Question: Let **L** be a set of line-segments (lines) in a two-dimensional (2D) [Euclidean space](http://en.wikipedia.org/wiki/Euclidean_space) that are randomly (say, using uniform random function) distributed, that is, some may overlap other(s). Overlapping could be on a point or a line segment (in this case it is partial or full overlapping). ***We aim to represent any frequency of overlapping lines in a reasonable way***. 

For those may think of [rendering](https://gis.stackexchange.com/questions/1493/rendering-overlapping-lines); it is not of our interest in this post.

To start, for example, our naive solution was to discretise all the lines into a grid, that is, aiming to generate a 2D raster map i.e., a matrix. In this way, we are able to count the number of visiting per cell for all lines in the set. The resulting map/matrix (as of our quick experiments) is promising.

![enter image description here](https://i.stack.imgur.com/JGFyp.jpg)

**{1} What is the closest topic (if exists) to the aim of our question?**  

*The answers may help us for improving our search keywords.*  

**{2} Any thoughts on better ideas or suggestions for improvement?**

Answer: 
We found a working solution as follows:

1. Define **P** as set of endpoints of all lines
2. Split all lines at the endpoints, **P**, producing **L**, set of newly generated lines
3. Counts all line occurrences in **L**

The above has been implemented easily in **Python** and works just fine.
It may seem counter-intuitive at first but it sounds like you are describing, at least in part, line-on-line overlay for which ArcGIS has overlay events available within Linear Referencing.