Some characterizations of families of surfaces using functional equations

In this article functional equations are used to characterize some families of surfaces. First, the most general surfaces in implicit form f(x, y, z) = 0, such that any arbitrary intersection with the planes z = z(0), y = y(0), and x = x(0) are linear combinations of sets of functions of the other two variables, are characterized. It is shown that only linear combinations of tensor products of univariate functions are possible for f(x, y, z). Second, we obtain the most general families of surfaces in explicit form such that their intersections with planes parallel to the planes y = 0 and x = 0 belong to two, not necessarily equal, parametric families of curves. Finally. functional equations are used to analyze the uniqueness of representation of Gordon-Coons surfaces. Some practical examples are used to illustrate the theoretical results.