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.

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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.