Research ArticleOpen AccessGoogle Scholar indexed
Construction and Application of Subdivision Surface Scheme Using Lagrange Interpolation Polynomial
Department of Mathematics, University of Sargodha, Sargodha, Pakistan
Department of Mathematics, University of Sargodha, Sargodha, Pakistan
Department of Mathematics, University of Sargodha, Sargodha, Pakistan
- 1 Department of Mathematics, University of Sargodha, Sargodha, Pakistan
- 2 Department of Mathematics, University of Sargodha, Sargodha, Pakistan
- 3 Department of Mathematics, University of Sargodha, Sargodha, Pakistan
Applied Mathematics·Volume 05 (2014)·Pages 387–397·Published 7 February 2014·DOI10.4236/am.2014.53040
Copy link · social · email
Abstract
This paper offers a general formula for surface subdivision rules for quad meshes by using 2-D Lagrange interpolating polynomial [1]. We also see that the result obtained is equivalent to the tensor product of (2 N + 4)-point n-ary interpolating curve scheme for N ≥ 0 and n ≥ 2. The simple interpolatory subdivision scheme for quadrilateral nets with arbitrary topology is presented by L. Kobbelt [2], which can be directly calculated from the proposed formula. Furthermore, some characteristics and applications of the proposed work are also discussed.
KeywordsSubdivision SchemeInterpolating Subdivision SchemeTensor Product SchemeAuxiliary PointsLagrange Interpolation Polynomial
- G. Dahlquist and A. Bjork, “Numerical Methods in Scientific Computing,” SIAM, Vol. 1, 2008. http://dx.doi.org/10.1137/1.9780898717785
- L. Kobbelt, “Interpolatory Subdivision on Open Quadrilateral Nets with Arbitrary Topology,” Computer Graphics Forum, Vol. 15, No. 3, 1996, pp. 409-420. http://dx.doi.org/10.1111/1467-8659.1530409
- J. A. Lian, “On A-Ary Subdivision for Curve Design: 4-Point and 6-Point Inerpolatory Schemes,” Application and Applied Mathematics: International Journal, Vol. 3, No. 1, 2008, pp. 18-29.
- J. A. Lian, “On A-Ary Subdivision for Curve Design. II. $3$-Point and $5$-Point Interpolatory Schemes,” Applications and Applied Mathematics: International Journal, Vol. 3, No. 2, 2008, pp. 176-187.
- J. A. Lian, “On A-Ary Subdivision for Curve Design. III. 2m-Point and (2m+1)-Point Interpolatory Schemes,” Applications and Applied Mathematics: International Journal, Vol. 4, No. 2, 2009, pp. 434-444.
- K. P. Ko, “A Study on Subdivision Scheme-Draft,” Dongseo University Busan South Korea, 2007. http://kowon.dongseo.ac.kr/$\sim$kpko/publication/2004book.pdf
- G. Mustafa and A. R. Najma, “The Mask of (2b+4)-Point n-Ary Subdivision Scheme,” Computing, Vol. 90, No. 1-2, 2010, pp. 1-14.
- 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, Chapter 4, Springer, Berlin, 2002, pp. 51-68.
- N. A. Dodgson, U. H. Augsdorfer, T. J. Cashman and M. A. Sabin, “Deriving Box-Spline Subdivision Schemes,” Springer-Verlag, Berlin, 2009, pp. 106-123.
- N. Dyn, D. Levin and J. Gregory, “A 4-Point Interpolatory Subdivision Scheme for Curve Design,” Computer Aided Geometric Design, Vol. 4, No. 4, 1987, pp. 257-268. http://dx.doi.org/10.1016/0167-8396(87)90001-X