Denominator contribution in a 3D Plot

Question

In an earlier post I asked for a 3D plot of Plot3D[Sqrt[((x^2 + y^2)/((x*y) + 1))], {x, -20, 20}, {y, -20, 20}] along with it's integer roots and in turn plot of those. (3D plot Versus actual values).

I would like now to see whether Mathematica can depict: To show the contribution of the +1 in the denominator, what if we were to graph it with just the x*y in the denominator like (x^2+y^2)/x y then do another one with a -1 like (x^2+y^2)/(x y-1), then have the resulting graphs interlaced/transposed on top of one another to see the contribution. Is there a function to allow this visibility?

Now let me expand a little more on this: Can I plot Plot3D[Sqrt[((x^2 + y^2)/((x*y) + n))] where -10<=n<=+10 and have Mathematica put that into motion to see how the function evolves around those values? That's 1 degree of freedom. Can that be combined with x^2-y^2 in the numerator (instead of the +)?

Thanks, Steve.

Answer 1

For the first part of what you asked, is the following what you had in mind?

Animate[Plot3D[Sqrt[(x^2 + y^2)/(x*y + n)], {x, -20, 20}, {y, -20, 20}, 
   PlotRange -> All], {n, -10, 10}]

So, just have a free parameter n and allow it to take values in a specific range whilst animating the plot. This results in the following

You can hit pause and then start shifting the value around to play with the graph.

And then you mentioned a change in the numerator. So, multiply by a constant and then allow for the constant to take values in the range $[-1,1]$. If I understood correctly the following gives what you asked for

Manipulate[Plot3D[Sqrt[(x^2 + c*y^2)/(x*y + n)], {x, -20, 20}, {y, -20, 20}, 
   PlotRange -> All], {n, -10, 10, Appearance -> "Open"}, 
  {{c, 1}, -1, 1, Appearance -> "Open"}]

which gives