Boy's surface is an example of a non-orientable surface similar to the Klein bottle. It contains no singularities (pinch-points), but it does cross through itself. The surface can be described implicitly by a polynomial of degree six; as such it is called a sextic surface. In 1901 Werner Boy discovered this object when by trying to immerse the real projective plane into ℝ3.
A common parameterization of Boy's surface in ℝ3 was by given by Apery in 1986 as:
xu,v =2⁢cos⁡v2⁢cos⁡2⁢u+cos⁡u⁢sin⁡2⁢v2−α 2⁢sin3⁢u⁢sin2⁢v,
yu,v =2⁢cos⁡v2⁢sin⁡2⁢u−sin⁡u⁢sin⁡2⁢v2−α 2⁢sin3⁢u⁢sin2⁢v,
zu,v = 3⁢cos⁡v22−α 2⁢sin3⁢u⁢sin2⁢v,
where α=1, u∈−π2, π2 , and v∈0, π. As the parameter α goes to zero, Boy's surface smoothly transforms into the Roman surface. Values in between 0 and 1 are interpreted as a mixture of the Roman surface and Boy's surface, which are topologically equivalent. Both surfaces can be obtained by attaching a Möbius strip to the circumference of a circle and stretching it until it forms a closed surface.
Another beautiful parameterization of Boy's surface was presented by Kusner and Bryant in 1988 which uses complex numbers. They first define g1, g2, g3 and g as:
g = g12+g22+g32,
where ℑ denotes the imaginary component of a complex number, and ℜ denotes the real part. The Cartesian parameterization is then given by:
x η= g1/g,
y η= g2/g,
Homotopy parameter, αRoman
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