Boy's Surface - Maple Help

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Boy's Surface${}$

Main Concept

Boy's surface is an example of a non-orientable surface similar to the Klein bottle. It contains no singularities (pinch-points), but it does cross through itself. The surface can be described implicitly by a polynomial of degree six; as such it is called a sextic surface. In 1901 Werner Boy discovered this object when by trying to immerse the real projective plane into ${\mathrm{ℝ}}^{3}$.

 Apery parameterization A common parameterization of Boy's surface in was by given by Apery in 1986 as:   ,   ,   ,   where $\mathrm{α}=1$, , and . As the parameter ${\mathrm{α}}_{}$ goes to zero, Boy's surface smoothly transforms into the Roman surface. Values in between 0 and 1 are interpreted as a mixture of the Roman surface and Boy's surface, which are topologically equivalent. Both surfaces can be obtained by attaching a Möbius strip to the circumference of a circle and stretching it until  it forms a closed surface.
 Kusner-Bryant parameterization Another beautiful parameterization of Boy's surface was presented by Kusner and Bryant in 1988 which uses complex numbers. They first define ${g}_{1}$, ${g}_{2}$, ${g}_{3}$ and $g$ as: ${g}_{1}=-\frac{3}{2}\cdot \mathrm{ℑ}\left(\frac{\mathrm{\eta }\left(1-{\mathrm{\eta }}^{4}\right)}{{\mathrm{\eta }}^{6}+\sqrt{5}{\mathrm{\eta }}^{3}-1}\right)$,   ,   $\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}{g}_{3}=\mathrm{ℑ}\left(\frac{1+{\mathrm{\eta }}^{6}}{{\mathrm{\eta }}^{6}+\sqrt{5}{\mathrm{\eta }}^{3}-1}\right)-\frac{1}{2}$,   ,   where $\mathrm{\Im }$ denotes the imaginary component of a complex number, and $\mathrm{\Re }$ denotes the real part. The Cartesian parameterization is then given by: , , . 

 Parameterization Homotopy parameter, ${\mathbf{α}}_{}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}$Roman   Boy's Transparency

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