Editing Birational geometry (section)

==== Birational equivalence of a plane conic ====
For example, the circle <math>X</math> with equation <math>x^2 + y^2 - 1 = 0</math> in the affine plane is a rational curve, because there is a rational map {{nowrap|''f'' : <math>\mathbb{A}^1</math> ⇢ ''X''}} given by
:<math>f(t) = \left( \frac{2t}{1+t^2}, \frac{1 - t^2}{1 + t^2}\right),</math>
which has a rational inverse ''g'': ''X'' ⇢ <math>\mathbb{A}^1</math> given by
:<math>g(x,y) = \frac{1-y}{x}.</math>
Applying the map ''f'' with ''t'' a [[rational number]] gives a systematic construction of [[Pythagorean triple]]s.

The rational map <math>f</math> is not defined on the locus where <math>1 + t^2 = 0</math>. So, on the complex affine line <math>\mathbb{A}^1_{\Complex}</math>, <math>f</math> is a morphism on the open subset <math>U = \mathbb{A}^1_{\Complex}-\{i, -i\}</math>, <math>f: U \to X</math>. Likewise, the rational map {{nowrap|''g'' : ''X'' ⇢ <math>\mathbb{A}^1</math>}} is not defined at the point (0,−1) in <math>X</math>.