CoCalc Public FilesSL2R.sagewsOpen with one click!
Author: Kyle Ormsby
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Description: Animations of the action of SL_2(R) on the upper half plane. See for details.
Compute Environment: Ubuntu 18.04 (Deprecated)
#hyperbolic transformation (translation) #a(t) is the prototypical hyperbolic family with fixed points 0 and infinity def a(t): return Matrix([[exp(t),0],[0,exp(-t)]]); #conjugate a(t) to produce a family with fixed points 0 and 1 def conj_a(t): c = Matrix([[1,0],[1,1]]); return c*a(t)*(c.inverse()); #act on a complex number by a matrix as a linear fractional transformation def LFT(A,z): a = A[0,0]; b = A[0,1]; c = A[1,0]; d = A[1,1]; return (a*z+b)/(c*z+d); #draw how the circle of radius r centered at 0 is transformed by conj_a(t) var('s'); def circ_a(t,r): complexval = LFT(conj_a(t),r*exp(i*s)); return parametric_plot((complexval.real(),complexval.imag()),(s,0,pi)); #animate the transformed circles by letting t range from 0 to 3 conj_a_circs = [(circ_a(t,1/32)+circ_a(t,1/16)+circ_a(t,3/32)+circ_a(t,1/8)) for t in srange(0,3,0.1,include_endpoint=True)]; hyperbolic = animate(conj_a_circs,xmin=-1,xmax=2,ymin=0,ymax=2);;'hyperbolic.gif');
#parabolic transformation (limit rotation) #n(t) is the prototypical parabolic family with fixed point 0 def n(t): return Matrix([[1,0],[t,1]]); #draw how some geodesics with endpoint at 0 are transformed by n(t) def circ_n(t,r): complexval = LFT(n(t),r*exp(i*s)-r); return parametric_plot((complexval.real(),complexval.imag()),(s,0,pi)); #animate the transformed geodesics by letting t range from 0 to 12 n_circs = [(circ_n(t,1/16)+circ_n(t,3/32)+circ_n(t,1/8)) for t in srange(0,12,0.25,include_endpoint=True)]; parabolic = animate(n_circs,xmin=-1,xmax=1,ymin=0,ymax=2);;'parabolic.gif')
#elliptic transformation (rotation) #k(t) is the prototypical elliptic family with fixed point i def k(t): return Matrix([[cos(t),-sin(t)],[sin(t),cos(t)]]); #draw how some geodesics passing through i are transformed by k(t) #theta is the angle between the line from the center c to i and the real axis def circ_k(t,theta): c = -cot(theta); r = csc(theta); complexval = LFT(k(t),r*exp(i*s)-c); return parametric_plot((complexval.real(),complexval.imag()),(s,0,pi)); #animate the the transformed geodesics by letting t range from 0 to 2*pi k_circs = [(circ_k(t,pi/6)+circ_k(t,pi/4)+circ_k(t,pi/3)) for t in srange(0,2*pi,pi/41,include_endpoint=True)]; elliptic = animate(k_circs,xmin=-3,xmax=3,ymin=0,ymax=3);;'elliptic.gif');