Explicit formulas for bicubic spline surface interpolation

In this paper, explicit formulas are developed for representing a uniform bicubic spline surface that passes through an array of data points. The interpolated surface in the closed case is topologically equivalent to a torus. Open surface cases are reduced to closed surface cases by introducing one or two rows of `free points' such that the spline surface wraps around its boundaries. Ordinary interpolation surfaces in open cases can thus be constructed with the same formulas. It turns to be more intuitive and effective to control and modify the shape of the resultant surfaces by adjusting `free points' than by the usual derivatives and twist vectors. The interpolation surface is obtained in a two step way and the procedure is very easy to implement. Experimental results demonstrate that the proposed formulas are practically useful.