Visualization of 4D Expanded form of $f(x,y) = y/x - x/y$

Question

I have a (not so) simple question: how do I visualize the following:

$$f(x,y,z) = ((yz)/x - x/(yz)) + (y/(xz) - (xz)/y) + (y/x - x/y)?$$

I am a csci person, but not a big math person, so I'm a bit at a loss as to how to approach this...Is there a software tool that can help me with this?

background:
This is a dimensional expansion of $f(x,y) = y/x - x/y$, which gives an interesting plot. I wanted to see what it looked like with an extra dimension.

Also, I hope Math.SE is the right place for this...I haven't been here before, so my apologies if I've stepped on any toes...

Answer 1

The issue is there are three independent variables, which would make this a four-dimensional graph. You'll have to either reduce one dimension, or find a new one to use, in other words:

1. Graph various level sets (implicitly defined surfaces of the form $f(x,y,z)=C$, for variously selected constants $C$), and then keep a collection of the images. This is roughly equivalent to visualizing a three-dimensional landscape topographically in two dimensions with contours.
2. Use the dimension of time for the dependent variable. This is essentially #1, but you put the images into an animation and use the $C\,$'s as a time parameter. This method could be computationally taxing, though it gives a more succinct summary of what's "going on."

One issue I see is that you have a number of reciprocals, which will make the image difficult to graph around the $xy$-, $yz$-, and $xz$-planes; you'll have to pick your bounds appropriately. (See also.)

Answer 2

You may want to try volume visualization. Try for instance VolVis and VTK.