Creating a 2D homogeneous plot of straight lines

I am trying to create a mesh (collection of straight intersecting lines) of given dimensions (a x a; a square mesh).

The fibers (straight lines) run from one edge to another. There is no pattern of distribuion of lines i.e, they are generated randomly. The problem I am facing is that the intersection points are not equally (more or less equally) spaced. There are regions in the mesh with clusters of intersection points and regions with few of them.

The sum of the lengths of all fibers is fixed.

I have tried a couple of ways to make the distribution homogeneous but have failed.

(in the eg (pic): the edge is divided into 11000 points; there are more points in the upper left sector of the mesh; also there are collections of intersection points close to each other, like one around (6000,3000))

Please suggest something. Thanks in advance.

[Rik]

What code did you use to generate this? It is likely much easier to adapt almost working code, than come up with new code.