**Boy's Surface can be described in many different ways**

*F. Apery defines a Boy's Surface as the real zero set of the
degree six polynomial
*

p(x,y,z) = 64 (1 - z)^{3} z^{3} - 48 (1 -
z)^{2} z^{2} (3x^{2} + 3y^{2} +
2z^{2}) + 12 (1 - z) z (27 (x^{2} +
y^{2})^{2} - 24 z^{2} (x^{2} +
y^{2}) + 36 Sqrt(2) y z (y^{2} - 3 x^{2}) +
4z^{4}) + (9x^{2} + 9y^{2} - 2z^{2})
(-81 (x^{2} + y^{2})^{2} - 72 z^{2}
(x^{2} + y^{2}) + 108 Sqrt(2) x z (x^{2} -
3y^{2}) + 4z^{4})

*which can be described by*

f = Cos(s) (1 - Sqrt(2) Sin(s) Cos(s) Sin(3t))^{-1}
[(Sqrt(2)/3) Cos(s) Cos(2t) + (2/3) Sin(s) Cos(t)]

g = Cos(s) (1 - Sqrt(2) Sin(s) Cos(s) Sin(3t))^{-1}
[(Sqrt(2)/3) Cos(s) Sin(2t) - (2/3) Sin(s) Sin(t)]

h = Cos(s) (1 - Sqrt(2) Sin(s) Cos(s) Sin(3t))^{-1} [Cos(s)]

s and t both range from 0 to Pi

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Notice the self-intersection curve on the degree six
Boy's Surface shown above. For this curve, F. Apery gives the
parametrization

f = 2 (3 - Cos(6t))^{-1} [(2/3) Sin(Pi/4 - 3t) Cos(t)]

g = 2 (3 - Cos(6t))^{-1} [(2/3) Sin(Pi/4 - 3t) Sin(t)]

h = 2 (3 - Cos(6t))^{-1} [1]

*The following parametrization was found by Petit and Souriau in 1981.
However, it has not been proven to be a Boy's Surface.*

a = 10 + 1.41 Sin(6t - Pi/3) + 1.98 Sin(3t - Pi/6)

b = 10 + 1.41 Sin(6t - Pi/3) - 1.98 Sin(3t - Pi/6)

c = a^{2} - b^{2}

d = (a^{2} + b^{2})^{1/2}

e = (Pi/8) Sin(3t)

f = c/d + a Cos(u) - Sin(u)

g = d + a Cos(u) + b Sin(u)

x = 3.3 (f Cos(t) - g Sin(e) Sin(t))

y = 3.3 (f Sin(t) + g Sin(e) Cos(t))

z = 4 (g Cos(e) - 10)

t ranges from 0 to Pi and u ranges from 0 to 2 Pi

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*This parametrization by R. Bryant uses the representation of
P*^{2} as the unit circle on the complex plane where the
mapping is invariante under z -> -1/(complex conjugate of z).

x = (g1^{2} + g2^{2} + g3^{2})^{-1} g1

y = (g1^{2} + g2^{2} + g3^{2})^{-1} g2

z = (g1^{2} + g2^{2} + g3^{2})^{-1} g3

where

g1 = -(3/2) Im[z (1 - z^{4}) (z^{6} + Sqrt(5)
z^{3} - 1)^{-1}]

g2 = -(3/2) Re[z (1 + z^{4}) (z^{6} + Sqrt(5)
z^{3} - 1)^{-1}]

g3 = -1/2 + Im[(1 + z^{6}) (z^{6} + Sqrt(5)
z^{3} - 1)^{-1}]

and

z = r Cos(t) + *i* r Sin(t)

r ranges from 0 to 1 and t ranges from 0 to 2 Pi

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See a mesh of this surface in Geomview

*This parametrization by J. F. Hughes consists of three homogeneous
polynomials of degree eight, defined on S*^{2}:

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**Source:** "Models of the Real Projective Plane" by
F. Apery (Chapter 2)

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