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.
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Famous quotes containing the words boy and/or surface:
“The boy seemed to have fallen
From shelf to shelf of someone’s rage.”
—John Ashbery (b. 1927)
“We’ve forgotten what it’s like not to be able to reach the light switch. We’ve forgotten a lot of the monsters that seemed to live in our room at night. Nevertheless, those memories are still there, somewhere inside us, and can sometimes be brought to the surface by events, sights, sounds, or smells. Children, though, can never have grown-up feelings until they’ve been allowed to do the growing.”
—Fred Rogers (20th century)