How to derive the equation of a spline inside a pipe

Let’s suppose a pipe with some curves. We put an elastic bar inside.
The diameter of the elastic bar is much smaller than the diameter of the pipe.
We know that the elastic bar bends to minimize the elastic energy, or to minimize the curvature in its entire length.
If I draw the section of this pipe, could be something like this:

My question is, how can I find the equation of this line?
I know it is a spline, but I don’t know a priori the coordinate of the interpolation point. How to find it?