INTERNATIONAL JOURNAL OF MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES Volume 8, 2014 ISSN: 1998-0140 9 C 1 Positive Bernstein-Bézier Rational Quartic Interpolation Maria Hussain, Malik Zawwar Hussain and Maryam Buttar Abstract – A positivity preserving 1 C rational quartic Bernstein-Bézier interpolation scheme is developed for scattered data. The constraints are developed on the Bézier ordinates to preserve the positive shape of scattered data. The weight functions are free to improve the shape of the surface. The developed scheme has more degrees of freedom as compared to existing rational positivity preserving interpolation schemes. It is local and equally applicable to data as well as data with derivatives. Keywords- 1 C Bernstein-Bézier rational quartic function, positivity, scattered data interpolation, weight functions, Bézier ordinates. I. INTRODUCTION Scattered data originates from a number of applications including earth sciences, meteorology, electronic imaging, reverse engineering, industrial design, 3D photography, ship building, aeronautics and medical modelling of human body parts. The data can be convex, monotone or positive partially or over the whole domain. However, the problem under consideration here is positivity of scattered data. The problem of positivity preserving interpolation of scattered data has been discussed by many authors. Some of the noticeable contributions are reviewed here. Malik Zawwar Hussain is a Professor in Department of Mathematics, University of the Punjab, Lahore, Pakistan (e-mail: malikzawwar.math@pu.edu.pk) Maria Hussain is an Assistant Professor in Department of Mathematics, Lahore College for Women University, Lahore-Pakistan(e-mail: mariahussain_1@yahoo.com) Maryam Buttar is a M.Phil. Student in Department of Mathematics, University of the Punjab, Lahore, Pakistan Brodlie, Asim and Unsworth [2] discussed the problem of visualization of data by using modified quadratic Shepard method. To assure the positivity of interpolant, sufficient conditions were imposed on quadratic basis functions. To interpolate the fractional data, they inhibited the interpolant within the limits [0,1]. The generalized form of the interpolant was produced. In [6], Hussain and Hussain developed a positivity preserving interpolation scheme which was based on Coons patches. A rational function with shape parameters (Hermite form) was used to interpolate each boundary of triangle. The side-vertex interpolation was also performed by a rational function (Hermite form). The final surface patch was the convex combination of these side-vertex interpolant. The authors in [6] developed data dependent constraints on shape parameters to preserve the positive shape of data. Hussain and Hussain [7] developed a 1 C Bernstein-Bézier rational cubic scheme to preserve the positive shape of scattered data. Luo and Peng [8] presented a range restricted 1 C rational spline interpolation scheme for scattered data. The rational spline interpolant was represented as a convex combination of three cubic Bézier triangular patches. The range restricted obstacles were the polynomial surfaces of degree up to three. Mulansky and Schmidt [9] proposed a range restricted 1 C quadratic spline interpolation scheme for scattered data. The domain was triangulated by the Powell-Sabin refinement. The solution to the problem came out as a solvable system of linear inequalities for the gradients as parameters. The inequalities were solved by a global quadratic optimization problem, where the thin plate functional was the objective function and the system of inequalities were the constraints. Herrmann, Mulansky and Schmidt [4] proposed a univariate and bivariate 1 C quadratic range restricted interpolation scheme for scattered data. In [9], the range