"On Multiplying Points: The Paired Algebras of Forms and Sites" Lyle Ramshaw Report #169, May 01, 2001. 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).

Let V be a finite-dimensional vector space. Everyone understands that it makes sense to multiply covectors, the elements of the dual space V* = Lin(V, R). For example, if x, y, and z are covectors, then the expression x² - 5yz denotes a "quadratic form" on the space V. Forms have lots of applications; for example, to put a Euclidean metric on V, we would choose a positive definite quadratic form as our measure of squared length.

But most people don't yet realize that it also makes sense to multiply vectors, the elements of V itself. If u, v, and w are vectors, then the expression u² - 5vw denotes an object that is the dual analog of a quadratic form. Let's call such an object a "quadratic site" over V. The sites over V of all degrees form an algebra, dual to the well-known algebra of forms on V.

What are sites good for? Consider, say, a cubic Bezier curve segment. It is the image, under a cubic function, of a closed interval on the parameter line, say the interval [R..S]. The best labels for the Bezier points of that cubic segment are the cubic sites R³, R² S, RS², and S³.