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Boy's Surface

Main Concept

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.

 

Apery parameterization

 

A common parameterization of Boy's surface in ℝ3 was by given by Apery in 1986 as:

 

xu,v =2cosv2cos2u+cosusin2v2α 2sin3usin2v,

 

yu,v =2cosv2sin2usinusin2v2α 2sin3usin2v,

 

zu,v = 3cosv22α 2sin3usin2v,

 

where α=1, uπ2, π2 , and v0, 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.

 

Kusner-Bryant parameterization

 

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:

g1=32ℑη1η4η6+5η31,

 

 g2=32ℜη1+ η4η6+5η31,

 

g3=ℑ1+η6η6+5η3112,

 

g = g12+g22+g32,

 

where  denotes the imaginary component of a complex number, and  denotes the real part. The Cartesian parameterization is then given by:

x η= g1/g,

 y η= g2/g,

 zη= g3/g.

 

Parameterization

 Homotopy parameter, αRoman   Boy's

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