Just some suggestions:
I am concerned you are talking about parameterization (as would be the natural thing in an engineering problem) rather than the intrinsic geometry of the curve itself (as a geometer should).
in your first case $v'(t) \propto v(t)$
in your second case $ \sum a_k(t)v^{(k)}(t) = 0$ for some time-dependent functions $a_k(t)$.
in your third case, it seems too much to have a third-order tangent everywhere but I'm not sure. who knows? You have certainly ruled it out.
I don't know how the everywhere existence of a particular catastrophe leads to a global restriction. There is a Gauss map from curves to projective space, or from surfaces to Grassmanian $\mathrm{Gr}(2,n)$ which encodes how the tangent plane changes as you move around the surface.
Whenever you write down an equation or do an approximations, polynomials have been the choice for describing these kind of relations. Therefore, algebraic geometry -- and algebraic varieties are always the main source of examples.
e.g. a Lie group is a variety
For me it's the fact that your theorem comes from a textbook, suggests the main geometers of the time knew your result under a different name.
I am researching the name of the map $C \to SO(k)$ from your curve to the first $k$ derivative at that point. There must be an analogue of the Gauss map there.
Just a bit from Wikipedia:
In algebraic geometry, ruled surfaces were originally defined as projective surfaces in projective space containing a straight line through any given point. This immediately implies that there is a projective line on the surface through any given point, and this condition is now often used as the definition of a ruled surface: ruled surfaces are defined to be abstract projective surfaces satisfying this condition that there is a projective line through any point. This is equivalent to saying that they are birational to the product of a curve and a projective line. Sometimes a ruled surface is defined to be one satisfying the stronger condition that it has a fibration over a curve with fibers that are projective lines. This excludes the projective plane, which has a projective line though every point but cannot be written as such a fibration.
Instance of Monge-Cayley-Salmon
