"- I have tried to show how one can construct the control grid so that the
patches are joined smoothly. All the 8 control points surrounding each corner of
the patches are placed in the same plane."

Thanks - this may help me re-think things.

Can you elaborate a little bit on the implied relationship between the small
vectors between the corner and control points, and the larger vector between
corners is?  And why?

I understand the necessity of having the same slope which I believe is what the
small patch shows, and can understand that in isolation, but I haven't yet made
the connection to understand the relation between that and the large vectors.

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