Editing Boy's surface (section)

===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]].