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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...

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2 Answers 2

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.)

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Thanks for the analysis! I'm going to attempt #2, but do you have any suggestions as to what tools exist (and would be best) that could be used to create the animation or images? This is all new to me, so I'm not sure what's available... Thanks so much! :) –  ZeeKay Apr 3 '12 at 19:37
    
also, I'd +1, but I don't have the rep :( –  ZeeKay Apr 3 '12 at 19:37

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

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+1, but the VolViz link seems to be broken. –  ysap Sep 7 '12 at 21:09
    
@ysap, fixed, thanks. –  lhf Sep 7 '12 at 23:39

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