Finding a parametrization of a hyperbola who has a fixed signature

Question

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?

Answer 1

If you want a rational parametrization, try $x = (s + 1/s)/2$, $y= (s - 1/s)/2$.

Answer 2

$$\text {As }(a^2+b^2)^2=(2ab)^2+(a^2-b^2)^2$$

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 $b=a\tanh y$