<s>[INST] Finding a parametrization of a hyperbola who has a fixed signature How do I find parametrization of the hyperbola $x^2-y^2=1$ which is the unit sphere of a quadratic form with signature $(1,-1)?$
The only parametrization that comes to mind is $x=\cosh t,y=\sinh t$. What is a quadratic form of a parametrization?
 [/INST] $$\text {As }(a^2+b^2)^2=(2ab)^2+(a^2-b^2)^2$$
For $ab\ne 0,$
$$\left(\frac{a^2+b^2}{2ab}\right)^2=1+\left(\frac{a^2-b^2}{2ab}\right)^2\implies (x,y)\text{ can be}\left(\frac{a^2+b^2}{2ab},\frac{a^2-b^2}{2ab}\right)$$
For $a^2-b^2\ne 0\implies a\ne \pm b$
$$\left(\frac{a^2+b^2}{a^2-b^2}\right)^2=1+\left(\frac{2ab}{a^2-b^2}\right)^2\implies (x,y)\text{ can be}\left(\frac{a^2+b^2}{a^2-b^2},\frac{2ab}{a^2-b^2}\right)$$
Try putting $a=b\tan\theta$ or $b\cot\theta$ or $b=a\tanh y$
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