Convex Surface Visualization Using Rational Bi- cubic Function
الموضوعات :Malik Zawwar Hussain 1 , Fareeha Saadia 2 , Maria Hussain 3
1 - Department of Mathematics, University of the Punjab, Lahore, Pakistan
2 - Department of Mathematics, University of the Punjab, Lahore, Pakistan
3 - Department of Mathematics, Lahore College for Women University, Pakistan
الکلمات المفتاحية: Rational bi-cubic function, Convex Surface, free parameters,
ملخص المقالة :
The rational cubic function with three parameters has been extended to rational bi-cubic function to visualize the shape of regular convex surface data. The rational bi-cubic function involves six parameters in each rectangular patch. Data dependent constraints are derived on four of these parameters to visualize the shape of convex surface data while other two are free to refine the shape of surface at user choice. The developed constraints on parameters act as sufficient conditions for visualization of convex surface data. Moreover, computationally simple and less time consuming as compared to exiting techniques.
[1] Asaturyan, S., Shape preserving surface interpolation scheme, Ph.D. Thesis, Department of Mathematics and Computer Science, University of Dundee, Scotland, UK, 1990.
[2] Asaturyan, S., Costantini P., and Manni C., Local shape preserving interpolation by space curves, IMA Journal of Numerical Analysis, 21(1), 301–325, 2001.
[3] Carnicer, J.M., Garcia-Esnaol, M., and Peña, J. M., Convexity of rational curves and total positivity, Journal of Computational and Applied Mathematics, 71(2), 365–382, 1996.
[4] Clements, J.C., A convexity preserving parametric rational cubic interpolant, Numerische Mathemati, 63(1), 165-171, 1992.
[5] Costantini, P., On monotone and convex spline interpolation, Mathematics of Computation, 46(173), 203-214, 1986.
[6] Costantini, P. and Fontanella F., Shape preserving bivariate interpolation, SIAM Journal of Numerical Analysis, 27(2), 488-506,1990.
[7] Dodd, S. L., McAllister, D. F., and Roulier J. A., Shape-preserving spline interpolation for specifying bivariate functions on grids, IEEE Computer Graphics and Applications, 3(6), 70-79,1983.
[8] Floater, M. S., A weak condition for the convexity of tensor-product Bézier and B-spline surfaces, Advances in Computational Mathematics, 2(1), 67–80,1994.
[9] Hussain, M.Z. and Hussain, M., Visualization of 3D data preserving convexity, Journal of Applied Mathematics and Computing, 2, 170-186, 2006.
[10] Hussain, M.Z. and Hussain, M., Convex surface interpolation, Lecture Notes in Computer Science, 4975, 475-482, 2008.
[11] Sarfraz, M., Al-Mulhem, M. and Ashraf, M., Preserving monotonic shape of data using piecewise rational cubic function, Computers & Graphics, 21(1), 5-14, 1997.
[12] Zhang, Y., Duan, Q., and Twizell, E. H., Convexity control of bivariate interpolation spline surfaces, Computers & Graphics, 31(5), 679-687, 2007.
