A common parameterization of Boy's surface in ${\mathrm{\ℝ}}^3$was by given by Apery in 1986 as:

$x\left(u\,v\right) \=\frac{\sqrt{2}{\cos \left(v\right)}^2\cos \left(2u\right)+\cos \left(u\right)\sin \left(2v\right)}{2-\mathrm{\alpha } \sqrt{2}\sin \left(3u\right)\sin \left(2v\right)}$,

$y\left(u\,v\right) \=\frac{\sqrt{2}{\cos \left(v\right)}^2\sin \left(2u\right)-\sin \left(u\right)\sin \left(2v\right)}{2-\mathrm{\alpha } \sqrt{2}\sin \left(3u\right)\sin \left(2v\right)}$,

$z\left(u\,v\right) \= \frac{3{\cos \left(v\right)}^2}{2-\mathrm{\alpha } \sqrt{2}\sin \left(3u\right)\sin \left(2v\right)}$,

where $\mathrm{\α}\=1$, $u\in \left[-\frac{\mathrm{\π}}{2}\, \frac{\mathrm{\π}}{2}\right]$, and $v\in \left[0\, \mathrm{\π}\right]$. 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.

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:

$g_1\=-\frac{3}{2}\cdot \mathrm{\ℑ}\left(\frac{\mathrm{\eta }\left(1-{\mathrm{\eta }}^4\right)}{{\mathrm{\eta }}^6\+\sqrt{5}{\mathrm{\eta }}^3-1}\right)$,

$g_2\=-\frac{3}{2}\cdot \mathrm{\ℜ}\left(\frac{\mathrm{\eta }\left(1\+ {\mathrm{\eta }}^4\right)}{{\mathrm{\eta }}^6\+\sqrt{5}{\mathrm{\eta }}^3-1}\right)$,

$g_3\=\mathrm{\ℑ}\left(\frac{1\+{\mathrm{\eta }}^6}{{\mathrm{\eta }}^6\+\sqrt{5}{\mathrm{\eta }}^3-1}\right)-\frac{1}{2}$,

$g \= g_1^2\+g_2^2\+g_3^2$,

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:

$x \left(\mathrm{\eta }\right)\= g_1\/g$,

$y \left(\mathrm{\eta }\right)\= g_2\/g$,

$z\left(\mathrm{\eta }\right)\= g_3\/g$.