Equations for Boy's Surface

Let

F = ( f(x, y, z), g(x, y, z), h(x, y, z) )

be a map from R^3 to R^3. If the functions

f, g, h

are homogeneous polynomials of even degree >= 2, then for all (x, y, z) in R^3 we have

F(x, y, z) = F(-(x, y, z)).

Thus the restiction of F to the unit 2-sphere sends antipodal points to the same image point--hence, F defines a new map from RP^2 to R^3. In the case of Boy's surface,

f(x, y, z) = [ (2x^2 - y^2 - z^2)(x^2 + y^2 + x^2) + 2yz(y^2 -
z^2) + zx(x^2 - z^2) + xy(y^2 - x^2) ]/2

g(x, y, z) = (Sqrt(3))/2 [ (y^2 - z^2)(x^2 + y^2 + z^2) + zx(z^2 -
x^2) + xy(y^2 - x^2) ]

h(x, y, z) = (x + y + z)[ (x + y + z)^3 + 4(y - x)(z - y)(x -
z)]

Pictures may be created parametrically in Maple by the following commands:

x:=cos(t)*sin(s);

y:=sin(t)*sin(s);

z:=cos(s);

f:=1/2*((2*x^2-y^2-z^2) + 2*y*z*(y^2-z^2) + z*x*(x^2-z^2)

 +x*y*(y^2-x^2));

g:= sqrt(3)/2*((y^2-z^2) + z*x*(z^2-x^2) + x*y*(y^2-x^2));

h:=(x+y+z)*((x+y+z)^3 + 4*(y-x)*(z-y)*(x-z));

plot3d( [h/8,f,g], s=0..Pi, t=0..Pi)

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