Some researches on trivariate Lagrange interpolation

In this paper, in order to go a step further research on the problem of trivariate Lagrange interpolation, we pose the concepts of sufficient intersection of algebraic surfaces and Lagrange interpolation along a space algebraic curve, and extend Cayley–Bacharach theorem in algebraic geometry from R2R2 to R3R3. By using the conclusion of the extended theorem, we deduce a general method of constructing properly posed set of nodes for Lagrange interpolation along a space algebraic curve, and give a series of corollaries for the practical applications. Moreover, we give a new method of constructing properly posed set of nodes for Lagrange interpolation along an algebraic surface, and as a result we make clear the geometrical structure of it.