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

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.