In the general context of **[Numbers and Geometry][1]** I was playing around with geometric visualizations of **structures in the natural numbers** and 
came up with a type of function graphs I haven't seen before. I'd like to know if they have been investigated before, under which name and what can be learned from them (by seeing some geometric patterns and symmetries that were not visible otherwise).

I call these function graphs *line graphs*. They are defined for arbitrary functions $f:X\times X \rightarrow Y$ with $X,Y = \mathbb{N},\mathbb{Z},\mathbb{Q},\mathbb{R}$ and are created by drawing a line from each point in $X\times X$ to the points $(f(x,y),0)$ and $(0,f(x,y))$.

This is how the [line graph for $f(x,y) = xy$][2] looks like (in different resolutions):
[![enter image description here][3]][3]
<br>
[![enter image description here][4]][4]

We see a maybe astonishing pattern emerge: a square grid (that might be typical for multiplication-like functions). (Note that the prime numbers are exactly those nodes on the x- resp. y-axis with degree 2.)

Other line graphs look quite different, of course:

[![enter image description here][5]][5]

(Can you guess at a glance which function's line graph this is?)

Which properties of a function can be read off the geometrical patterns (symmetries) of its line graph - and how? The other way around: Which geometrical patterns (symmetries) can be predicted just by looking at the definition of a function $f$?

*[If you like to play around with line graphs you can do it [here][2].]*

----------

Related question: By construction of line graphs, every function $f$ has a "reverse" function $f^*$ associated to it, which is defined by the point at which the line going through $(x,y)$ and $(f(x,y),0)$ crosses the y-axis, which is by definition at $(0,f^*(x,y))$. 

We have $f^*(x,y) = \frac{f(x,y)\times y}{f(x,y) - x}$ and $f(x,y) = \frac{f^*(x,y)\times x}{f^*(x,y) - y}$

For $f(x,y) = x y$, we have $f^*(x,y) = y^2/(y-1)$ (which by the way does not depend on $x$):

[![enter image description here][6]][6]

Has this construction of an associated function $f^*$ been investigated before? Might it be interesing to investigate the relationship between $f$ and $f^*$? 


  [1]: https://www.youtube.com/watch?v=NdgQQfQLtWw
  [2]: http://syspedia.de/line-graphs/
  [3]: https://i.sstatic.net/RbiCp.png
  [4]: https://i.sstatic.net/mvwR4.png
  [5]: https://i.sstatic.net/cBbqh.png
  [6]: https://i.sstatic.net/mPZ23.png