Algebraic aspects of geometric continuity

Let C ( M , τ) denote the set of all scalar valued functions with connection matrix M at τ. We show that C ( M , τ) 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.