Abstract
We present the best bounds on the distance between 3-direction quartic box spline surface patch and its control net by means of analysis and computing for the basis functions of 3-direction quartic box spline surface. Both the local bounds and the global bounds are given by the maximum norm of the first differences or second differences or mixed differences of the control points of the surface patch.
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References
C K Chui. Multivariate Splines, In: CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 54, SIAM, 1988.
C K Chui, M J Lai. Computation of box splines and B-splines on triangulations of nonuniform rectangualr partions, Approx Theory Appl, 1987, 3: 37–62.
C de Boor, K Hollig, S Riemenschneider. Box Spline, Springer-Verlag, 1993.
C de Boor, R De Vore. Approximation by smooth multivariate splines, Trans Amer Math Soc, 1983, 276: 775–788.
Y Huang, H Su. The bound on derivatives of rational Bézier curves, Comput Aided Geom Design, 2006, 23: 698–702.
M J Lai. Fortran subroutines for B-nets of box splines on three- and four-directional meshes, Numer Algorithms, 1992, 2: 33–38.
M J Lai, L L Schumaker. Spline Functions on Triangulations, Cambrige University Press, 2007.
C T Loop. Smooth subdivision surfaces based on triangles, M.S. Thesis, Unversity of Utah, 1987.
D Lutterkort, J Peters. Optimized refinable enclosures of multivariate polynomial pieces, Comput Aided Geom Design, 2001, 18: 851–863.
D Nairn, J Peters, D Lutterkort. Sharp, quantitative bounds on the distance between a polynomial piece and its Bézier control polygon, Comput Aided Geom Design, 1999, 16: 613–631.
U Reif. Best bounds on the approximation of polynomials and splines by their control structure, Comput Aided Geom Design, 2000, 17: 579–589.
J Stam. Evaluation of Loop Subdivision Surfaces, SIGGRAPH’99 Course Notes, 1999.
Z Wu, F Lin, S H Soon, C K Yun. Evaluation of difference bounds for computing rational Bézier curves and surfaces, Comput Graph, 2004, 28: 551–558.
R J Zhang, G J Wang. Sharp bounds on the approximation of a Bézier polynomial by its quasicontrol polygon, Comput Aided Geom Design, 2006, 23: 1–16.
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Supported by the National Natural Science Foundation of China (61170324 and 61100105).
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Zeng, Xm., Zhou, Gr. & Yang, Lq. Best bounds on the distance between 3-direction quartic box spline surface and its control net. Appl. Math. J. Chin. Univ. 28, 147–157 (2013). https://doi.org/10.1007/s11766-013-2986-0
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DOI: https://doi.org/10.1007/s11766-013-2986-0