Editing Pythagorean triple (section)

==Geometry of Euclid's formula==

===Rational points on a unit circle===
[[Image:Pythagorean triple and rational point on unit triangle 1.svg|thumb|right|3,4,5 maps to x,y point (4/5,3/5) on the unit circle]]
[[Image:Stereographic projection of rational points.svg|thumb|right|The [[rational point]]s on a circle correspond, under [[stereographic projection]], to the rational points of the line.]]
Euclid's formula for a Pythagorean triple

:<math>a = m^2-n^2,\quad b=2mn,\quad c=m^2+n^2</math>

can be understood in terms of the geometry of [[rational point]]s on the [[unit circle]] {{harv|Trautman|1998}}.

In fact, a point in the [[Cartesian plane]] with coordinates {{math|(''x'', ''y'')}} belongs to the unit circle if {{math|''x''{{sup|2}} + ''y''{{sup|2}} {{=}} 1}}. The point is ''rational'' if {{math|''x''}} and {{math|''y''}} are [[rational number]]s, that is, if there are [[coprime integers]] {{math|''a'', ''b'', ''c''}} such that 
:<math>\biggl(\frac{a}{c}\biggr)^2\! + \biggl(\frac{b}{c}\biggr)^2=1.</math>

By multiplying both members by {{math|''c''{{sup|2}}}}, one can see that the rational points on the circle are in one-to-one correspondence with the primitive Pythagorean triples.

The unit circle may also be defined by a [[parametric equation]]
:<math>x=\frac{1-t^2}{1+t^2},\quad y=\frac{2t}{1+t^2}.</math>
Euclid's formula for Pythagorean triples and the inverse relationship {{math|1=''t'' = ''y'' / (''x'' + 1)}} mean that, except for {{math|(−1, 0)}}, a point {{math|(''x'', ''y'')}} on the circle is rational if and only if the corresponding value of {{math|''t''}} is a rational number.  Note that {{math|1=''t'' = ''y'' / (''x'' + 1) = ''b'' / (''a'' + ''c'') = ''n'' / ''m''}} is also the [[Tangent half-angle formula|tangent of half of the angle]] that is opposite the triangle side of length {{mvar|b}}.

===Stereographic approach===
[[Image:Stereoprojzero.svg|thumb|right|Stereographic projection of the unit circle onto the {{math|''x''}}-axis. Given a point {{math|''P''}} on the unit circle, draw a line from {{math|''P''}} to the point {{math|''N'' {{=}} (0, 1)}} (the ''north pole''). The point {{math|''P''}}′ where the line intersects the {{math|''x''}}-axis is the stereographic projection of {{math|''P''}}. Inversely, starting with a point {{math|''P''}}′ on the {{math|''x''}}-axis, and drawing a line from {{math|''P''}}′ to {{math|''N''}}, the inverse stereographic projection is the point {{math|''P''}} where the line intersects the unit circle.]]
There is a correspondence between [[Group of rational points on the unit circle|points on the unit circle with rational coordinates]] and primitive Pythagorean triples. At this point, Euclid's formulae can be derived either by methods of [[trigonometry]] or equivalently by using the [[stereographic projection]].

For the stereographic approach, suppose that {{math|''P''}}′ is a point on the {{math|''x''}}-axis with rational coordinates

:<math>P' = \left(\frac{m}{n},0\right).</math>

Then, it can be shown by basic algebra that the point {{math|''P''}} has coordinates

:<math>
P = \left(
 \frac{2\left(\frac{m}{n}\right)}{\left(\frac{m}{n}\right)^2+1},
 \frac{\left(\frac{m}{n}\right)^2-1}{\left(\frac{m}{n}\right)^2+1}
\right) =
\left(
 \frac{2mn}{m^2+n^2},
 \frac{m^2-n^2}{m^2+n^2}
\right).</math>

This establishes that each [[rational point]] of the {{math|''x''}}-axis goes over to a rational point of the unit circle. The converse, that every rational point of the unit circle comes from such a point of the {{math|''x''}}-axis, follows by applying the inverse stereographic projection. Suppose that {{math|''P''(''x'', ''y'')}} is a point of the unit circle with {{math|''x''}} and {{math|''y''}} rational numbers. Then the point {{math|''P''}}′ obtained by stereographic projection onto the {{math|''x''}}-axis has coordinates

:<math>\left(\frac{x}{1-y},0\right)</math>

which is rational.

In terms of [[algebraic geometry]], the [[algebraic variety]] of rational points on the unit circle is [[birational]] to the [[affine line]] over the rational numbers. The unit circle is thus called a [[rational curve]], and it is this fact which enables an explicit parameterization of the (rational number) points on it by means of rational functions.