze and construct noninterpolatory Hermite subdivision schemes, a class of vector subdivision schemes which can be applied to iteratively refine Hermite data in a not necessarily interpolatory fashion. We also study symmetry properties of such subdivision schemes which are crucial for application in free-form subdivision surfaces. A key step in our mathematical analysis of Hermite type subdivision schemes is that we make use of the strong convergence theory of refinement equations to convert a prescribed geometric condition on the subdivision scheme—namely, the subdivision scheme is of Hermite type—to an algebraic condition on the subdivision mask. The latter algebraic condition can then be used in a computational framework to construct specific schemes.
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Additional Information
  • Bin Han
  • Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1
  • MR Author ID: 610426
  • Email: bhan@math.ualberta.ca
  • Thomas P.-Y. Yu
  • Affiliation: Department of Mathematical Science, Rensselaer Polytechnic Institute, Troy, New York 12180-3590
  • MR Author ID: 644909
  • Email: yut@rpi.edu
  • Yonggang Xue
  • Affiliation: Department of Mathematical Science, Rensselaer Polytechnic Institute, Troy, New York 12180-3590
  • Email: xuey@rpi.edu
  • Received by editor(s): April 15, 2003
  • Received by editor(s) in revised form: December 10, 2003
  • Published electronically: September 10, 2004
  • Additional Notes: The first author’s research was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC Canada) under grant G121210654
    The second author’s research was supported in part by an NSF CAREER Award (CCR 9984501)
  • © Copyright 2004 American Mathematical Society
  • Journal: Math. Comp. 74 (2005), 1345-1367
  • MSC (2000): Primary 41A05, 41A15, 41A63, 42C40, 65T60, 65F15
  • DOI: https://doi.org/10.1090/S0025-5718-04-01704-1
  • MathSciNet review: 2137006