A bivariate rational interpolation based on scattered data on parallel lines

In many practical problems, such as geological exploration, forging technology and medical imaging, among others, it has been detected that the scattered data are usually arranged in parallel lines. In this paper, a new approach to construct a bivariate rational interpolation over triangulation is presented, based on scattered data in parallel lines. The main advantage of this method comparing with the present interpolation methods have two points: (1) the interpolation function is carried out by a simple and explicit mathematical representation through the parameter α; (2) the shape of the interpolating surface can be modified by using the parameter for the unchanged interpolating data. Moreover, a local shape control method is employed to control the shape of surfaces. In the special case, the method of “Barycenter Value Control” is studied, and numerical examples are presented to show the performance of the method.

► A method to construct bivariate rational spline over triangulation is proposed.
► The interpolation function has simple and explicit mathematical representation.
► The interpolation have a free parameter and the convenient basis functions.
► The shape of the surfaces can be modified by selecting suitable parameter.