Conditions of Gⁿ continuity between surfaces

Geometric continuity between parametric surfaces is an important issue in CAGD. So far researches have been concentrated on it for N-dimensional surfaces whose common boundary is of N - 1 dimensions. No results have ever been obtained for the case where the common boundary of the two surfaces is of L dimensions, 0 ≤ L ≤ N, or the two adjacent surfaces are of unequal dimensions. In this paper, the conditions of high-order geometric continuity between parametric surfaces are studied for these general cases. By analysing the structure of Gⁿ transformations, general solutions as well as an iterative method for determining adjustable functions are proposed. Some equivalent conditions of Gⁿ continuity (nth order geometric continuity) are derived with an emphasis on conditions for triangular and rectangular Bezier surfaces. Since the Gⁿ conditions for polynomial parametric surfaces are expressed in the form of explicit relationship of control points of Bezier surfaces, they are very useful both in theory and application for CAD/CAM and CAGD. Finally, Gⁿ conditions between q-dimensional submanifolds of N-dimensional and M-dimensional surfaces are studied, thus a complete theoretical basis is laid for the problem of geometric continuity.