Hopf fibration Wikipedia Analyse
Replacement of fiber (curve) by fiberC (Circle)
#====================================
# Wikipedia: Hopf_fibration
#====================================
x1(e1,e2,n) = sin(n) * cos((e2+e1)/2)
x2(e1,e2,n) = sin(n) * sin((e2+e1)/2)
x3(e1,e2,n) = cos(n) * cos((e2-e1)/2)
x4(e1,e2,n) = cos(n) * sin((e2-e1)/2)
#====================================
# Base of fibers: sphere S2
#====================================
xS2(e1,n) = sin(2n) * cos(e1)
yS2(e1,n) = sin(2n) * sin(e1)
zS2(e1,n) = cos(2n)
#====================================
# Stereographic projection
#====================================
xR3(e1,e2,n) = x1(e1,e2,n) / (1-x4(e1,e2,n))
yR3(e1,e2,n) = x2(e1,e2,n) / (1-x4(e1,e2,n))
zR3(e1,e2,n) = x3(e1,e2,n) / (1-x4(e1,e2,n))
Pfiber= (xR3(e1,e2,n), yR3(e1,e2,n), zR3(e1,e2,n))
fiber= curve((xR3(e1,e2,n), yR3(e1,e2,n), zR3(e1,e2,n)),e2,0,4pi)
#====================================
# fiber as Circle. If n==0 then fiber is zAxis
#--------------------------------------------------------
# Replacement of fiber (curve) by fiberC (Circle)
#--------------------------------------------------------
#
# F0: e2=e1-pi : z=0
# F1: e2=e1+pi : z=0
# Centerpoint: C=(F0+F1)/2 (by symmetry)
# Radius : R=length(C-F0)
#
# F0 = ( sin(e1),-cos(e1), 0) * sin(n)/(1+cos(n))
# F1 = (-sin(e1), cos(e1), 0) * sin(n)/(1-cos(n))
# C = (-sin(e1), cos(e1), 0) * cot(n)
# R = 1/sin(n)
# I did not calculate the direction. It's obvious -n.
#====================================
C = (cot(n);pi/2+e1;0)
R = 1/sin(n)
fiberC = Circle(C, R, vector((1;e1;-n)) )
fiber0 = segment((0,0,-10),(0,0,10))