Boy Surface #1Written by Paul BourkeJanuary 2004 Projective plane without singularities. x = A (cos(u) cos(2v) + B sin(u) cos(v)) cos(u) / (B  sin(2u) sin(3v)) y = A (cos(u) sin(2v)  B sin(u) sin(v)) cos(u) / (B  sin(2u) sin(3v)) z = B cos(u) cos(u) / (B  sin(2u) sin(3v)) where A = 2/3 and B = sqrt(2) and 0 <= u,v <= pi
Animation of a transition from a Steiner (Roman) surface into the boy surface.
BoyLike SurfaceWritten by Paul BourkeOctober 1992 Projective plane without singularities.
Boy Surface, #2Written by Paul BourkeJanuary 2004 Projective plane without singularities. x = cos(t) sin(s) y = sin(t) sin(s) z = cos(s) x^{} = A ((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})) y^{} = B ((y^{2}  z^{2}) + z x (z^{2}  x^{2}) + x y (y^{2}  x^{2})) z^{} = C (x + y + z) ((x + y + z)^{3} + 4 (y  x) (z  y) (x  z))
where A = 1/2, B = sqrt(3)/2, and C = 1/8
