Conditions of Gn continuity of two Bezier patches sharing common edge

[mralexandermiller]

Hi everyone! The task is as follows. I work in the Autodesk Alias program, which builds Bezier surfaces with specified properties according to some internal algorithms. For some of my purposes, I would like to deal with these internal algorithms in order to be able to perform such construction and modification in other programs, including Wolfram Mathematica. So, if you are given two Bezier surfaces with a common edge, then, among other things, the algorithms of the program can modify one of these surfaces (or both surfaces) in such a way that there will be geometric continuity of a given degree (G1, G2 or G3). The program can perform exactly the same operation with Bezier curves. The mathematical foundations for such modifications of curves and surfaces are known. but if I figured out the modification of Bezier curves (redefining the coordinates of the control points of the curve / curves) on my own, then I could not figure out the modification of surfaces on my own, despite the fact that I have a literature with the mathematical foundations of such transformations. In other words, I simply do not understand this mathematical language, which is very difficult for me. Below I am attaching the article, which is the most successful in my opinion, which details the conditions for such continuity. My question is, how can I apply the knowledge presented in this article in order, for example, to redefine the coordinates of control points of one of two Bezier surfaces of degree 5 by 5 that have a common edge? Thanks in advance.