Rewrite the following surface so that I can graph it.

Question

$z = \dfrac{1+x^2}{1+y^2}$

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I want the part of the surface above the square $|x|+|y|\leq 1$

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OR we can write this square as $-y<x<y$ and $-1<x<-1$

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I have spent hours trying to get this graph on MAC grapher and other online java tools.

I need a parameterization with bounds independent of variables.

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Example : $x =f(u, v), y = g(u, v), z = r(u, v)$ . and $a<u<b, n<v<m$

Where $a, b, n, m$ are integers.

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If you can get the graph in any other way, that will be totally welcome, I just need this graph

Answer 1

The square $|x|+|y|\leq1$ can be parametrized by $(x,y)=(t-u,t+u),\quad -\frac{1}{2}\leq t\leq\frac{1}{2},\quad -\frac{1}{2}\leq u\leq\frac{1}{2}$. Thus the desired surface can be parametrized by

$(x,y,z)=(t-u,t+u,\frac{1+(t-u)^2}{1+(t+u)^2})$ with the same bounds on $t$ and $u$.