Attached Link: http://www.euclideanspace.com/maths/algebra/vectors/angleBetween/index.htm

> Quote - Can you suggest how I can get the proper angles, given the plot points of the curve?  I'm afraid my formal math training is limited and I'm pressing against the edges of my knowledge, here.  :lol:

The simplest way to think of it is that you want to rotate each link onto the curve (starting with the first and working along the chain).
The math for this is quite simple (because again you start with things aligned to the primary axes) and so you're effectively working in 2 dimensions when calculating the required angle between the current link and the target segment of the curve (you need to take notice of the signs, especially once you reach the lowest point).
ISTR that this is why my chain is rigged as it is because it was never intended for general posing, but only to make fitting it to a catenary easy.

Useful reference ->http://www.j3d.org/matrix_faq/matrfaq_latest.html
Vector math -> http://chortle.ccsu.edu/VectorLessons/vectorIndex.html

Picking a convenient set of axes (i.e. frame of reference) is always a good first step as it simplifies things as it's much easier to work with tihngs aligned to the primrary (possibly orthognal) axes and then rotate to match the "real world" axes at the very end than it is to work with arbitary orientation throughout.
You should get into the habit of looking for such a convenient FoR as the first step in a geometry problem.