A Unified Interpolating Subdivision Scheme for Curves/Surfaces by Using Newton Interpolating Polynomial

ABSTRACT

This paper presents a general formula for (2*m** +** *2)-point *n*-ary interpolating subdivision scheme for curves for any integer *m *≥* *0 and *n *≥* *2 by using Newton interpolating polynomial. As a consequence, the proposed work is extended for surface case, which is equivalent to the tensor product of above proposed curve case. These formulas merge several notorious curve/surface schemes. Furthermore, visual performance of the subdivision schemes is also presented.

Cite this paper

F. Khan, I. Mukhtar and N. Batool, "A Unified Interpolating Subdivision Scheme for Curves/Surfaces by Using Newton Interpolating Polynomial,"*Open Journal of Applied Sciences*, Vol. 3 No. 3, 2013, pp. 263-269. doi: 10.4236/ojapps.2013.33033.

F. Khan, I. Mukhtar and N. Batool, "A Unified Interpolating Subdivision Scheme for Curves/Surfaces by Using Newton Interpolating Polynomial,"

References

[1] N. Dyn, D. Levin and J. A. Gregory, “A 4-Point Interpolatory Subdivision Scheme for Curve Design,” Computer Aided Geometric Design, Vol. 24, No. 4, 1987, pp. 257-268. doi:10.1016/0167-8396(87)90001-X

[2] C. Beccari, G. Casciola and L. Romani, “An interpolating 4-Point Ternary Non-Stationary Subdivision Scheme with Tension Control,” Computer Aided Geometric De sign, Vol. 24, No. 4, 2007, pp. 210-219. doi:10.1016/j.cagd.2007.02.001

[3] N. Dyn, “Interpolatory Subdivision Scheme and Analysis of Convergence and Smoothness by the Foralism of Laurent Polynomials,” In: A.Iske, E. Quak and M. S. Floater, Eds., Tutorials on Multiresolution in Geometric Model ling, Springer, Berlin, 2002, pp. 51-68 (Chapter 2 and 3).

[4] M. F. Hassan and N. A. Dodgson, “Ternary and Three Point Univariate Subdivision Schemes,” In: A. Cohen, J. L. Marrien and L. L. Schumaker, Eds., Curve and Surface Fitting: Saint-Malo, 2002, Nashboro Press, Brentwood, 2003, pp. 199-208.

[5] M. F. Hassan, I. P. Ivrissimitzis and N. A. Dodgson, “An Interpolating 4-Point Ternary Stationary Subdivision Scheme,” Computer Aided Geometric Design, Vol. 19, No. 1, 2002, pp. 1-18. doi:10.1016/S0167-8396(01)00084-X

[6] M. K. Jena, P. Shunmugaraj and P. C. Das, “A Non-Stationary Subdivision Scheme for Curve Interpolation,” Anziam Journal, Vol. 44, 2003, pp. 216-235.

[7] G. Mustafa and F. Khan, “Ternary Six-Point Interpolatig Subdivision Scheme,” Lobachevskii Journal of Mathematics, Vol. 29, No. 3, 2008, pp. 153-163. doi:10.1134/S1995080208030062

[8] A. Weissman, “A 6-Point Interpolatory Subdivision Scheme for Curve Design,” M.Sc. Thesis, Tel Aviv University, Tel Aviv, 1990.

[9] E. Catmull and J. Clark, “Recursively Generated B-Spline Surfaces on Arbitrary Topological Meshes,” Computer Aided Design, Vol. 10, No. 6, 1978, pp. 350-355. doi:10.1016/0010-4485(78)90110-0

[10] Doo and M. Sabin, “Bahaviour of Recursive Division Durfaces near Extraordinary Points,” Computer Aided Design, Vol. 10, No. 6, 1978, pp. 356-360. doi:10.1016/0010-4485(78)90111-2

[11] L. Kobbelt, “Interpolatory Subdivision on Open Quadri lateral Nets with Arbitrary Topology,” Computer Graphics Forum, Vol. 15, No. 3, 1996, pp. 409-420. doi:10.1111/1467-8659.1530409

[12] G. Deslauries and S. Dubuc, “Symmetric Iterative Interpolation Process,” Constructive Approximation, Vol. 5, No. 1, 1989, pp. 49-68. doi:10.1007/BF01889598

[13] J. A. Lian, “On A-Aray Subdivision for Curve Design: 4-Point and 6-Point Interpolatory Scheme,” Applications and Applied Mathematics: An International Journal, Vol. 3, No. 1, 2008, pp. 18-29.

[14] M. Sabin, “Eigenanalysis and Artifacts of Subdivision Curves and Surfaces,” In: A. Iske, E. Quak and M. S. Floater, Eds., Tutorials on Multiresolution in Geometric Modelling, Springer, Berlin, 2002, pp. 51-68 (Chapter 4).

[15] N. A. Dodgson, U. H. Augsdorfer, T. J. Cashman and M. A. Sabin, “Deriving Box-Spline Subdivision Schemes,” Springer-Verlag, Berlin, LNCS-5654, 2009, pp. 106-123.

[1] N. Dyn, D. Levin and J. A. Gregory, “A 4-Point Interpolatory Subdivision Scheme for Curve Design,” Computer Aided Geometric Design, Vol. 24, No. 4, 1987, pp. 257-268. doi:10.1016/0167-8396(87)90001-X

[2] C. Beccari, G. Casciola and L. Romani, “An interpolating 4-Point Ternary Non-Stationary Subdivision Scheme with Tension Control,” Computer Aided Geometric De sign, Vol. 24, No. 4, 2007, pp. 210-219. doi:10.1016/j.cagd.2007.02.001

[3] N. Dyn, “Interpolatory Subdivision Scheme and Analysis of Convergence and Smoothness by the Foralism of Laurent Polynomials,” In: A.Iske, E. Quak and M. S. Floater, Eds., Tutorials on Multiresolution in Geometric Model ling, Springer, Berlin, 2002, pp. 51-68 (Chapter 2 and 3).

[4] M. F. Hassan and N. A. Dodgson, “Ternary and Three Point Univariate Subdivision Schemes,” In: A. Cohen, J. L. Marrien and L. L. Schumaker, Eds., Curve and Surface Fitting: Saint-Malo, 2002, Nashboro Press, Brentwood, 2003, pp. 199-208.

[5] M. F. Hassan, I. P. Ivrissimitzis and N. A. Dodgson, “An Interpolating 4-Point Ternary Stationary Subdivision Scheme,” Computer Aided Geometric Design, Vol. 19, No. 1, 2002, pp. 1-18. doi:10.1016/S0167-8396(01)00084-X

[6] M. K. Jena, P. Shunmugaraj and P. C. Das, “A Non-Stationary Subdivision Scheme for Curve Interpolation,” Anziam Journal, Vol. 44, 2003, pp. 216-235.

[7] G. Mustafa and F. Khan, “Ternary Six-Point Interpolatig Subdivision Scheme,” Lobachevskii Journal of Mathematics, Vol. 29, No. 3, 2008, pp. 153-163. doi:10.1134/S1995080208030062

[8] A. Weissman, “A 6-Point Interpolatory Subdivision Scheme for Curve Design,” M.Sc. Thesis, Tel Aviv University, Tel Aviv, 1990.

[9] E. Catmull and J. Clark, “Recursively Generated B-Spline Surfaces on Arbitrary Topological Meshes,” Computer Aided Design, Vol. 10, No. 6, 1978, pp. 350-355. doi:10.1016/0010-4485(78)90110-0

[10] Doo and M. Sabin, “Bahaviour of Recursive Division Durfaces near Extraordinary Points,” Computer Aided Design, Vol. 10, No. 6, 1978, pp. 356-360. doi:10.1016/0010-4485(78)90111-2

[11] L. Kobbelt, “Interpolatory Subdivision on Open Quadri lateral Nets with Arbitrary Topology,” Computer Graphics Forum, Vol. 15, No. 3, 1996, pp. 409-420. doi:10.1111/1467-8659.1530409

[12] G. Deslauries and S. Dubuc, “Symmetric Iterative Interpolation Process,” Constructive Approximation, Vol. 5, No. 1, 1989, pp. 49-68. doi:10.1007/BF01889598

[13] J. A. Lian, “On A-Aray Subdivision for Curve Design: 4-Point and 6-Point Interpolatory Scheme,” Applications and Applied Mathematics: An International Journal, Vol. 3, No. 1, 2008, pp. 18-29.

[14] M. Sabin, “Eigenanalysis and Artifacts of Subdivision Curves and Surfaces,” In: A. Iske, E. Quak and M. S. Floater, Eds., Tutorials on Multiresolution in Geometric Modelling, Springer, Berlin, 2002, pp. 51-68 (Chapter 4).

[15] N. A. Dodgson, U. H. Augsdorfer, T. J. Cashman and M. A. Sabin, “Deriving Box-Spline Subdivision Schemes,” Springer-Verlag, Berlin, LNCS-5654, 2009, pp. 106-123.