Boy's Surface - Parametrization of Boy's Surface

Parametrization of Boy's Surface

Boy's surface can be parametrized in several ways. One parametrization, discovered by Rob Kusner and Robert Bryant, is the following: given a complex number z whose magnitude is less than or equal to one, let

\begin{align} g_1 &= -{3 \over 2} \mathrm{Im} \left\\ g_2 &= -{3 \over 2} \mathrm{Re} \left\\ g_3 &= \mathrm{Im} \left - {1 \over 2}\\ \end{align}

so that

where x, y, and z are the desired Cartesian coordinates 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 "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.

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