Editing Boy's surface (section)

==Parametrization==
[[File:BoysSurfaceKusnerBryant.svg|thumb|A view of the Kusner–Bryant parametrization of the Boy's surface]]
Boy's surface can be parametrized in several ways. One parametrization, discovered by Rob Kusner and [[Robert Bryant (mathematician)|Robert Bryant]],<ref name="Wells1988">{{cite book|contribution=Surfaces in conformal geometry (Robert Bryant)|author=Raymond O'Neil Wells|title=The Mathematical Heritage of Hermann Weyl (May 12–16, 1987, Duke University, Durham, North Carolina)|volume=48|url=https://books.google.com/books?id=e0MECAAAQBAJ&pg=PA227|year=1988|series=Proc. Sympos. Pure Math.|publisher=American Mathematical Soc.|isbn=978-0-8218-1482-6|pages=227–240|doi=10.1090/pspum/048/974338}}</ref> is the following: given a complex number ''w'' whose [[magnitude (mathematics)|magnitude]] is less than or equal to one (<math> \| w \| \le 1</math>), let
:<math>\begin{align}
  g_1 &= -{3 \over 2} \operatorname{Im} \left[ {w \left(1 - w^4\right) \over w^6 + \sqrt{5} w^3 - 1} \right]\\[4pt]
  g_2 &= -{3 \over 2} \operatorname{Re} \left[ {w \left(1 + w^4\right) \over w^6 + \sqrt{5} w^3 - 1} \right]\\[4pt]
  g_3 &= \operatorname{Im} \left[ {1 + w^6 \over w^6 + \sqrt{5} w^3 - 1}  \right] - {1 \over 2}\\
\end{align}</math>

and then set 
:<math>\begin{pmatrix}x\\ y\\ z\end{pmatrix} = \frac{1}{g_1^2 + g_2^2 + g_3^2} \begin{pmatrix}g_1\\ g_2\\ g_3\end{pmatrix}</math>

we then obtain the [[Cartesian coordinates]] ''x'', ''y'', and ''z''   of a point on the Boy's surface.

If one performs an inversion of this parametrization centered on the triple point, one obtains a complete [[minimal surface]] with three [[End (topology)|ends]] (that's how this parametrization was discovered naturally).  This implies that the Bryant–Kusner parametrization of Boy's surfaces is "optimal" in the sense that it is the "least bent" immersion of a [[projective plane]] into [[three-space]].

===Property of Bryant–Kusner parametrization===
{{Wikibooks|Famous Theorems of Mathematics|Boy's surface}}
If ''w'' is replaced by the negative reciprocal of its [[complex conjugate]], <math display="inline">-{1 \over w^\star},</math> then the functions ''g''<sub>1</sub>, ''g''<sub>2</sub>, and ''g''<sub>3</sub> of ''w'' are left unchanged.

By replacing {{math|''w''}} in terms of its real and imaginary parts {{math|1=''w'' = ''s'' + ''it''}}, and expanding resulting parameterization, one may obtain a parameterization of Boy's surface in terms of [[rational function]]s of {{math|''s''}} and {{math|''t''}}. This shows that Boy's surface is not only an [[algebraic surface]], but even a [[rational surface]]. The remark of the preceding paragraph shows that the [[generic fiber]] of this parameterization consists of two points (that is that almost every point of Boy's surface may be obtained by two parameters values).

===Relation to the real projective plane===
Let <math>P(w) = (x(w), y(w), z(w))</math> be the Bryant–Kusner parametrization of Boy's surface. Then
:<math> P(w) = P\left(-{1 \over w^\star} \right). </math>

This explains the condition <math>\left\| w \right\| \le 1</math> on the parameter: if <math>\left\| w \right\| < 1,</math> then <math display="inline"> \left\| - {1 \over w^\star} \right\| > 1 .</math> However, things are slightly more complicated for <math> \left\| w \right\| = 1.</math>  In this case, one has <math display="inline">-{1 \over w^\star} = -w .</math> This means that, if <math> \left \| w \right\| = 1, </math> the point of the Boy's surface is obtained from two parameter values: <math>P(w) = P(-w).</math> In other words, the Boy's surface has been parametrized by a disk such that pairs of diametrically opposite points on the [[perimeter]] of the disk are equivalent. This shows that the Boy's surface is the image of the [[real projective plane]], RP<sup>2</sup> by a [[smooth function|smooth map]]. That is, the parametrization of the Boy's surface is an [[immersion (mathematics)|immersion]] of the real projective plane into the [[Euclidean space]].