Emergent

Personally, I'd just make five levels, corresponding to the five Platonic solids.  It's kind of cool that there are only five; to me, it seems to lend them additional importance.

I like the subdivision stuff too, but my (totally subjective / aesthetic) opinion is that it's more useful for approximating spheres.

I also think linear algebra is very important to appreciate some of the things you'll do in course on ODEs.  I'd say the most important concept for this purpose is eigendecomposition (eigenvectors and eigenvalues) and the spectral theorem.  For instance, local equilibria of ODEs are characterized by the eigendecomposition of the Jacobian; and ODEs like the heat and wave equations are themselves solved by eigendecomposing a linear operator (the sines/cosines are its eigenvectors).

For chokepoint detection, see this thread:

http://www.gamedev.net/topic/601508-choke-point-detection/

I haven't looked at your code.  However, the first thing I do whenever I write any local optimization routine is to check the derivatives with finite differences.  Unless something more obvious pops out, I'd recommend starting with that.  (And it's a useful test to have anyway.)

How many points are you requesting as input?  (I understand how two points specify a line.  But if you have three or more points, then I would like to understand how you want to interpret them.  For example, are you drawing a polyline?  Or do you just have two points?)

You write,

 i wish calculate the slope of the line for each point with the tangent.

The slope of a line is the same everywhere; it does not depend on what point you look at.

Also, what do you mean by "tangent?"  Do you mean the "tan" function from trigonometry?  I ask because the "tangent" to a curve at a point is a line that (a) passes through that point, and (b) has the same slope as the curve at that point.  But the tangent to a line, at any point, is just the line itself...

How i can calculate the slope of a line from his points?

If your line contains the points (x1,y1) and (x2, y2), then its slope is,

m = (y2 - y1)/(x2 - x1) .