This paper addresses the issue of interpolation for 3D surfaces, required in fields like robotics, data analysis and visualization, and animation. It proposes an interpolation algorithm that is able to generate 3D surfaces, developed on quadratic grids, in addition to maintaining shape characteristics such as monotonicity and positivity, and avoiding as much as possible undesired fluctuations. The interpolation function concatenates parameterized trigonometric polynomials over a rectangular grid, ensuring GC¹ continuity, and a unique interpolation order, three, over the whole domain. The most important part of the paper investigates sufficient conditions to be obeyed by the sets of free parameters in order to preserve the original positivity and monotonicity properties. Their inference is the result of a thorough calculus exposed in almost full detail, and is concluded by two theorems. The last part of the paper contains graphical illustrations to consolidate the theoretical results, using several datasets, with many comparative graphics and tables presenting parameters’ numerical values obtained on the grid cells. Computer performance of the method is compared to the results of two earlier methods devised by the same team of authors and the progress concerning the speed of the algorithms is outlined.

The material is very elaborate and accurate, the solution is simple and imaginative, and it does not demand high-level mathematics. The results can be easily verified. The work is well documented; it is an extension of previous results of the authors and denotes their involvement in the improvement of this important topic.