Examples
Applying the above transformation to the lemniscate of Bernoulli
gives us
the equation of a hyperbola; since inversion is a birational transformation and the hyperbola is a rational curve, this shows the lemniscate is also a rational curve, which is to say a curve of genus zero. If we apply it to the Fermat curve xn + yn = 1, where n is odd, we obtain
Any rational point on the Fermat curve has a corresponding rational point on this curve, giving an equivalent formulation of Fermat's Last Theorem.
Read more about this topic: Inverse Curve
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