## Parameterising a trajectory to pass through fixed points

Hi, all!

I'm new here, so I'm not quite sure if this is the right forum section for my problem, but I'm sure one of the mods will be kind enough to move it if it's not.

I'm at a bit of a loss as to how to do the following elegantly—I've had some success with a slow, "nested" optimisation for what I'm trying to do, but I'm not happy with the results.

I have a trajectory described by:

$\frac{{dx}}{{dt}} = f(x,y,a,b,c)$

$\frac{{dy}}{{dt}} = g(x,y,a,b,c)$

where $a$, $b$, and $c$ are parameters. The functions $f$ and $g$ cannot be calculated analytically (iteration is required), but what happens internally is unimportant anyway, I think.

What I'm trying to do is find out which combinations of parameters $a$, $b$, and $c$ will make the trajectory pass through two fixed points, say $(x_1,y_1)$ and $(x_2,y_2)$.

Of course, this will more than likely lead to many solutions to the problem, and I will choose a set of parameters that minimises another function, say $h(x_1,y_1,x_2,y_2,a,b,c)$.

Does anyone have any idea how I'd go about finding appropriate combinations of parameters?

My weapon of choice is MATLAB, in case that happens to have built-in functions for what I need.

Any help, or even just a pointer in the right direction, would be greatly appreciated.

Thanks!
NChE