Abstract
We present a non-stationary, non-uniform scheme for two-point Hermite subdivision. The novelty of this approach relies on a geometric interpretation of the subdivision steps—related to generalized Bernstein bases—which permits to overcome the usually unavoidable analytical difficulties. The main advantages consist in extra smoothness conditions, which in turn produce highly regular limit curves, and in an elegant structure of the subdivision—described by three de Casteljau type matrices. As a by-product, the scheme is inherently shape preserving.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Carnicer J., Mainar E., Pena J.M.: Critical length for design purposes and extended Chebyshev spaces. Constr. Approx. 20, 55–71 (2004)
Carnicer J., Pena J.M.: Shape-preserving representations and optimality of the Bernstein basis. Adv. Comput. Math. 1, 173–196 (1993)
Costantini P.: Curve and surface construction using variable degree polynomial splines. Comput. Aided Geom. Des. 17, 426–446 (2000)
Costantini P., Lyche T., Manni C.: On a class of weak Tchebycheff systems. Numer. Math. 101, 333–354 (2005)
Costantini P., Manni C.: Geometric construction of generalized cubic splines. Rend. Mat. 26, 327–338 (2006)
Costantini P., Manni C.: On constrained nonlinear Hermite subdivision. Constr. Approx. 28, 291–331 (2008)
Costantini, P., Manni, C.: Curve and surface construction using Hermite subdivision schemes. J. Comput. Appl. Math. doi:10.1016/j.cam.2009.02.096 (2009)
de Boor C.: Cutting corners always work. Comput. Aided Geom. Des. 4, 125–131 (1987)
de Boor C.: Local corner cutting corners and the smoothnes of the limiting curve. Comput. Aided Geom. Des. 7, 389–397 (1990)
Delbourgo R., Gregory J.A.: Shape preserving piecewise rational interpolation. SIAM J. Sci. Stat. Comput. 6, 967–976 (1985)
Dubuc S.: Scalar and Hermite subdivision schemes. Appl. Comp. Harmon. Anal. 21, 376–394 (2006)
Dubuc S., Merrien J.-L.: A 4-point Hermite subdivision scheme. In: Lyche, T., Schumaker, L.L. (eds) Mathematical Methods for Curves and Surfaces, Innovative Applied Mathematics, pp. 113–122. Vanderbilt University Press, Nashville (2001)
Goodman T.N.T., Mazure M.L.: Blossoming beyond extended Chebyshev spaces. J. Approx. Theory 109, 48–81 (2001)
Gori L., Pitolli F.: A class of totally positive refinable functions. Rend. Mat. 20, 305–322 (2000)
Hoschek J., Lasser D.: Fundamentals of Computer Aided Geometric Design. A.K. Peters, Wellesley (1993)
Lyche T., Merrien J.L.: C 1 Interpolatory subdivision with shape constraints for curves. SIAM J. Numer. Anal. 44, 1095–1121 (2006)
Mazure M.L.: Chebyshev spaces and Bernstein bases. Constr. Approx. 22, 347–363 (2005)
Merrien J.-L.: A family of Hermite interpolants by bisection algorithms. Numer. Algorithms 2, 187–200 (1992)
Merrien J.-L.: Interpolants d’Hermite C 2 obtenus par subdivision. M2AN Math. Model. Numer. Anal. 33, 55–65 (1999)
Merrien J.-L., Sablonniere P.: Monotone and convex C 1 Hermite interpolants generated by a subdivision scheme. Constr. Approx. 19, 279–298 (2003)
Pelosi, F., Sablonniere, P.: C 1 GP Hermite Interpolants Generated by a Subdivision Scheme. J. Comput. Appl. Math. doi:10.1016/j.cam.2007.09.013 (2007)
Peña, J.M. (eds): Shape Preserving Representations in Computer-Aided Geometric Design. Nova Science Publishers, Inc, Commack (1999)
Sablonniere P.: Bernstein-type bases and corner cutting algorithms for C 1 Merrien’s curves. Adv. Comput. Math. 20, 229–246 (2004)
Schweikert D.G.: An interpolation curve using a spline in tension. J. Math. Phys. 45, 312–317 (1966)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Costantini, P., Manni, C. A geometric approach for Hermite subdivision. Numer. Math. 115, 333–369 (2010). https://doi.org/10.1007/s00211-009-0280-0
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00211-009-0280-0