def _draw_funddom_d(coset_reps,format="MP",z0=I):
r""" Draw a fundamental domain for self in the circle model
INPUT:
- ''format'' -- (default 'Disp') How to present the f.d.
= 'S' -- Display directly on the screen
- z0 -- (default I) the upper-half plane is mapped to the disk by z-->(z-z0)/(z-z0.conjugate())
EXAMPLES::
sage: G=MySubgroup(Gamma0(3))
sage: G._draw_funddom_d()
"""
# The fundamental domain consists of copies of the standard fundamental domain
pi=RR.pi()
from sage.plot.plot import (Graphics,line)
g=Graphics()
bdcirc=_circ_arc(0 ,2 *pi,0 ,1 ,1000 )
g=g+bdcirc
# Corners
x1=-RR(0.5) ; y1=RR(sqrt(3 )/2)
x2=RR(0.5) ; y2=RR(sqrt(3 )/2)
z_inf=1
l1 = _geodesic_between_two_points_d(x1,y1,x1,infinity)
l2 = _geodesic_between_two_points_d(x2,y2,x2,infinity)
c0 = _geodesic_between_two_points_d(x1,y1,x2,y2)
tri=c0+l1+l2
g=g+tri
for A in coset_reps:
[a,b,c,d]=A
if(a==1 and b==0 and c==0 and d==1 ):
continue
if(a<0 ):
a=-a; b=-b; c=-c; d=-1
if(c==0 ): # then this is easier
l1 = _geodesic_between_two_points_d(x1+b,y1,x1+b,infinity)
l2 = _geodesic_between_two_points_d(x2+b,y2,x2+b,infinity)
c0 = _geodesic_between_two_points_d(x1+b,y1,x2+b,y2)
# c0=line(L0); l1=line(L1); l2=line(L2); l3=line(L3)
tri=c0+l1+l2
g=g+tri
else:
den=(c*x1+d)**2 +c**2 *y1**2
x1_t=(a*c*(x1**2 +y1**2 )+(a*d+b*c)*x1+b*d)/den
y1_t=y1/den
den=(c*x2+d)**2 +c**2 *y2**2
x2_t=(a*c*(x2**2 +y2**2 )+(a*d+b*c)*x2+b*d)/den
y2_t=y2/den
inf_t=a/c
c0=_geodesic_between_two_points_d(x1_t,y1_t,x2_t,y2_t)
c1=_geodesic_between_two_points_d(x1_t,y1_t,inf_t,0.0 )
c2=_geodesic_between_two_points_d(x2_t,y2_t,inf_t,0.0 )
tri=c0+c1+c2
g=g+tri
g.xmax(1 )
g.ymax(1 )
g.xmin(-1 )
g.ymin(-1 )
g.set_aspect_ratio(1 )
return g
def signature_function_of_integral_matrix(V, prec=53):
"""
Computes the signature function sigma of V via numerical methods.
Returns two lists, the first representing a partition of [0, 1]:
x_0 = 0 < x_1 < x_2 < ... < x_n = 1
and the second list consisting of the values [v_0, ... , v_(n-1)]
of sigma on the interval (x_i, x_(i+1)). Currently, the value of
sigma *at* x_i is not computed.
"""
poly = alexander_poly_from_seifert(V)
RR = RealField(prec)
CC = ComplexField(prec)
pi = RR.pi()
I = CC.gen()
partition = [RR(0)] + [a for a, e in roots_on_unit_circle(poly, prec)
] + [RR(1)]
n = len(partition) - 1
values = []
for i in range(n):
omega = exp((2 * pi * I) * (partition[i] + partition[i + 1]) / 2)
A = (1 - omega) * V + (1 - omega.conjugate()) * V.transpose()
values.append(signature_via_numpy(A))
assert list(reversed(values)) == values
return partition, values
def _draw_funddom(coset_reps,format="S"):
r""" Draw a fundamental domain for G.
INPUT:
- ``format`` -- (default 'Disp') How to present the f.d.
- ``S`` -- Display directly on the screen
EXAMPLES::
sage: G=MySubgroup(Gamma0(3))
sage: G._draw_funddom()
"""
pi=RR.pi()
pi_3 = pi / RR(3.0)
from sage.plot.plot import (Graphics,line)
from sage.functions.trig import (cos,sin)
g=Graphics()
x1=RR(-0.5) ; y1=RR(sqrt(3 )/2 )
x2=RR(0.5) ; y2=RR(sqrt(3 )/2 )
xmax=RR(20.0)
l1 = line([[x1,y1],[x1,xmax]])
l2 = line([[x2,y2],[x2,xmax]])
l3 = line([[x2,xmax],[x1,xmax]]) # This is added to make a closed contour
c0=_circ_arc(RR(pi/3.0) ,RR(2.0*pi)/RR(3.0) ,0 ,1 ,100 )
tri=c0+l1+l3+l2
g=g+tri
for A in coset_reps:
[a,b,c,d]=A
if(a==1 and b==0 and c==0 and d==1 ):
continue
if(a<0 ):
a=RR(-a); b=RR(-b); c=RR(-c); d=RR(-d)
else:
a=RR(a); b=RR(b); c=RR(c); d=RR(d)
if(c==0 ): # then this is easier
L0 = [[cos(pi_3*RR(i/100.0))+b,sin(pi_3*RR(i/100.0))] for i in range(100 ,201 )]
L1 = [[x1+b,y1],[x1+b,xmax]]
L2 = [[x2+b,y2],[x2+b,xmax]]
L3 = [[x2+b,xmax],[x1+b,xmax]]
c0=line(L0); l1=line(L1); l2=line(L2); l3=line(L3)
tri=c0+l1+l3+l2
g=g+tri
else:
den=(c*x1+d)**2 +c**2 *y1**2
x1_t=(a*c*(x1**2 +y1**2 )+(a*d+b*c)*x1+b*d)/den
y1_t=y1/den
den=(c*x2+d)**2 +c**2 *y2**2
x2_t=(a*c*(x2**2 +y2**2 )+(a*d+b*c)*x2+b*d)/den
y2_t=y2/den
inf_t=a/c
c0=_geodesic_between_two_points(x1_t,y1_t,x2_t,y2_t)
c1=_geodesic_between_two_points(x1_t,y1_t,inf_t,0. )
c2=_geodesic_between_two_points(x2_t,y2_t,inf_t,0.0)
tri=c0+c1+c2
g=g+tri
return g
def _geodesic_between_two_points(x1, y1, x2, y2):
r""" Geodesic path between two points hyperbolic upper half-plane
INPUTS:
- ''(x1,y1)'' -- starting point (0<y1<=infinity)
- ''(x2,y2)'' -- ending point (0<y2<=infinity)
- ''z0'' -- (default I) the point in the upper corresponding
to the point 0 in the disc. I.e. the transform is
w -> (z-I)/(z+I)
OUTPUT:
- ''ca'' -- a polygonal approximation of a circular arc centered
at c and radius r, starting at t0 and ending at t1
EXAMPLES::
sage: l=_geodesic_between_two_points(0.1,0.2,0.0,0.5)
"""
pi = RR.pi()
from sage.plot.plot import line
from sage.functions.trig import arcsin
# logging.debug("z1=%s,%s" % (x1,y1))
# logging.debug("z2=%s,%s" % (x2,y2))
if abs(x1 - x2) < 1e-10:
# The line segment [x=x1, y0<= y <= y1]
return line([[x1, y1], [x2, y2]]) # [0,0,x0,infinity]
c = RR(y1 ** 2 - y2 ** 2 + x1 ** 2 - x2 ** 2) / RR(2 * (x1 - x2))
r = RR(sqrt(y1 ** 2 + (x1 - c) ** 2))
r1 = RR(y1 / r)
r2 = RR(y2 / r)
if abs(r1 - 1) < 1e-12:
r1 = RR(1.0)
elif abs(r2 + 1) < 1e-12:
r2 = -RR(1.0)
if abs(r2 - 1) < 1e-12:
r2 = RR(1.0)
elif abs(r2 + 1) < 1e-12:
r2 = -RR(1.0)
if x1 >= c:
t1 = RR(arcsin(r1))
else:
t1 = RR(pi) - RR(arcsin(r1))
if x2 >= c:
t2 = RR(arcsin(r2))
else:
t2 = RR(pi) - arcsin(r2)
# tmid = (t1 + t2) * RR(0.5)
# a0 = min(t1, t2)
# a1 = max(t1, t2)
# logging.debug("c,r=%s,%s" % (c,r))
# logging.debug("t1,t2=%s,%s"%(t1,t2))
return _circ_arc(t1, t2, c, r)
def draw_transformed_triangle_H(A, xmax=20):
r"""
Draw the modular triangle translated by A=[a,b,c,d]
"""
#print "A=",A,type(A)
pi = RR.pi()
pi_3 = pi / RR(3.0)
from sage.plot.plot import (Graphics, line)
from sage.functions.trig import (cos, sin)
x1 = RR(-0.5)
y1 = RR(sqrt(3) / 2)
x2 = RR(0.5)
y2 = RR(sqrt(3) / 2)
a, b, c, d = A #[0,0]; b=A[0,1]; c=A[1,0]; d=A[1,1]
if a < 0:
a = RR(-a)
b = RR(-b)
c = RR(-c)
d = RR(-d)
else:
a = RR(a)
b = RR(b)
c = RR(c)
d = RR(d)
if c == 0: # then this is easier
if a * d <> 0:
a = a / d
b = b / d
L0 = [[
a * cos(pi_3 * RR(i / 100.0)) + b, a * sin(pi_3 * RR(i / 100.0))
] for i in range(100, 201)]
L1 = [[a * x1 + b, a * y1], [a * x1 + b, xmax]]
L2 = [[a * x2 + b, a * y2], [a * x2 + b, xmax]]
L3 = [[a * x2 + b, xmax], [a * x1 + b, xmax]]
c0 = line(L0)
l1 = line(L1)
l2 = line(L2)
l3 = line(L3)
tri = c0 + l1 + l3 + l2
else:
den = (c * x1 + d)**2 + c**2 * y1**2
x1_t = (a * c * (x1**2 + y1**2) + (a * d + b * c) * x1 + b * d) / den
y1_t = y1 / den
den = (c * x2 + d)**2 + c**2 * y2**2
x2_t = (a * c * (x2**2 + y2**2) + (a * d + b * c) * x2 + b * d) / den
y2_t = y2 / den
inf_t = a / c
#print "A=",A
#print "arg1=",x1_t,y1_t,x2_t,y2_t
c0 = _geodesic_between_two_points(x1_t, y1_t, x2_t, y2_t)
#print "arg1=",x1_t,y1_t,inf_t
c1 = _geodesic_between_two_points(x1_t, y1_t, inf_t, 0.)
#print "arg1=",x2_t,y2_t,inf_t
c2 = _geodesic_between_two_points(x2_t, y2_t, inf_t, 0.0)
tri = c0 + c1 + c2
return tri
def _geodesic_between_two_points(x1, y1, x2, y2):
r""" Geodesic path between two points hyperbolic upper half-plane
INPUTS:
- ''(x1,y1)'' -- starting point (0<y1<=infinity)
- ''(x2,y2)'' -- ending point (0<y2<=infinity)
- ''z0'' -- (default I) the point in the upper corresponding
to the point 0 in the disc. I.e. the transform is
w -> (z-I)/(z+I)
OUTPUT:
- ''ca'' -- a polygonal approximation of a circular arc centered
at c and radius r, starting at t0 and ending at t1
EXAMPLES::
sage: l=_geodesic_between_two_points(0.1,0.2,0.0,0.5)
"""
pi = RR.pi()
from sage.plot.plot import line
from sage.functions.trig import arcsin
#print "z1=",x1,y1
#print "z2=",x2,y2
if (abs(x1 - x2) < 1E-10):
# The line segment [x=x1, y0<= y <= y1]
return line([[x1, y1], [x2, y2]]) #[0,0,x0,infinity]
c = RR(y1**2 - y2**2 + x1**2 - x2**2) / RR(2 * (x1 - x2))
r = RR(sqrt(y1**2 + (x1 - c)**2))
r1 = RR(y1 / r)
r2 = RR(y2 / r)
if (abs(r1 - 1) < 1E-12):
r1 = RR(1.0)
elif (abs(r2 + 1) < 1E-12):
r2 = -RR(1.0)
if (abs(r2 - 1) < 1E-12):
r2 = RR(1.0)
elif (abs(r2 + 1) < 1E-12):
r2 = -RR(1.0)
if (x1 >= c):
t1 = RR(arcsin(r1))
else:
t1 = RR(pi) - RR(arcsin(r1))
if (x2 >= c):
t2 = RR(arcsin(r2))
else:
t2 = RR(pi) - arcsin(r2)
tmid = (t1 + t2) * RR(0.5)
a0 = min(t1, t2)
a1 = max(t1, t2)
#print "c,r=",c,r
#print "t1,t2=",t1,t2
return _circ_arc(t1, t2, c, r)
def _circ_arc(t0, t1, c, r, num_pts=500):
r""" Circular arc
INPUTS:
- ''t0'' -- starting parameter
- ''t1'' -- ending parameter
- ''c'' -- center point of the circle
- ''r'' -- radius of circle
- ''num_pts'' -- (default 100) number of points on polygon
OUTPUT:
- ''ca'' -- a polygonal approximation of a circular arc centered
at c and radius r, starting at t0 and ending at t1
EXAMPLES::
sage: ca=_circ_arc(0.1,0.2,0.0,1.0,100)
"""
from sage.plot.plot import parametric_plot
from sage.functions.trig import cos, sin
from sage.all import var
t00 = t0
t11 = t1
## To make sure the line is correct we reduce all arguments to the same branch,
## e.g. [0,2pi]
pi = RR.pi()
while t00 < 0.0:
t00 = t00 + RR(2.0 * pi)
while t11 < 0:
t11 = t11 + RR(2.0 * pi)
while t00 > 2 * pi:
t00 = t00 - RR(2.0 * pi)
while t11 > 2 * pi:
t11 = t11 - RR(2.0 * pi)
xc = CC(c).real()
yc = CC(c).imag()
# L0 =
# [[RR(r*cos(t00+i*(t11-t00)/num_pts))+xc,RR(r*sin(t00+i*(t11-t00)/num_pts))+yc]
# for i in range(0 ,num_pts)]
t = var("t")
if t11 > t00:
ca = parametric_plot((r * cos(t) + xc, r * sin(t) + yc), (t, t00, t11))
else:
ca = parametric_plot((r * cos(t) + xc, r * sin(t) + yc), (t, t11, t00))
return ca
def _circ_arc(t0, t1, c, r, num_pts=5000):
r""" Circular arc
INPUTS:
- ''t0'' -- starting parameter
- ''t1'' -- ending parameter
- ''c'' -- center point of the circle
- ''r'' -- radius of circle
- ''num_pts'' -- (default 100) number of points on polygon
OUTPUT:
- ''ca'' -- a polygonal approximation of a circular arc centered
at c and radius r, starting at t0 and ending at t1
EXAMPLES::
sage: ca=_circ_arc(0.1,0.2,0.0,1.0,100)
"""
from sage.plot.plot import line, parametric_plot
from sage.functions.trig import (cos, sin)
from sage.all import var
t00 = t0
t11 = t1
## To make sure the line is correct we reduce all arguments to the same branch,
## e.g. [0,2pi]
pi = RR.pi()
while (t00 < 0.0):
t00 = t00 + RR(2.0 * pi)
while (t11 < 0):
t11 = t11 + RR(2.0 * pi)
while (t00 > 2 * pi):
t00 = t00 - RR(2.0 * pi)
while (t11 > 2 * pi):
t11 = t11 - RR(2.0 * pi)
xc = CC(c).real()
yc = CC(c).imag()
num_pts = 3
t = var('t')
if t11 > t00:
ca = parametric_plot((r * cos(t) + xc, r * sin(t) + yc), (t, t00, t11))
else:
ca = parametric_plot((r * cos(t) + xc, r * sin(t) + yc), (t, t11, t00))
#L0 = [[RR(r*cos(t00+i*(t11-t00)/num_pts))+xc,RR(r*sin(t00+i*(t11-t00)/num_pts))+yc] for i in range(0 ,num_pts)]
#ca=line(L0)
return ca
def draw_transformed_triangle_H(A,xmax=20):
r"""
Draw the modular triangle translated by A=[a,b,c,d]
"""
#print "A=",A,type(A)
pi=RR.pi()
pi_3 = pi / RR(3.0)
from sage.plot.plot import (Graphics,line)
from sage.functions.trig import (cos,sin)
x1=RR(-0.5) ; y1=RR(sqrt(3 )/2 )
x2=RR(0.5) ; y2=RR(sqrt(3 )/2 )
a,b,c,d = A #[0,0]; b=A[0,1]; c=A[1,0]; d=A[1,1]
if a<0:
a=RR(-a); b=RR(-b); c=RR(-c); d=RR(-d)
else:
a=RR(a); b=RR(b); c=RR(c); d=RR(d)
if c==0: # then this is easier
if a*d<>0:
a=a/d; b=b/d;
L0 = [[a*cos(pi_3*RR(i/100.0))+b,a*sin(pi_3*RR(i/100.0))] for i in range(100 ,201 )]
L1 = [[a*x1+b,a*y1],[a*x1+b,xmax]]
L2 = [[a*x2+b,a*y2],[a*x2+b,xmax]]
L3 = [[a*x2+b,xmax],[a*x1+b,xmax]]
c0=line(L0); l1=line(L1); l2=line(L2); l3=line(L3)
tri=c0+l1+l3+l2
else:
den=(c*x1+d)**2 +c**2 *y1**2
x1_t=(a*c*(x1**2 +y1**2 )+(a*d+b*c)*x1+b*d)/den
y1_t=y1/den
den=(c*x2+d)**2 +c**2 *y2**2
x2_t=(a*c*(x2**2 +y2**2 )+(a*d+b*c)*x2+b*d)/den
y2_t=y2/den
inf_t=a/c
#print "A=",A
#print "arg1=",x1_t,y1_t,x2_t,y2_t
c0=_geodesic_between_two_points(x1_t,y1_t,x2_t,y2_t)
#print "arg1=",x1_t,y1_t,inf_t
c1=_geodesic_between_two_points(x1_t,y1_t,inf_t,0. )
#print "arg1=",x2_t,y2_t,inf_t
c2=_geodesic_between_two_points(x2_t,y2_t,inf_t,0.0)
tri=c0+c1+c2
return tri
def draw_funddom(coset_reps, format="S"):
r""" Draw a fundamental domain for G.
INPUT:
- ``format`` -- (default 'Disp') How to present the f.d.
- ``S`` -- Display directly on the screen
EXAMPLES::
sage: G=MySubgroup(Gamma0(3))
sage: G._draw_funddom()
"""
pi = RR.pi()
pi_3 = pi / RR(3.0)
from sage.plot.plot import (Graphics, line)
from sage.functions.trig import (cos, sin)
g = Graphics()
x1 = RR(-0.5)
y1 = RR(sqrt(3) / 2)
x2 = RR(0.5)
y2 = RR(sqrt(3) / 2)
xmax = RR(20.0)
l1 = line([[x1, y1], [x1, xmax]])
l2 = line([[x2, y2], [x2, xmax]])
l3 = line([[x2, xmax], [x1,
xmax]]) # This is added to make a closed contour
c0 = _circ_arc(RR(pi / 3.0), RR(2.0 * pi) / RR(3.0), 0, 1, 100)
tri = c0 + l1 + l3 + l2
g = g + tri
for A in coset_reps:
if list(A) == [1, 0, 0, 1]:
continue
tri = draw_transformed_triangle_H(A, xmax=xmax)
g = g + tri
return g
def roots_on_unit_circle(poly, prec=53):
"""
For a palindromic polynomial p(x) of even degree, return all the
roots on the unit circle in the form
(argument/(2 pi), multiplicity)
"""
assert is_palindromic(poly) and poly.degree() % 2 == 0
assert poly.parent().is_exact()
# Deal with these corner cases at some point if needed.
assert poly(1) != 0 and poly(-1) != 0
ans = []
RR = RealField(prec)
pi = RR.pi()
g = compact_form(poly)
for f, e in g.factor():
roots = [r for r in f.roots(RR, False) if -2 < r < 2]
args = [arccos(r / 2) / (2 * pi) for r in roots]
args += [1 - a for a in args]
ans += [(arg, e) for arg in args]
return sorted(ans)
def draw_funddom(coset_reps,format="S"):
r""" Draw a fundamental domain for G.
INPUT:
- ``format`` -- (default 'Disp') How to present the f.d.
- ``S`` -- Display directly on the screen
EXAMPLES::
sage: G=MySubgroup(Gamma0(3))
sage: G._draw_funddom()
"""
pi=RR.pi()
pi_3 = pi / RR(3.0)
from sage.plot.plot import (Graphics,line)
from sage.functions.trig import (cos,sin)
g=Graphics()
x1=RR(-0.5) ; y1=RR(sqrt(3 )/2 )
x2=RR(0.5) ; y2=RR(sqrt(3 )/2 )
xmax=RR(20.0)
l1 = line([[x1,y1],[x1,xmax]])
l2 = line([[x2,y2],[x2,xmax]])
l3 = line([[x2,xmax],[x1,xmax]]) # This is added to make a closed contour
c0=_circ_arc(RR(pi/3.0) ,RR(2.0*pi)/RR(3.0) ,0 ,1 ,100 )
tri=c0+l1+l3+l2
g=g+tri
for A in coset_reps:
if list(A)==[1,0,0,1]:
continue
tri=draw_transformed_triangle_H(A,xmax=xmax)
g=g+tri
return g
def kbarbar(weight):
# The weight part of the analytic conductor
return psi(RR(weight)/2).exp() / (2*RR.pi())
# This file is used to determine positions for discs in the image associated to a group
from collections import defaultdict, Counter
from colorsys import hsv_to_rgb
from sage.all import RR, ZZ, QQ
from sage.misc.lazy_attribute import lazy_attribute
from sage.misc.cachefunc import cached_function
from itertools import combinations
import heapq
eps = RR(0.00001) # tolerance
pi = RR.pi()
def distxy(P, Q):
return ((P[0] - Q[0])**2 + (P[1] - Q[1])**2).sqrt()
def distrt(P, Q):
return (P[0]**2 + Q[0]**2 - 2 * P[0] * Q[0] * (P[1] - Q[1]).cos()).sqrt()
def distray(P, Q):
# Distance to the ray starting at Q and going out
R0, theta = Q
if P[0] * (P[1] - theta).cos() >= R0:
# The minimum distance to the theta line occurs on the ray, so we can just use the right triangle
return P[0] * (P[1] - theta).sin().abs()
else:
# The minimum distance occurs at the inner radius
return distrt(P, Q)
def _geodesic_between_two_points_d(x1,y1,x2,y2,z0=I):
r""" Geodesic path between two points represented in the unit disc
by the map w = (z-I)/(z+I)
INPUTS:
- ''(x1,y1)'' -- starting point (0<y1<=infinity)
- ''(x2,y2)'' -- ending point (0<y2<=infinity)
- ''z0'' -- (default I) the point in the upper corresponding
to the point 0 in the disc. I.e. the transform is
w -> (z-I)/(z+I)
OUTPUT:
- ''ca'' -- a polygonal approximation of a circular arc centered
at c and radius r, starting at t0 and ending at t1
EXAMPLES::
sage: l=_geodesic_between_two_points_d(0.1,0.2,0.0,0.5)
"""
pi=RR.pi()
from sage.plot.plot import line
from sage.functions.trig import (cos,sin)
# First compute the points
if(y1<0 or y2<0 ):
raise ValueError,"Need points in the upper half-plane! Got y1=%s, y2=%s" %(y1,y2)
if(y1==infinity):
P1=CC(1 )
else:
P1=CC((x1+I*y1-z0)/(x1+I*y1-z0.conjugate()))
if(y2==infinity):
P2=CC(1 )
else:
P2=CC((x2+I*y2-z0)/(x2+I*y2-z0.conjugate()))
# First find the endpoints of the completed geodesic in D
if(x1==x2):
a=CC((x1-z0)/(x1-z0.conjugate()))
b=CC(1 )
else:
c=RR(y1**2 -y2**2 +x1**2 -x2**2 )/RR(2 *(x1-x2))
r=RR(sqrt(y1**2 +(x1-c)**2 ))
a=c-r
b=c+r
a=CC((a-z0)/(a-z0.conjugate()))
b=CC((b-z0)/(b-z0.conjugate()))
if( abs(a+b) < 1E-10 ): # On a diagonal
return line([[P1.real(),P1.imag()],[P2.real(),P2.imag()]])
th_a=a.argument()
th_b=b.argument()
# Compute the center of the circle in the disc model
if( min(abs(b-1 ),abs(b+1 ))< 1E-10 and min(abs(a-1 ),abs(a+1 ))>1E-10 ):
c=b+I*(1 -b*cos(th_a))/sin(th_a)
elif( min(abs(b-1 ),abs(b+1 ))> 1E-10 and min(abs(a-1 ),abs(a+1 ))<1E-10 ):
c=a+I*(1 -a*cos(th_b))/RR(sin(th_b))
else:
cx=(sin(th_b)-sin(th_a))/sin(th_b-th_a)
c=cx+I*(1 -cx*cos(th_b))/RR(sin(th_b))
# First find the endpoints of the completed geodesic
r=abs(c-a)
t1=CC(P1-c).argument()
t2=CC(P2-c).argument()
#print "t1,t2=",t1,t2
return _circ_arc(t1,t2,c,r)
def _geodesic_between_two_points_d(x1, y1, x2, y2, z0=I):
r""" Geodesic path between two points represented in the unit disc
by the map w = (z-I)/(z+I)
INPUTS:
- ''(x1,y1)'' -- starting point (0<y1<=infinity)
- ''(x2,y2)'' -- ending point (0<y2<=infinity)
- ''z0'' -- (default I) the point in the upper corresponding
to the point 0 in the disc. I.e. the transform is
w -> (z-I)/(z+I)
OUTPUT:
- ''ca'' -- a polygonal approximation of a circular arc centered
at c and radius r, starting at t0 and ending at t1
EXAMPLES::
sage: l=_geodesic_between_two_points_d(0.1,0.2,0.0,0.5)
"""
pi = RR.pi()
from sage.plot.plot import line
from sage.functions.trig import (cos, sin)
# First compute the points
if (y1 < 0 or y2 < 0):
raise ValueError, "Need points in the upper half-plane! Got y1=%s, y2=%s" % (
y1, y2)
if (y1 == infinity):
P1 = CC(1)
else:
P1 = CC((x1 + I * y1 - z0) / (x1 + I * y1 - z0.conjugate()))
if (y2 == infinity):
P2 = CC(1)
else:
P2 = CC((x2 + I * y2 - z0) / (x2 + I * y2 - z0.conjugate()))
# First find the endpoints of the completed geodesic in D
if (x1 == x2):
a = CC((x1 - z0) / (x1 - z0.conjugate()))
b = CC(1)
else:
c = RR(y1**2 - y2**2 + x1**2 - x2**2) / RR(2 * (x1 - x2))
r = RR(sqrt(y1**2 + (x1 - c)**2))
a = c - r
b = c + r
a = CC((a - z0) / (a - z0.conjugate()))
b = CC((b - z0) / (b - z0.conjugate()))
if (abs(a + b) < 1E-10): # On a diagonal
return line([[P1.real(), P1.imag()], [P2.real(), P2.imag()]])
th_a = a.argument()
th_b = b.argument()
# Compute the center of the circle in the disc model
if (min(abs(b - 1), abs(b + 1)) < 1E-10
and min(abs(a - 1), abs(a + 1)) > 1E-10):
c = b + I * (1 - b * cos(th_a)) / sin(th_a)
elif (min(abs(b - 1), abs(b + 1)) > 1E-10
and min(abs(a - 1), abs(a + 1)) < 1E-10):
c = a + I * (1 - a * cos(th_b)) / RR(sin(th_b))
else:
cx = (sin(th_b) - sin(th_a)) / sin(th_b - th_a)
c = cx + I * (1 - cx * cos(th_b)) / RR(sin(th_b))
# First find the endpoints of the completed geodesic
r = abs(c - a)
t1 = CC(P1 - c).argument()
t2 = CC(P2 - c).argument()
#print "t1,t2=",t1,t2
return _circ_arc(t1, t2, c, r)
def draw_funddom_d(coset_reps, format="MP", z0=I, verbose=0):
r""" Draw a fundamental domain for self in the circle model
INPUT:
- ''format'' -- (default 'Disp') How to present the f.d.
= 'S' -- Display directly on the screen
- z0 -- (default I) the upper-half plane is mapped to the disk by z-->(z-z0)/(z-z0.conjugate())
EXAMPLES::
sage: G=MySubgroup(Gamma0(3))
sage: G._draw_funddom_d()
"""
# The fundamental domain consists of copies of the standard fundamental domain
pi = RR.pi()
from sage.plot.plot import (Graphics, line)
g = Graphics()
bdcirc = _circ_arc(0, 2 * pi, 0, 1, 1000)
g = g + bdcirc
# Corners
x1 = -RR(0.5)
y1 = RR(sqrt(3) / 2)
x2 = RR(0.5)
y2 = RR(sqrt(3) / 2)
z_inf = 1
l1 = _geodesic_between_two_points_d(x1, y1, x1, infinity)
l2 = _geodesic_between_two_points_d(x2, y2, x2, infinity)
c0 = _geodesic_between_two_points_d(x1, y1, x2, y2)
tri = c0 + l1 + l2
g = g + tri
for A in coset_reps:
a, b, c, d = A
if a == 1 and b == 0 and c == 0 and d == 1:
continue
if a < 0:
a = -a
b = -b
c = -c
d = -d
if verbose > 0:
print "a,b,c,d=", a, b, c, d
if c == 0: # then this is easier
l1 = _geodesic_between_two_points_d(x1 + b, y1, x1 + b, infinity)
l2 = _geodesic_between_two_points_d(x2 + b, y2, x2 + b, infinity)
c0 = _geodesic_between_two_points_d(x1 + b, y1, x2 + b, y2)
# c0=line(L0); l1=line(L1); l2=line(L2); l3=line(L3)
tri = c0 + l1 + l2
g = g + tri
else:
den = (c * x1 + d)**2 + c**2 * y1**2
x1_t = (a * c * (x1**2 + y1**2) +
(a * d + b * c) * x1 + b * d) / den
y1_t = y1 / den
den = (c * x2 + d)**2 + c**2 * y2**2
x2_t = (a * c * (x2**2 + y2**2) +
(a * d + b * c) * x2 + b * d) / den
y2_t = y2 / den
inf_t = a / c
if verbose > 0:
print "x1_t=", x1_t
print "y1_t=", y1_t
print "x2_t=", x2_t
print "y2_t=", y2_t
print "inf_t=", inf_t
c0 = _geodesic_between_two_points_d(x1_t, y1_t, x2_t, y2_t)
c1 = _geodesic_between_two_points_d(x1_t, y1_t, inf_t, 1.0)
c2 = _geodesic_between_two_points_d(x2_t, y2_t, inf_t, 1.0)
tri = c0 + c1 + c2
g = g + tri
g.xmax(1)
g.ymax(1)
g.xmin(-1)
g.ymin(-1)
g.set_aspect_ratio(1)
return g
def _draw_funddom(coset_reps, format="S"):
r""" Draw a fundamental domain for G.
INPUT:
- ``format`` -- (default 'Disp') How to present the f.d.
- ``S`` -- Display directly on the screen
EXAMPLES::
sage: G=MySubgroup(Gamma0(3))
sage: G._draw_funddom()
"""
pi = RR.pi()
pi_3 = pi / RR(3.0)
from sage.plot.plot import (Graphics, line)
from sage.functions.trig import (cos, sin)
g = Graphics()
x1 = RR(-0.5)
y1 = RR(sqrt(3) / 2)
x2 = RR(0.5)
y2 = RR(sqrt(3) / 2)
xmax = RR(20.0)
l1 = line([[x1, y1], [x1, xmax]])
l2 = line([[x2, y2], [x2, xmax]])
l3 = line([[x2, xmax], [x1,
xmax]]) # This is added to make a closed contour
c0 = _circ_arc(RR(pi / 3.0), RR(2.0 * pi) / RR(3.0), 0, 1, 100)
tri = c0 + l1 + l3 + l2
g = g + tri
for A in coset_reps:
[a, b, c, d] = A
if (a == 1 and b == 0 and c == 0 and d == 1):
continue
if (a < 0):
a = RR(-a)
b = RR(-b)
c = RR(-c)
d = RR(-d)
else:
a = RR(a)
b = RR(b)
c = RR(c)
d = RR(d)
if (c == 0): # then this is easier
L0 = [[cos(pi_3 * RR(i / 100.0)) + b,
sin(pi_3 * RR(i / 100.0))] for i in range(100, 201)]
L1 = [[x1 + b, y1], [x1 + b, xmax]]
L2 = [[x2 + b, y2], [x2 + b, xmax]]
L3 = [[x2 + b, xmax], [x1 + b, xmax]]
c0 = line(L0)
l1 = line(L1)
l2 = line(L2)
l3 = line(L3)
tri = c0 + l1 + l3 + l2
g = g + tri
else:
den = (c * x1 + d)**2 + c**2 * y1**2
x1_t = (a * c * (x1**2 + y1**2) +
(a * d + b * c) * x1 + b * d) / den
y1_t = y1 / den
den = (c * x2 + d)**2 + c**2 * y2**2
x2_t = (a * c * (x2**2 + y2**2) +
(a * d + b * c) * x2 + b * d) / den
y2_t = y2 / den
inf_t = a / c
c0 = _geodesic_between_two_points(x1_t, y1_t, x2_t, y2_t)
c1 = _geodesic_between_two_points(x1_t, y1_t, inf_t, 0.)
c2 = _geodesic_between_two_points(x2_t, y2_t, inf_t, 0.0)
tri = c0 + c1 + c2
g = g + tri
return g
@cached_function
def DirGroup(m):
return DirichletGroup_conrey(m)
@cached_function
def primitivize(label):
m, n = [ZZ(a) for a in label.split(".")]
char = DirichletCharacter_conrey(DirGroup(m), n).primitive_character()
return "%d.%d" % (char.modulus(), char.number())
logpi = RR.pi().log()
log2pi = (2 * RR.pi()).log()
def log_L_inf(s, mu, nu):
return ((sum([psi((s + elt) / 2) for elt in mu]) -
len(mu) * logpi) / 2 +
sum([psi(s + elt) for elt in nu]) -
len(nu) * log2pi)
@cached_function
def conductor_an(GR, GC):
return (2*log_L_inf(1/2, GR, GC).real()).exp()
class read_lfunctions:
def __init__(self, in_headers, out_headers):