Hopf fibration Wikipedia Analyse
- Author:
- Thijs
Replacement of fiber (curve) by fiberC (Circle)
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# Wikipedia: Hopf_fibration
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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)
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# Base of fibers: sphere S2
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xS2(e1,n) = sin(2n) * cos(e1)
yS2(e1,n) = sin(2n) * sin(e1)
zS2(e1,n) = cos(2n)
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# Stereographic projection
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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)
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# fiber as Circle. If n==0 then fiber is zAxis
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# Replacement of fiber (curve) by fiberC (Circle)
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#
# F0: e2=e1-pi : z=0
# F1: e2=e1+pi : z=0
# Centerpoint: C=(F0+F1)/2 (by symmetry)
# Radius : R=length(C-F0)
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# 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.
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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))