Is there a parametrization of a hyperbola x2−y2=1 by functions x(t) and y(t) both birational?Consider the hyperbola x2−y2=1. I am aware of some parametrizations like:1. (x(t),y(t))=(t2+12t,t2−12t)2. (x(t),y(t))=(t2+1t2−1,2tt2−1)3. (x(t),y(t))=(cosht,sinht)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?

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Ausspruchx807 · Accepted Answer

If (f(t),g(t)) is a parameterization with f and g rational and g−1 is rational, then:(f(g−1(s)),s)is a parameterization and f∘g−1 is rationalBut f∘g−1(s)=1+s2 is not a rational function.This works if we even just want (f(t),g(t))  to parameterize an subset of the curve x2−y2=1, for t in some interval (a,b).This same argument shows that f can’t be birational if g is rational.

Is there a parametrization of a hyperbola

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