Frequency of partially or fully overlapping lines

Question

Let L be a set of line-segments (lines) in a two-dimensional (2D) 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; 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.

{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 1

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.

Answer 2

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.