Is there a parametrization of a hyperbola $x^2-y^2=1$ by functions x(t) and y(t) both birational?

Consider the hyperbola $x^2-y^2=1$. I am aware of some parametrizations like:

1. $(x(t),y(t))=(\frac{t^2+1}{2t},\frac{t^2-1}{2t})$;
2. $(x(t),y(t))=(\frac{t^2+1}{t^2-1},\frac{2t}{t^2-1})$;
3. $(x(t),y(t))=(\cosh t,\sinh t)$;
4. $(x(t),y(t))=(\sec(t),\tan(t))$;

The first and the second are by rational functions $x(t)$ and $y(t)$. But the functions are not birational(i.e. with rational inverse of each).

Is there a parametrization where:

• $x(t)$ is rational with inverse also rational, and
• $y(t)$ is rational with inverse also rational?

Is possible, to find a parametrization where both are rational and at least one of the has inverse rational?

Solution 1:

If $(f(t),g(t))$ is a parameterization with $f$ and $g$ rational and $g^{-1}$ is rational, then:

$$\left(f\left(g^{-1}(s)\right),s\right)$$ is a parameterization and $f\circ g^{-1}$ is rational.

But $f\circ g^{-1}(s)=\sqrt{1+s^2}$ is not a rational function.

This works if we even just want $(f(t),g(t))$ to parameterize an subset of the curve $x^2-y^2=1,$ for $t$ in some interval $(a,b).$

This same argument shows that $f$ can’t be birational if $g$ is rational.