There is a multiplication operation on points that your teachers failed to tell you about, either because they didn't know about it or because they judged it to be unimportant. But that multiplication turns out to have important applications in computer-aided geometric design (CAGD). Among other things, it provides the best labels for Bezier control points --- better even than the labels provided by polar forms (a.k.a. blossoms).
In computer-aided geometric design (CAGD),a beautiful technology has
emerged for manipulating algebraic curves and surfaces,associated with the
names Bernstein,B Žezier de Casteljau, and de Boor.I have spent an embarrassing fraction of the last fifteen years exploring the roots of that technology
trying to clarify the mathematics at its core.
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