Unification and Application of 3-point Approximating Subdivision Schemes of Varying Arity

Affiliation(s)

Department of Mathematics The Islamia University of Bahawalpur Bahawalpur, PAKISTAN.

Dept. of Computer Science & Technology Tsinghua University Beijing 100084, P. R. CHINA.

Department of Mathematics The Islamia University of Bahawalpur Bahawalpur, PAKISTAN.

Dept. of Computer Science & Technology Tsinghua University Beijing 100084, P. R. CHINA.

ABSTRACT

In this paper, we propose and analyze a subdivision scheme which unifies 3-point approximating subdivision schemes of any arity in its compact form and has less support, computational cost and error bounds. The usefulness of the scheme is illustrated by considering different examples along with its comparison with the established subdivision schemes. Moreover, B-splines of degree 4and well known 3-point schemes [1, 2, 3, 4, 6, 11, 12, 14, 15] are special cases of our proposed scheme.

In this paper, we propose and analyze a subdivision scheme which unifies 3-point approximating subdivision schemes of any arity in its compact form and has less support, computational cost and error bounds. The usefulness of the scheme is illustrated by considering different examples along with its comparison with the established subdivision schemes. Moreover, B-splines of degree 4and well known 3-point schemes [1, 2, 3, 4, 6, 11, 12, 14, 15] are special cases of our proposed scheme.

Cite this paper

nullGhaffar, A. , Mustafa, G. and Qin, K. (2012) Unification and Application of 3-point Approximating Subdivision Schemes of Varying Arity.*Open Journal of Applied Sciences*, **2**, 48-52. doi: 10.4236/ojapps.2012.24B012.

nullGhaffar, A. , Mustafa, G. and Qin, K. (2012) Unification and Application of 3-point Approximating Subdivision Schemes of Varying Arity.

References

[1] M. Aslam, G. Mustafa, and A. Ghaffar, “(2n?1)-point ternary approximating and interpolatingsubdivision schemes,” Journal of Applied Mathematics, Article ID 832630, 12 pages,2011.

[2] G. M. Chaikin, “An algorithm for high-speed curve generation, Computer Graphics andImage Processing,”vol. 3, 1974, pp. 346-349.

[3] S. Daniel and P. Shunmugaraj, “Three point stationary and non-stationary subdivisionschemes,” 3rd International Conference on Geometric Modeling & Imaging,DOI:10.1109/GMAI.2008.13

[4] M. F. Hassan and N. A. Dodgson, “Ternary and three-point univariate subdivision schemes,” in:A. Cohen, J. L. Marrien, L. L. Schumaker (Eds.), Curve and Surface Fitting: Sant-Malo2002, Nashboro Press, Brentwood, pp. 199-208, 2003.

[5] M. F. Hassan, I. P.Ivrissimitzis, N. A. Dodgson,and M. A. Sabin, “An interpolating4-point ternary stationary subdivision scheme,” Computer Aided Geometric Design,vol. 19, pp.1-18, 2002.

[6] K. Hormann and M. A. Sabin, “A family of subdivision schemes with cubic precision,” ComputerAided Geometric Design, vol.25, 2008,pp. 41-52.

[7] F. Khan, G. Mustafa, “A new 4-point quaternary approximating subdivision scheme,”Abstractand Applied Analysis, doi:10.1155/2009/301967 (2009).

[8] J.-A. Lian, “On a-ary subdivision for curve design: III. 2m-point and (2m + 1)-point interpolatoryschemes,” Applications and Applied Mathematics: An International Journal, vol. 4(2),2009, pp. 434-444.

[9] J.-A. Lian, “On a-ary subdivision for curve design: I. 4-point and 6-point interpolatoryschemes,” Applications and Applied Mathematics: An International Journal, vol. 3(1), 2008, pp.18-29.

[10] J.-A. Lian, “On a-ary subdivision for curve design: II. 3-point and 5-point interpolatoryschemes,” Applications and Applied Mathematics: An International Journal, vol. 3(2), 2008, pp. 176-187.

[11] G. Mustafa, F. Khan and A. Ghaffar, The m-point approximating subdivision scheme,Lobachevskii Journal of Mathematics, vol. 30(2), 2009, pp. 138-145.

[12] G. Mustafa, A. Ghaffar and F. Khan, “The Odd-Point Ternary Approximating Schemes,”American Journal of Computational Mathematics, vol. 1(2), pp. 111-118, 2011. doi: 10.4236/ajcm.2011.12011

[13] G. Mustafa & M. S. Hashmi, Subdivision depth computation for n-ary subdivisioncurves/surfaces, Vis Comput, vol. 26, , 2010, pp. 841-851.

[14] S. S. Siddiqi and N. Ahmad, A new three point approximating C2 subdivision scheme,Applied Mathematics Letters, vol. 20, 2007, pp. 707-711.

[15] S. S. Siddiqi and Rehan, K, Modified form of binary and ternary 3-point subdivisionscheme, Applied Mathematics and Computation, vol. 216, 2010, pp. 970-982.

[1] M. Aslam, G. Mustafa, and A. Ghaffar, “(2n?1)-point ternary approximating and interpolatingsubdivision schemes,” Journal of Applied Mathematics, Article ID 832630, 12 pages,2011.

[2] G. M. Chaikin, “An algorithm for high-speed curve generation, Computer Graphics andImage Processing,”vol. 3, 1974, pp. 346-349.

[3] S. Daniel and P. Shunmugaraj, “Three point stationary and non-stationary subdivisionschemes,” 3rd International Conference on Geometric Modeling & Imaging,DOI:10.1109/GMAI.2008.13

[4] M. F. Hassan and N. A. Dodgson, “Ternary and three-point univariate subdivision schemes,” in:A. Cohen, J. L. Marrien, L. L. Schumaker (Eds.), Curve and Surface Fitting: Sant-Malo2002, Nashboro Press, Brentwood, pp. 199-208, 2003.

[5] M. F. Hassan, I. P.Ivrissimitzis, N. A. Dodgson,and M. A. Sabin, “An interpolating4-point ternary stationary subdivision scheme,” Computer Aided Geometric Design,vol. 19, pp.1-18, 2002.

[6] K. Hormann and M. A. Sabin, “A family of subdivision schemes with cubic precision,” ComputerAided Geometric Design, vol.25, 2008,pp. 41-52.

[7] F. Khan, G. Mustafa, “A new 4-point quaternary approximating subdivision scheme,”Abstractand Applied Analysis, doi:10.1155/2009/301967 (2009).

[8] J.-A. Lian, “On a-ary subdivision for curve design: III. 2m-point and (2m + 1)-point interpolatoryschemes,” Applications and Applied Mathematics: An International Journal, vol. 4(2),2009, pp. 434-444.

[9] J.-A. Lian, “On a-ary subdivision for curve design: I. 4-point and 6-point interpolatoryschemes,” Applications and Applied Mathematics: An International Journal, vol. 3(1), 2008, pp.18-29.

[10] J.-A. Lian, “On a-ary subdivision for curve design: II. 3-point and 5-point interpolatoryschemes,” Applications and Applied Mathematics: An International Journal, vol. 3(2), 2008, pp. 176-187.

[11] G. Mustafa, F. Khan and A. Ghaffar, The m-point approximating subdivision scheme,Lobachevskii Journal of Mathematics, vol. 30(2), 2009, pp. 138-145.

[12] G. Mustafa, A. Ghaffar and F. Khan, “The Odd-Point Ternary Approximating Schemes,”American Journal of Computational Mathematics, vol. 1(2), pp. 111-118, 2011. doi: 10.4236/ajcm.2011.12011

[13] G. Mustafa & M. S. Hashmi, Subdivision depth computation for n-ary subdivisioncurves/surfaces, Vis Comput, vol. 26, , 2010, pp. 841-851.

[14] S. S. Siddiqi and N. Ahmad, A new three point approximating C2 subdivision scheme,Applied Mathematics Letters, vol. 20, 2007, pp. 707-711.

[15] S. S. Siddiqi and Rehan, K, Modified form of binary and ternary 3-point subdivisionscheme, Applied Mathematics and Computation, vol. 216, 2010, pp. 970-982.