Boy’s Surfaces

Boy’s surface is a nonorientable surface found by Werner Boy in 1901. Boy’s surface is an immersion of the real projective plane in 3-dimensional without infinities and singularities, but it meets itself in a triple point (self-intersect). The images below are generated using 3D-XplorMath, and with the “optimal” Bryant-Kusner parametrization when a= 0.5, -1.45 < u < 0, and 0 < v < 2\pi :

Boy's Surface (Bryant-Kusner)

Other viewpoints :

Boy's Surface (Bryant-Kusner) other viewpoints

Projective Plane

There are many ways to make a model of the Boy’s surface using the projective plane, one of them is to take a disc, and join together opposite points on the edge with self intersection (note : In fact, this can not be done in three dimensions without self intersections); so the disc must pass through itself somewhere. The Boy’s surface can be obtained by sewing a corresponding band (Möbius band) round the edge of a disc.

The \mathbb{R}^{3} Parametrization of Boy’s surface

Rob Kusner and Robert Bryant discovered the beautiful parametrization of the Boy’s surface on a given complex number z, where \left | z \right |\leq 1, so that giving the Cartesian coordinates \left ( X,Y,Z \right ) of a point on the surface.

In 1986 Apéry gave  the analytic equations for the general method of nonorientable surfaces. Following this standard form, \mathbb{R}^{3} parametrization of the Boy’s surface can also be written as a smooth deformation given by the equations :

x\left ( u,v \right )= \frac {\sqrt{2}\cos\left ( 2u \right ) \cos^{2}\left ( v \right )+\cos \left ( u \right )\sin\left ( 2v \right ) }{D},

y\left ( u,v \right )= \frac {\sqrt{2}\sin\left ( 2u \right ) \cos^{2}\left ( v \right )-\sin \left ( u \right )\sin\left ( 2v \right ) }{D},

z\left ( u,v \right )= \frac {3 \cos^{2}\left ( v \right )}{D}.

where

D= 2-a\sqrt{2}\sin\left ( 3u \right )\sin\left ( 2v \right ),

a varies from 0 to 1.

Here are some links related to Boy’s surface :

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