Abstract: "Let C(M, [tau]) denote the set of all scalar valued functions with connection matrix M at [tau]. We show that C(M, [tau]) is closed under multiplication and division if and only if M is a reparametrization matrix. We conclude that reparametrization is the most general form of geometric continuity for which the shape parameters remain invariant under lifting and projection. We go on to show that Frenet frame continuity is also invariant under projection, even though the shape parameters are not preserved. We also investigate curves which are not smooth, but which become smooth under projection."

Abstract: "Tensor products are widely used in computer graphics and computer aided geometric design for representing freeform surfaces. Standard tensor product surface patches retain vestiges of the original rectilinear shape of their domains. This feature is not always desired. We introduce a variant of the tensor product, called the tensor product slice, which can be used to create multi-sided surface patches of non-rectilinear shape. We give examples to show that this variant retains many desirable properties of standard tensor products."