Theories of contact specified by connection matrices Original Research Article

We begin by characterizing notions of geometric continuity represented by connection matrices. Next we present a set of geometric properties that must be satisfied by all reasonable notions of geometric continuity. These geometric requirements are then reinterpreted as an equivalent collection of algebraic constraints on corresponding sets of connection matrices. We provide a general technique for constructing sets of connection matrices satisfying these criteria and apply this technique to generate many examples of novel notions of geometric continuity. Using these constraints and construction techniques, we show that there is no notion of geometric continuity between reparametrization continuity of order 3, (G3), and Frenet frame continuity of order 3, (F3); that there are several notions of geometric continuity between G4 and F4; and that the number of different notions of geometric continuity between Gn and Fn grows at least exponentially with n.