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**Laurent** What you need to do is plot a few level curves, like $\displaystyle u=-3,-2,-1,1,2,3,\ldots$ and same for $\displaystyle v$. It is easier for $\displaystyle v$: we have $\displaystyle y=\frac{v}{2x}$ (hyperbola). For $\displaystyle u$, if you don't know conics (these are also hyperbola), you can plot the level curves on your calculator or computer (equation $\displaystyle y=\pm\sqrt{x^2-u}$ (either + or -)), together with level curves for v and see what relation there is between both (you'll need an orthonormal basis to see it).