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.
Read more about this topic: Boy's Surface
Famous quotes containing the words boy and/or surface:
“From his childhood onwards this boy will be surrounded by sycophants and flatterers.... In due course, following the precedent which has already been set, he will be sent on a tour of the world and probably rumours of a morganatic marriage alliance will follow, and the end of it will be the country will be called upon to pay the bill.”
—James Keir Hardie (1856–1915)
“I have passed down the river before sunrise on a summer morning, between fields of lilies still shut in sleep; and when, at length, the flakes of sunlight from over the bank fell on the surface of the water, whole fields of white blossoms seemed to flash open before me, as I floated along, like the unfolding of a banner, so sensible is this flower to the influence of the sun’s rays.”
—Henry David Thoreau (1817–1862)