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.

Read more about this topic:  Boy's Surface

Famous quotes containing the words boy and/or surface:

    “The boy stood on the burning deck,
    Whence all but he had fled;
    The flame that lit the battle’s wreck,
    Shone round him o’er the dead.

    Yet beautiful and bright he stood,
    As born to rule the storm;
    A creature of heroic blood,
    A proud though childlike form.”
    —Felicia Dorothea Hemans (1783–1835)

    “Brave men are all vertebrates; they have their softness on the surface and their toughness in the middle.”
    —Gilbert Keith Chesterton (1874–1936)