def test_hyperelliptic_regular_places(self):
R = QQ['x,y']
x, y = R.gens()
X = RiemannSurface(y**2 - (x + 1) * (x - 1) * (x - 2) * (x + 2))
omegas = differentials(X)
# the places above x=0 are regular
places = X(0)
for P in places:
a, b = P.x, P.y
for omega in omegas:
omega_P = omega.centered_at_place(P)
val1 = omega(a, b)
val2 = omega_P(CC(0))
self.assertLess(abs(val1 - val2), 1e-8)
# the places above x=oo are regular: P = (1/t, \pm 1/t**2 + O(1))
# (however, all places at infinity are treated as discriminant)
#
# in this particular example, omega[0] = 1/(2*y). At the places at
# infinity, these are equal to \mp 0.5, respectively. (the switch in
# sign comes from the derivative dxdt = -1/t**2)
places = X('oo')
for P in places:
sign = P.puiseux_series.ypart[-2]
for omega in omegas:
omega_P = omega.centered_at_place(P)
val1 = -sign * 0.5
val2 = omega_P(CC(0))
self.assertLess(abs(val1 - val2), 1e-8)
def analytically_continue(self, xi, yi, xip1):
r"""Analytically continue the y-fibre `yi` lying above `xi` to the y-fibre lying
above `xip1`.
We analytically continue by simply evaluating the ordered puiseux
series computed during initialization of the Riemann surface path.
Parameters
----------
xi : complex
The starting complex x-value.
yi: complex[:]
The starting complex y-fibre lying above `xi`.
xip1: complex
The target complex x-value.
Returns
-------
yi : complex[:]
The corresponding y-fibre lying above `xi`.
"""
# XXX HACK - need to coerce input to CC for puiseux series to evaluate
xi = CC(xi)
xip1 = CC(xip1)
# return the current fibre if the step size is too small
if numpy.abs(xip1 - xi) < 1e-15:
return yi
# simply evaluate the ordered puiseux series at xip1
alpha = CC(0) if self.target_point == infinity else CC(
self.target_point)
yip1 = [pj(xip1 - alpha) for pj in self.puiseux_series]
yip1 = numpy.array(yip1, dtype=complex)
return yip1
def value(self, z, embedding=0):
if self.prec == 0:
return 0
else:
q = exp(2 * CC.pi() * CC(0, 1) * z)
return sum(
self.coefficient_embedding(n, embedding) * q**n
for n in range(self.prec))
def __init__(self, A, B, C, options):
"""
Initialize HyperbolicTriangle:
Examples::
sage: from sage.plot.hyperbolic_triangle import HyperbolicTriangle
sage: print HyperbolicTriangle(0, 1/2, I, {})
Hyperbolic triangle (0.000000000000000, 0.500000000000000, 1.00000000000000*I)
"""
A, B, C = (CC(A), CC(B), CC(C))
self.path = []
sides = options.pop('sides', [1, 2, 3])
verbose = options.pop('verbose', 0)
sides.sort()
if sides == [1]:
if verbose > 0:
print "Drawing A - B!"
self._hyperbolic_arc(A, B, True)
elif sides == [2]:
if verbose > 0:
print "Drawing B - C!"
self._hyperbolic_arc(B, C, True)
elif sides == [3]:
if verbose > 0:
print "Drawing C - A!"
self._hyperbolic_arc(C, A, True)
elif sides == [1, 2]:
if verbose > 0:
print "Drawing A - B! & B - C!"
self._hyperbolic_arc(A, B, True)
self._hyperbolic_arc(B, C, False)
elif sides == [1, 3]:
if verbose > 0:
print "Drawing C - A! & A - B"
self._hyperbolic_arc(C, A, True)
self._hyperbolic_arc(A, B, False)
elif sides == [2, 3]:
if verbose > 0:
print "Drawing B - C! & C - A"
self._hyperbolic_arc(B, C, True)
self._hyperbolic_arc(C, A, False)
else:
self._hyperbolic_arc(A, B, True)
self._hyperbolic_arc(B, C, False)
self._hyperbolic_arc(C, A, False)
BezierPath.__init__(self, self.path, options)
self.A, self.B, self.C = (A, B, C)
self._pts = [A, B, C]
def check_ap2_slow(rec):
# Check a_{p^2} = a_p^2 - chi(p) for primes up to 31
ls = rec['lfunction_label'].split('.')
level, weight, chi = map(int, [ls[0], ls[1], ls[-2]])
char = DirichletGroup_conrey(level, CC)[chi]
Z = rec['an_normalized[0:1000]']
for p in prime_range(31+1):
if level % p != 0:
# a_{p^2} = a_p^2 - chi(p)
charval = CC(2*char.logvalue(int(p)) * CC.pi()*CC.gens()[0]).exp()
else:
charval = 0
if (CC(*Z[p**2 - 1]) - (CC(*Z[p-1])**2 - charval)).abs() > 1e-11:
return False
return True
def check_ap2_slow(rec):
# Check a_{p^2} = a_p^2 - chi(p) for primes up to 31
ls = rec['lfunction_label'].split('.')
level, weight, chi = map(int, [ls[0], ls[1], ls[-2]])
char = DirichletGroup_conrey(level, CC)[chi]
Z = rec['an_normalized[0:1000]']
for p in prime_range(31+1):
if level % p != 0:
# a_{p^2} = a_p^2 - chi(p)
charval = CC(2*char.logvalue(int(p)) * CC.pi()*CC.gens()[0]).exp()
else:
charval = 0
if (CC(*Z[p**2 - 1]) - (CC(*Z[p-1])**2 - charval)).abs() > 1e-11:
return False
return True
def test_elliptic_curve(a, b, alpha=7, angles=12):
from sage.all import cos, sin, pi
E = EllipticCurve(QQ, [a, b])
rationalpoints = E.point_search(10)
EQbar = EllipticCurve(QQbar, [a, b])
infx, infy, _ = rationalpoints[0]
inf = EQbar(infx, infy)
print inf
R = PolynomialRing(QQ, "w")
w = R.gen()
f = w**3 + a * w + b
iaj = InvertAJlocal(f, 256, method='gauss-legendre')
iaj.set_basepoints([(mpmath.mpf(infx), mpmath.mpf(infy))])
for eps in [
cos(theta * 2 * pi / angles) + I * sin(theta * 2 * pi / angles)
for theta in range(0, angles)
]:
px = infx + eps
py = iaj.sign * sqrt(-E.defining_polynomial().subs(x=px, y=0, z=1))
P = EQbar(px, py)
try:
(v, errorv) = iaj.to_J(px, infx)
qx = (alpha * P - (alpha - 1) * inf).xy()[0]
qxeps = CC(qx) + (mpmath.rand() + I * mpmath.rand()) / 1000
(t1, errort1) = iaj.solve(alpha * v, 100, iaj.error, [qxeps])
qxguess = t1[0, 0]
print mpmath.fabs(qxguess - qx) < 2**(-mpmath.mp.prec / 2)
except RuntimeWarning as detail:
print detail
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 parameterize(self, omega):
r"""Returns the differential omega parameterized on the path.
Given a differential math:`\omega = \omega(x,y)dx`, `parameterize`
returns the differential
.. math::
\omega_\gamma(s) = \omega(\gamma_x(s),\gamma_y(s)) \gamma_x'(s)
where :math:`s \in [0,1]` and :math:`\gamma_x,\gamma_y` and the x- and
y-components of the path `\gamma` using this analytic continuator.
Parameters
----------
omega : Differential
A differential (one-form) on the Riemann surface.
Returns
-------
omega_gamma : function
The differential parameterized on the curve for s in the interval
[0,1].
"""
# localize the differential at the discriminant place
P = self.target_place
omega_local = omega.localize(P)
omega_local = omega_local.laurent_polynomial().change_ring(CC)
# extract relevant information about the Puiseux series
p = P.puiseux_series
x0 = complex(self.gamma.x0)
y0 = complex(self.gamma.y0[0])
alpha = 0 if self.target_point == infinity else self.target_point
xcoefficient = complex(p.xcoefficient)
e = numpy.int(p.ramification_index)
# the parameter of the path s \in [0,1] does not necessarily match with
# the local coordinate t of the place. perform the appropriate scaling
# on the integral.
tprim = complex((x0 - alpha) / xcoefficient)**(1. / e)
unity = [numpy.exp(2.j * numpy.pi * k / abs(e)) for k in range(abs(e))]
tall = [unity[k] * tprim for k in range(abs(e))]
ytprim = numpy.array([p.eval_y(tk) for tk in tall],
dtype=numpy.complex)
k = numpy.argmin(numpy.abs(ytprim - y0))
tcoefficient = tall[k]
# XXX HACK - CC coercion
tcoefficient = CC(tcoefficient)
def omega_gamma(s):
s = CC(s)
dtds = -tcoefficient
val = omega_local(tcoefficient * (1 - s)) * dtds
return complex(val)
return numpy.vectorize(omega_gamma, otypes=[complex])
def signtocolour(sign):
"""
Assigns an rgb string colour to a complex number based on its argument.
"""
argument = cmath.phase(CC(str(sign)))
r = int(255.0 * (math.cos((1.0 * math.pi / 3.0) - (argument / 2.0))) ** 2)
g = int(255.0 * (math.cos((2.0 * math.pi / 3.0) - (argument / 2.0))) ** 2)
b = int(255.0 * (math.cos(argument / 2.0)) ** 2)
return("rgb(" + str(r) + "," + str(g) + "," + str(b) + ")")
def embedding_maker(self, rec, lang):
emb_list = []
embeddings = rec["embeddings"]
if lang == "magma":
for z in embeddings:
z_str = "ComplexField(15)!%s" % z
emb_list.append(z_str)
return "[%s]" % ", ".join(emb_list)
if lang == "sage":
return "%s" % [CC(z) for z in embeddings]
def parse_complex_number(z):
"""convert a string representing a complex number to another string looking like "(x,y)"
"""
from sage.all import CC
try:
# need to convert from unicode to orginary string type
x, y = CC(string2number(str(z)))
return "({},{})".format(x, y)
except (TypeError, SyntaxError):
print("Unable to parse {} as complex number".format(z))
return "(0,0)"
def is_number(a):
if isinstance(a, (int, float, long, complex,
sage.rings.all.CommutativeRingElement)):
return True
elif hasattr(a, 'parent'):
numgen = sage.rings.number_field.number_field.NumberField_generic
parent = a.parent()
return CC.has_coerce_map_from(parent) or \
isinstance(parent, numgen) or \
(hasattr(parent, "is_field") and hasattr(parent, "is_finite")
and parent.is_field() and parent.is_finite())
else:
return False
def test_f2_regular_places(self):
X = self.X2
omegas = differentials(X)
# the places above x=1 are regular
places = X(1)
for P in places:
a, b = P.x, P.y
for omega in omegas:
omega_P = omega.centered_at_place(P)
val1 = omega(a, b)
val2 = omega_P(CC(0))
self.assertLess(abs(val1 - val2), 1e-8)
def check_amn_slow(self, rec, verbose=False):
"""
Check that a_{pn} = a_p * a_n for p < 32 prime, n prime to p
"""
Z = [0] + [CC(*elt) for elt in rec['an_normalized']]
for pp in prime_range(len(Z)-1):
for k in range(1, (len(Z) - 1)//pp + 1):
if gcd(k, pp) == 1:
if (Z[pp*k] - Z[pp]*Z[k]).abs() > 1e-13:
if verbose:
print "amn failure", k, pp, Z[pp*k], Z[pp]*Z[k]
return False
return True
def __init__(self, riemann_surface, complex_path, y0, ncheckpoints=16):
# if the complex path leads to a discriminant point then get the exact
# representation of said discrimimant point
target_point = complex_path(1)
if target_point in [numpy.Infinity, infinity]:
target_point = infinity
elif abs(CC(target_point)) > 1e12:
target_point = infinity
else:
discriminant_point = riemann_surface.path_factory.closest_discriminant_point(
target_point)
if abs(CC(target_point - discriminant_point)) < 1e-12:
target_point = discriminant_point
else:
# if it's not discriminant then try to coerce to QQ or QQbar
try:
target_point = QQ(target_point)
except TypeError:
try:
target_point = QQbar(target_point)
except TypeError:
pass
# compute and store the ordered puiseux series needed to analytically
# continue as well as the target place for parameterization purposes
puiseux_series, target_place = ordered_puiseux_series(
riemann_surface, complex_path, y0, target_point)
self.puiseux_series = puiseux_series
self.target_point = target_point
self.target_place = target_place
# now that the machinery is set up we can instantiate the base object
RiemannSurfacePathPrimitive.__init__(self,
riemann_surface,
complex_path,
y0,
ncheckpoints=ncheckpoints)
def _path_to_infinite_place(self, P):
r"""Returns a path to a place at an infintiy of the surface.
A place at infinity is one where the `x`-projection of the place is the
point `x = \infty` of the complex Riemann sphere. An infinite place is
a type of discirminant place.
Parameters
----------
P : Place
The target infinite place.
Returns
-------
gamma : RiemannSurfacePath
A path from the base place to the place at infinity.
"""
# determine a place Q from where we can reach the target place P. first,
# pick an appropriate x-point over which a Q is chosen
x0 = self.base_point
if numpy.real(x0) < 0:
xa = 5 * x0 # move away from the origin
else:
xa = -5 # arbitrary choice away from the origin
# next, determine an appropriate y-part from where we can reach the
# place at infinity
p = P.puiseux_series
center, coefficient, ramification_index = p.xdata
ta = CC(xa / coefficient).nth_root(abs(ramification_index))
ta = ta if ramification_index > 0 else 1 / ta
p.extend_to_t(ta)
ya = complex(p.eval_y(ta))
# construct the place Q and compute the path going from P0 to Q
Q = self.riemann_surface(xa, ya)
gamma_P0_to_Q = self.path_to_place(Q)
# construct the path going from Q to P
xend = complex(gamma_P0_to_Q.get_x(1.0))
yend = array(gamma_P0_to_Q.get_y(1.0), dtype=complex)
gamma_x = ComplexRay(xend)
gamma_Q_to_P = RiemannSurfacePathPuiseux(self.riemann_surface, gamma_x,
yend)
gamma = gamma_P0_to_Q + gamma_Q_to_P
return gamma
def insert_dirichlet_L_functions(start, end):
print "Putting Dirichlet L-functions into database."
start = max(3, start)
for q in range(start, end):
print "Working on modulus", q
sys.stdout.flush()
G = DirichletGroup(q)
for n in range(len(G)):
chi = G[n]
if chi.is_primitive():
L = lc.Lfunction_from_character(chi)
z = L.find_zeros_via_N(1)[0]
Lfunction_data = {}
Lfunction_data['first_zero'] = float(z)
Lfunction_data[
'description'] = "Dirichlet L-function for character number " + str(
n) + " modulo " + str(q)
Lfunction_data['degree'] = 1
Lfunction_data['signature'] = (1, 0)
if chi.is_odd():
Lfunction_data['mu'] = [
(1.0, 0.0),
]
else:
Lfunction_data['mu'] = [
(0.0, 0.0),
]
Lfunction_data['level'] = q
coeffs = []
for k in range(0, 10):
coeffs.append(CC(chi(k)))
Lfunction_data['coeffs'] = [(float(x.real()), float(x.imag()))
for x in coeffs]
Lfunction_data['special'] = {
'type': 'dirichlet',
'modulus': q,
'number': n
}
Lfunctions.insert(Lfunction_data)
def specialValueTriple(L, s, sLatex_analytic, sLatex_arithmetic):
''' Returns [L_arithmetic, L_analytic, L_val]
Currently only used for genus 2 curves
and Dirichlet characters.
Eventually want to use for all L-functions.
'''
number_of_decimals = 10
val = None
if hasattr(L, "lfunc_data"):
s_alg = s + p2sage(L.lfunc_data['analytic_normalization'])
if 'values' in L.lfunc_data.keys():
for x in p2sage(L.lfunc_data['values']):
# the numbers here are always half integers
# so this comparison is exact
if x[0] == s_alg:
val = x[1]
break
if val is None:
if L.fromDB:
val = "not computed"
else:
val = L.sageLfunction.value(s)
# We must test for NaN first, since it would show as zero otherwise
# Try "RR(NaN) < float(1e-10)" in sage -- GT
lfunction_value_tex_arithmetic = L.texname_arithmetic.replace(
's)', sLatex_arithmetic + ')')
lfunction_value_tex_analytic = L.texname.replace('(s',
'(' + sLatex_analytic)
try:
if CC(val).real().is_NaN():
Lval = "\\infty"
elif val.abs() < 1e-10:
Lval = "0"
else:
Lval = latex(
round(val.real(), number_of_decimals) +
round(val.imag(), number_of_decimals) * I)
except (TypeError, NameError):
Lval = val # if val is text
return [lfunction_value_tex_analytic, lfunction_value_tex_arithmetic, Lval]
def insert_EC_L_functions(start=1, end=100):
curves = C.ellcurves.curves
for N in range(start, end):
print("Processing conductor", N)
sys.stdout.flush()
query = curves.find({'conductor': N, 'number': 1})
for curve in query:
E = EllipticCurve([int(x) for x in curve['ainvs']])
L = lc.Lfunction_from_elliptic_curve(E)
first_zeros = L.find_zeros_via_N(curve['rank'] + 1)
if len(first_zeros) > 1:
if not first_zeros[-2] == 0:
print("problem")
z = float(first_zeros[-1])
Lfunction_data = {}
Lfunction_data['first_zero'] = z
Lfunction_data[
'description'] = 'Elliptic curve L-function for curve ' + str(
curve['label'][:-1])
Lfunction_data['degree'] = 2
Lfunction_data['signature'] = [0, 1]
Lfunction_data['eta'] = [
(1.0, 0),
]
Lfunction_data['level'] = N
Lfunction_data['special'] = {
'type': 'elliptic',
'label': curve['label'][:-1]
}
coeffs = []
for k in range(1, 11):
coeffs.append(CC(E.an(k) / sqrt(k)))
Lfunction_data['coeffs'] = [(float(x.real()), float(x.imag()))
for x in coeffs]
Lfunctions.insert(Lfunction_data)
def check_ap2_slow(self, rec, verbose=False):
"""
Check a_{p^2} = a_p^2 - chi(p) for primes up to 31
"""
ls = rec['label'].split('.')
level, weight, chi = map(int, [ls[0], ls[1], ls[-2]])
char = DirichletGroup_conrey(level, CC)[chi]
Z = rec['an_normalized']
for p in prime_range(31+1):
if level % p != 0:
# a_{p^2} = a_p^2 - chi(p)
charval = CC(2*char.logvalue(int(p)) * CC.pi()*CC.gens()[0]).exp()
else:
charval = 0
if (CC(*Z[p**2 - 1]) - (CC(*Z[p-1])**2 - charval)).abs() > 1e-13:
if verbose:
print "ap2 failure", p, CC(*Z[p**2 - 1]), CC(*Z[p-1])**2 - charval
return False
return True
def specialValueString(L, s, sLatex, normalization="analytic"):
''' Returns the LaTex to dislpay for L(s)
Will eventually be replaced by specialValueTriple.
'''
number_of_decimals = 10
val = None
if hasattr(L, "lfunc_data"):
s_alg = s + p2sage(L.lfunc_data['analytic_normalization'])
for x in p2sage(L.lfunc_data['values']):
# the numbers here are always half integers
# so this comparison is exact
if x[0] == s_alg:
val = x[1]
break
if val is None:
if L.fromDB:
val = "not computed"
else:
val = L.sageLfunction.value(s)
if normalization == "arithmetic":
lfunction_value_tex = L.texname_arithmetic.replace('s)', sLatex + ')')
else:
lfunction_value_tex = L.texname.replace('(s', '(' + sLatex)
# We must test for NaN first, since it would show as zero otherwise
# Try "RR(NaN) < float(1e-10)" in sage -- GT
if CC(val).real().is_NaN():
return "\\[{0}=\\infty\\]".format(lfunction_value_tex)
elif val.abs() < 1e-10:
return "\\[{0}=0\\]".format(lfunction_value_tex)
elif normalization == "arithmetic":
return (lfunction_value_tex,
latex(
round(val.real(), number_of_decimals) +
round(val.imag(), number_of_decimals) * I))
else:
return "\\[{0} \\approx {1}\\]".format(
lfunction_value_tex,
latex(
round(val.real(), number_of_decimals) +
round(val.imag(), number_of_decimals) * I))
def _hyperbolic_arc_d(self, z0, z3, first=False):
"""
Function to construct Bezier path as an approximation to
the hyperbolic arc between the complex numbers z0 and z3 in the
hyperbolic plane.
"""
w0 = self._cayley_transform(z0)
w3 = self._cayley_transform(z3)
if w0.real()==0 and w3.real() == 0:
y = (w0.imag() + w0.imag())/2
#self.path.append([(0,w0.imag()),CC(0,y), (0,w3.imag())])
self._graphics.add_primitive(BezierPath([(0,w0.imag()),CC(0,y), (0,w3.imag())],self._options))
return
z0,z3 = CC(z0), CC(z3)
if z0.imag() == 0 and z3.imag() == 0:
p = (z0.real()+z3.real())/2
r = abs(z0-p)
zm = CC(p, r)
self._hyperbolic_arc_d(z0, zm, first)
self._hyperbolic_arc_d(zm, z3)
return
else:
p = (abs(z0)*abs(z0)-abs(z3)*abs(z3))/(z0-z3).real()/2 # center of the circle in H
r = abs(z0-p) # radius of the circle in H
zm = ((z0+z3)/2-p)/abs((z0+z3)/2-p)*r+p
t = (8*zm-4*(z0+z3)).imag()/3/(z3-z0).real()
z1 = z0 + t*CC(z0.imag(), (p-z0.real()))
z2 = z3 - t*CC(z3.imag(), (p-z3.real()))
zm = self._cayley_transform(zm)
w1 = self._cayley_transform(z1)
w2 = self._cayley_transform(z2)
c = self._cayley_transform(CC(p,0)) # center of circle on the unit disk.
r = abs(w1-c) # radius
theta = t
self._graphics.add_primitive(Arc((zm.real(),zm.imag()),r,sector=(-t,t),options=self._options))
def check_amn_slow(rec):
Z = [0] + [CC(*elt) for elt in rec['an_normalized[0:1000]']]
for m, n in pairs:
if (Z[m * n] - Z[m] * Z[n]).abs() > 1e-11:
return False
return True
def gaussum(n, N, prec=53):
CC = ComplexField(prec)
return sum(CC(exp(2 * CC.pi() * CC(0, 1) * n * m ** 2 / N)) for m in range(N))
def signtocolour(sign):
argument = cmath.phase(CC(str(sign)))
r = int(255.0 * (math.cos((1.0 * math.pi / 3.0) - (argument / 2.0))) ** 2)
g = int(255.0 * (math.cos((2.0 * math.pi / 3.0) - (argument / 2.0))) ** 2)
b = int(255.0 * (math.cos(argument / 2.0)) ** 2)
return("rgb(" + str(r) + "," + str(g) + "," + str(b) + ")")
def add_trace_and_norm_ladic(g, D, alpha_geo, verbose=True):
if verbose:
print "add_trace_and_norm_ladic()"
#load Fields
L = D['L']
if L is QQ:
QQx = PolynomialRing(QQ, "x")
L = NumberField(QQx.gen(), "b")
prec = D['prec']
CCap = ComplexField(prec)
# load endo
alpha = D['alpha']
rosati = bound_rosati(alpha_geo)
if alpha.base_ring() is not L:
alpha_K = copy(alpha)
alpha = Matrix(L, 2, 2)
#shift alpha field from K to L
for i, row in enumerate(alpha_K.rows()):
for j, elt in enumerate(row):
if elt not in L:
assert elt.base_ring().absolute_polynomial(
) == L.absolute_polynomial()
alpha[i, j] = L(elt.list())
else:
alpha[i, j] = L(elt)
# load algx_poly
algx_poly_coeff = D['algx_poly']
#sometimes, by mistake the algx_poly is defined over K where K == L, but with a different name
for i, elt in enumerate(algx_poly_coeff):
if elt not in L:
assert elt.base_ring().absolute_polynomial(
) == L.absolute_polynomial()
algx_poly_coeff[i] = L(elt.list())
else:
algx_poly_coeff[i] = L(elt)
x_poly = vector(CCap, D['x_poly'])
for i in [0, 1]:
assert almost_equal(
x_poly[i],
algx_poly_coeff[i]), "%s != %s" % (algx_poly_coeff[i], x_poly[i])
# load P
P0 = vector(L, [D['P'][0], D['P'][1]])
for i, elt in enumerate(P0):
if elt not in L:
assert elt.base_ring().absolute_polynomial(
) == L.absolute_polynomial()
P0[i] = L(elt.list())
else:
P0[i] = L(elt)
if verbose:
print "P0 = %s" % (P0, )
# load image points, P1 and P2
L_poly = PolynomialRing(L, "xL")
xL = L_poly.gen()
Xpoly = L_poly(algx_poly_coeff)
if Xpoly.is_irreducible():
M = Xpoly.root_field("c")
else:
# this avoids bifurcation later on in the code, we don't want to be always checking if M is L
M = NumberField(xL, "c")
# trying to be sure that we keep the same complex_embedding...
M_complex_embedding = 0
if L.gen() not in QQ:
M_complex_embedding = None
Lgen_CC = toCCap(L.gen(), prec=prec)
for i, _ in enumerate(M.complex_embeddings()):
if norm(Lgen_CC - M(L.gen()).complex_embedding(prec=prec, i=i)
) < CCap(2)**(-0.7 * Lgen_CC.prec()) * Lgen_CC.abs():
M_complex_embedding = i
assert M_complex_embedding is not None, "\nL = %s\n%s = %s\n%s" % (
L, L.gen(), Lgen_CC, M.complex_embeddings())
M_poly = PolynomialRing(M, "xM")
xM = M_poly.gen()
# convert everything to M
P0_M = vector(M, [elt for elt in P0])
alpha_M = Matrix(M, [[elt for elt in row] for row in alpha.rows()])
Xpoly_M = M_poly(Xpoly)
for i in [0, 1]:
assert almost_equal(x_poly[i],
Xpoly_M.list()[i],
ithcomplex_embedding=M_complex_embedding
), "%s != %s" % (Xpoly_M.list()[i], x_poly[i])
P1 = vector(M, 2)
P1_ap = vector(CCap, D['R'][0])
P2 = vector(M, 2)
P2_ap = vector(CCap, D['R'][1])
M_Xpoly_roots = Xpoly_M.roots()
assert len(M_Xpoly_roots) > 0
Ypoly_M = prod([xM**2 - g(root)
for root, _ in M_Xpoly_roots]) # also \in L_poly
assert sum(m for _, m in Ypoly_M.roots(M)) == Ypoly_M.degree(
), "%s != %s\n%s\n%s" % (sum(m for _, m in Ypoly_M.roots(M)),
Ypoly_M.degree(), Ypoly_M, Ypoly_M.roots(M))
if len(M_Xpoly_roots) == 1:
# we have a double root
P1[0] = M_Xpoly_roots[0][0]
P2[0] = M_Xpoly_roots[0][0]
ae_prec = prec * 0.4
else:
assert len(M_Xpoly_roots) == 2
ae_prec = prec
# we have two distinct roots
P1[0] = M_Xpoly_roots[0][0]
P2[0] = M_Xpoly_roots[1][0]
if not_equal(P1_ap[0], P1[0],
ithcomplex_embedding=M_complex_embedding):
P1[0] = M_Xpoly_roots[1][0]
P2[0] = M_Xpoly_roots[0][0]
assert almost_equal(
P1_ap[0],
P1[0],
ithcomplex_embedding=M_complex_embedding,
prec=ae_prec), "\n%s = %s \n != %s" % (
P1[0], toCCap(
P1[0], ithcomplex_embedding=M_complex_embedding), CC(P1_ap[0]))
assert almost_equal(
P2_ap[0],
P2[0],
ithcomplex_embedding=M_complex_embedding,
prec=ae_prec), "\n%s = %s \n != %s" % (
P2[0], toCCap(
P2[0], ithcomplex_embedding=M_complex_embedding), CC(P2_ap[0]))
# figure out the right square root
# pick the default branch
P1[1] = sqrt(g(P1[0]))
P2[1] = sqrt(g(P2[0]))
if 0 in [P1[1], P2[1]]:
print "one of image points is a Weirstrass point"
print P1
print P2
raise ZeroDivisionError
#switch if necessary
if not_equal(P1_ap[1],
P1[1],
ithcomplex_embedding=M_complex_embedding,
prec=ae_prec):
P1[1] *= -1
if not_equal(P2_ap[1],
P2[1],
ithcomplex_embedding=M_complex_embedding,
prec=ae_prec):
P2[1] *= -1
# double check
for i in [0, 1]:
assert almost_equal(P1_ap[i],
P1[i],
ithcomplex_embedding=M_complex_embedding,
prec=ae_prec), "%s != %s" % (P1_ap[i], P1[i])
assert almost_equal(P2_ap[i],
P2[i],
ithcomplex_embedding=M_complex_embedding,
prec=ae_prec), "%s != %s" % (P2_ap[i], P2[i])
# now alpha, P0 \in L
# P1, P2 \in L
if verbose:
print "P1 = %s\nP2 = %s" % (P1, P2)
print "Computing the trace and the norm ladically\n"
trace_and_norm = trace_and_norm_ladic(L,
M,
P0_M,
P1,
P2,
g,
2 * alpha_M,
16 * rosati,
primes=ceil(prec * L.degree() /
61))
else:
trace_and_norm = trace_and_norm_ladic(L,
M,
P0_M,
P1,
P2,
g,
2 * alpha_M,
16 * rosati,
primes=ceil(prec * L.degree() /
61))
# Convert the coefficients to polynomials
trace_numerator, trace_denominator, norm_numerator, norm_denominator = [
L_poly(coeff) for coeff in trace_and_norm
]
assert trace_numerator(P0[0]) == trace_denominator(
P0[0]) * -algx_poly_coeff[1], "%s/%s (%s) != %s" % (
trace_numerator, trace_denominator, P0[0], -algx_poly_coeff[1])
assert norm_numerator(P0[0]) == norm_denominator(
P0[0]) * algx_poly_coeff[0], "%s/%s (%s) != %s" % (
norm_numerator, norm_denominator, P0[0], algx_poly_coeff[0])
buffer = "# x1 + x2 = degree %d/ degree %d\n" % (
trace_numerator.degree(), trace_denominator.degree())
buffer += "# = (%s) / (%s) \n" % (trace_numerator, trace_denominator)
buffer += "# max(%d, %d) <= %d\n\n" % (
trace_numerator.degree(), trace_denominator.degree(), 16 * rosati)
buffer += "# x1 * x2 = degree %d/ degree %d\n" % (
norm_numerator.degree(), norm_denominator.degree())
buffer += "# = (%s) / (%s) \n" % (norm_numerator, norm_denominator)
buffer += "# max(%d, %d) <= %d\n" % (
norm_numerator.degree(), norm_denominator.degree(), 16 * rosati)
if verbose:
print buffer
print "\n"
assert max(trace_numerator.degree(),
trace_denominator.degree()) <= 16 * rosati
assert max(norm_numerator.degree(),
norm_denominator.degree()) <= 16 * rosati
if verbose:
print "Veritfying if x1*x2 and x1 + x2 are correct..."
verified = verify_algebraically(g,
P0,
alpha,
trace_and_norm,
verbose=verbose)
if verbose:
print "\nDoes it act on the tangent space as expected? %s\n" % verified
print "Done add_trace_and_norm_ladic()"
return verified, [
trace_numerator.list(),
trace_denominator.list(),
norm_numerator.list(),
norm_denominator.list()
]
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 _hyperbolic_arc_d(self, z0, z3, first=False):
"""
Function to construct Bezier path as an approximation to
the hyperbolic arc between the complex numbers z0 and z3 in the
hyperbolic plane.
"""
w0 = self._cayley_transform(z0)
w3 = self._cayley_transform(z3)
if self._verbose>0:
print "in plane z0,z3=",z0,z3
print "in disc: ",w0,w3
npts = self._npts
if z0 == infinity or z0==CC(infinity):
zm = [z3 + CC(0,j+0.5) for j in range(npts-2)]
wm = [self._cayley_transform(x) for x in zm]
pts = [w3]
pts.extend(wm)
pts.append(w0)
opt = self._options
opt['fill']=False
self._graphics.add_primitive(BezierPath([[(x.real(),x.imag()) for x in pts ]],opt))
return
if z3 == infinity or z3==CC(infinity):
zm = [z0 + CC(0,j+0.5) for j in range(npts-2)]
wm = [self._cayley_transform(x) for x in zm]
pts = [w0]
pts.extend(wm)
pts.append(w3)
opt = self._options
opt['fill']=False
self._graphics.add_primitive(BezierPath([[(x.real(),x.imag()) for x in pts ]],opt))
#self._graphics.add_primitive(Line([w1.real(),w2.real()],[w1.imag(),w2.imag()],self._options))
#self.path.append([(0,w0.imag()),CC(0,y), (0,w3.imag())])
return
x0=z0.real(); y0 = z0.imag()
x3=z3.real(); y3 = z3.imag()
if y0 == 0 and y3 == 0:
p = (z0.real()+z3.real())/2
r = abs(z0-p)
zm = CC(p, r)
self._hyperbolic_arc_d(z0, zm, first)
self._hyperbolic_arc_d(zm, z3)
return
else:
if abs(x0-x3)<1e-10: ## on the same vertical line
xmid = (x0+x3)*0.5; h = y3-y0
zm = [ CC(xmid,y0+t*h/(npts-1)) for t in range(npts) ]
else:
p = RR((x0+x3)*(x3-x0)+(y0+y3)*(y3-y0))/(2*(x3-x0))
r = RR((p - x0)**2 + y0**2).sqrt() # radius of the circle in H
zm = ((z0+z3)/2-p)/abs((z0+z3)/2-p)*r+p # midpoint (at least approximately) of geodesic between z0 and z3
t0 = CC(z0 - p).argument()
t3 = CC(z3 - p).argument()
if self._verbose>1:
print "x0,x3=",x0,x3
print "t0,t3=",t0,t3
print "r=",r
print "opt=",self._options
if x0 <= x3:
zm = [p + r*CC(0,(t0+t*(t3-t0)/(npts-1))).exp() for t in range(npts)]
else:
zm = [p + r*CC(0,(t0+t*(t3-t0)/(npts-1))).exp() for t in range(npts)]
#print "zm=",zm
#zm.insert(0,z0)
#zm.append(z3)
pts = [self._cayley_transform(x) for x in zm]
opt = self._options
opt['fill']=False
#w0 = self._cayley_transform(z0)
#w3 = self._cayley_transform(z3)
if self._verbose>2:
print "pts=",pts
self._graphics.add_primitive(BezierPath([[(x.real(),x.imag()) for x in pts ]],opt))
return
#print "z0_test=",(p+r*exp(t0*I))
#print "z3_test=",(p+r*exp(t3*I))
#t = (8*zm-4*(z0+z3)).imag()/3/(z3-z0).real()
# I have no idea what these points should represent....
#z1 = z0 + t*CC(z0.imag(), (p-z0.real()))
#z2 = z3 - t*CC(z3.imag(), (p-z3.real()))
wm = [self._cayley_transform(x) for x in zm]
pp = self._cayley_transform(CC(p,0))
w1 = self._cayley_transform(z0)
w2 = self._cayley_transform(z3)
c = self._cayley_transform(CC(p,0)) # center of circle on the unit disk.
if self._verbose>2:
print "p,r=",p,r
print "zm=",zm
#print "t=",t
#print "tt=",(8*zm-4*(z0+z3)).imag()/3/(z3-z0).real()
print "C(c)=",pp
print "C(zm)=",wm
print "C(z0)=",w1
print "C(z3)=",w2
#print "z2=",z2
r = abs(w1-c) # radius
t0 = CC(w0 - pp).argument()
t3 = CC(w3 - pp).argument()
t = abs(t0-t3)
if self._verbose>0:
print "adding a:rc ",zm.real(),zm.imag(),r,r,t,t0,t3
self._graphics.add_primitive(Line([w1.real(),w2.real(),wm.real()],[w1.imag(),w2.imag(),wm.imag()],{'thickness':2,'alpha':1,
'rgbcolor':'blue',
'legend_label':""}))
self._graphics.add_primitive(Point([w1.real(),w2.real(),wm.real()],[w1.imag(),w2.imag(),wm.imag()],{'size':10,'alpha':1,
'faceted':True,
'rgbcolor':'red',
'legend_label':""}))
def __init__(self, A, B, C, **options):
"""
Initialize HyperbolicTriangle under the map (z-z0)/(z-\bar(z0)):
Examples::
sage: from sage.plot.hyperbolic_triangle import HyperbolicTriangle
sage: print HyperbolicTriangle(0, 1/2, I, {})
Hyperbolic triangle (0.000000000000000, 0.500000000000000, 1.00000000000000*I)
"""
A, B, C = (CC(A), CC(B), CC(C))
self.path = []
self._options = {}
self._graphics = Graphics()
self._verbose = options.pop('verbose', None)
Z0 = options['center'] #.get('center',C(0,1))
options.pop('center', None)
self._npts = options.pop('npts', 10)
sides = options.pop('sides', [1, 2, 3])
#options.pop('fill',None)
self._options.update(options)
self._z0 = CC(Z0)
self._z0bar = CC(Z0).conjugate()
verbose = self._verbose
sides.sort()
if sides == [1]:
if verbose > 0:
print "Drawing A - B!"
self._hyperbolic_arc_d(A, B, True)
elif sides == [2]:
if verbose > 0:
print "Drawing B - C!"
self._hyperbolic_arc_d(B, C, True)
elif sides == [3]:
if verbose > 0:
print "Drawing C - A!"
self._hyperbolic_arc_d(C, A, True)
elif sides == [1, 2]:
if verbose > 0:
print "Drawing A - B! & B - C!"
self._hyperbolic_arc_d(A, B, True)
self._hyperbolic_arc_d(B, C, False)
elif sides == [1, 3]:
if verbose > 0:
print "Drawing C - A! & A - B"
self._hyperbolic_arc_d(C, A, True)
self._hyperbolic_arc_d(A, B, False)
elif sides == [2, 3]:
if verbose > 0:
print "Drawing B - C! & C - A"
self._hyperbolic_arc_d(B, C, True)
self._hyperbolic_arc_d(C, A, False)
else:
self._hyperbolic_arc_d(A, B, True)
self._hyperbolic_arc_d(B, C, False)
self._hyperbolic_arc_d(C, A, False)
#self._hyperbolic_arc_d(A, B, True);
#self._hyperbolic_arc_d(B, C);
#self._hyperbolic_arc_d(C, A);
#BezierPath.__init__(self, self.path, options)
#super(HyperbolicTriangleDisc,self).__init__(options)
self.A, self.B, self.C = (A, B, C)
def set_table(self, fnr=-1, cusp=0, prec=9):
r"""
Setup a table with coefficients for function nr. fnr in self,
at cusp nr. cusp. If the real or imaginary parts are less than
1e-`prec`, then set them to zero.
"""
table = {'nrows': self.num_coeff}
if fnr < 0:
colrange = range(self.dim)
table['ncols'] = self.dim + 1
elif fnr < self.dim:
colrange = [fnr]
table['ncols'] = 2
table['data'] = []
table['negc'] = 0
realnumc = 0
if self.num_coeff == 0:
self.table = table
return
if self.symmetry != -1:
for n in range(self.num_coeff):
row = [n]
for k in colrange:
if self.dim == 1:
c = None
try:
c = self.coeffs[k][cusp].get(n, None)
except (KeyError, IndexError):
mwf_logger.critical(
"Got coefficient in wrong format for id={0}".format(self._maassid))
# mwf_logger.debug("{0},{1}".format(k,c))
if c is not None:
realnumc += 1
row.append(pretty_coeff(c, prec=prec))
else:
for j in range(self.dim):
c = ((self.coeffs.get(j, {})).get(0, {})).get(n, None)
if c is not None:
row.append(pretty_coeff(c, prec=prec))
realnumc += 1
table['data'].append(row)
else:
table['negc'] = 1
# in this case we need to have coeffs as dict.
if not isinstance(self.coeffs, dict):
self.table = {}
return
for n in range(len(self.coeffs.keys() / 2)):
row = [n]
if self.dim == 1:
for k in range(table['ncols']):
#cpositive and cnegative
cp = self.coeffs.get(n, 0)
cn = self.coeffs.get(-n, 0)
row.append((cp, cn))
realnumc += 1
else:
for j in range(self.dim):
c = (self.coeffs.get(j, {})).get(n, None)
if c is not None:
c1 = c.get(n, None)
cn1 = c.get(-n, None)
c1 = CC(c1)
cn1 = CC(cn1)
row.append((c1, cn1))
realnumc += 1
table['data'].append(row)
self.table = table
mwf_logger.debug("realnumc={0}".format(realnumc))
class HyperbolicTriangleDisc(object): #]GraphicPrimitive):
r"""
Hyperbolic triangles in the disc model of the hyperbolic plane.
Note that we are given coordinates in the upper half-plane and map them to the disc.
"""
@options(center=CC(0, 1))
def __init__(self, A, B, C, **options):
"""
Initialize HyperbolicTriangle under the map (z-z0)/(z-\bar(z0)):
Examples::
sage: from sage.plot.hyperbolic_triangle import HyperbolicTriangle
sage: print HyperbolicTriangle(0, 1/2, I, {})
Hyperbolic triangle (0.000000000000000, 0.500000000000000, 1.00000000000000*I)
"""
A, B, C = (CC(A), CC(B), CC(C))
self.path = []
self._options = {}
self._graphics = Graphics()
self._verbose = options.pop('verbose', None)
Z0 = options['center'] #.get('center',C(0,1))
options.pop('center', None)
self._npts = options.pop('npts', 10)
sides = options.pop('sides', [1, 2, 3])
#options.pop('fill',None)
self._options.update(options)
self._z0 = CC(Z0)
self._z0bar = CC(Z0).conjugate()
verbose = self._verbose
sides.sort()
if sides == [1]:
if verbose > 0:
print "Drawing A - B!"
self._hyperbolic_arc_d(A, B, True)
elif sides == [2]:
if verbose > 0:
print "Drawing B - C!"
self._hyperbolic_arc_d(B, C, True)
elif sides == [3]:
if verbose > 0:
print "Drawing C - A!"
self._hyperbolic_arc_d(C, A, True)
elif sides == [1, 2]:
if verbose > 0:
print "Drawing A - B! & B - C!"
self._hyperbolic_arc_d(A, B, True)
self._hyperbolic_arc_d(B, C, False)
elif sides == [1, 3]:
if verbose > 0:
print "Drawing C - A! & A - B"
self._hyperbolic_arc_d(C, A, True)
self._hyperbolic_arc_d(A, B, False)
elif sides == [2, 3]:
if verbose > 0:
print "Drawing B - C! & C - A"
self._hyperbolic_arc_d(B, C, True)
self._hyperbolic_arc_d(C, A, False)
else:
self._hyperbolic_arc_d(A, B, True)
self._hyperbolic_arc_d(B, C, False)
self._hyperbolic_arc_d(C, A, False)
#self._hyperbolic_arc_d(A, B, True);
#self._hyperbolic_arc_d(B, C);
#self._hyperbolic_arc_d(C, A);
#BezierPath.__init__(self, self.path, options)
#super(HyperbolicTriangleDisc,self).__init__(options)
self.A, self.B, self.C = (A, B, C)
def _cayley_transform(self, z):
#print "z=",z,z==infyinity,type(z),type(infinity)
if z == infinity or z == CC(infinity):
return CC(1, 0)
return (CC(z) - self._z0) / (CC(z) - self._z0bar)
def _hyperbolic_arc_d(self, z0, z3, first=False):
"""
Function to construct Bezier path as an approximation to
the hyperbolic arc between the complex numbers z0 and z3 in the
hyperbolic plane.
"""
w0 = self._cayley_transform(z0)
w3 = self._cayley_transform(z3)
if self._verbose > 0:
print "in plane z0,z3=", z0, z3
print "in disc: ", w0, w3
npts = self._npts
if z0 == infinity or z0 == CC(infinity):
zm = [z3 + CC(0, j + 0.5) for j in range(npts - 2)]
wm = [self._cayley_transform(x) for x in zm]
pts = [w3]
pts.extend(wm)
pts.append(w0)
opt = self._options
opt['fill'] = False
self._graphics.add_primitive(
BezierPath([[(x.real(), x.imag()) for x in pts]], opt))
return
if z3 == infinity or z3 == CC(infinity):
zm = [z0 + CC(0, j + 0.5) for j in range(npts - 2)]
wm = [self._cayley_transform(x) for x in zm]
pts = [w0]
pts.extend(wm)
pts.append(w3)
opt = self._options
opt['fill'] = False
self._graphics.add_primitive(
BezierPath([[(x.real(), x.imag()) for x in pts]], opt))
#self._graphics.add_primitive(Line([w1.real(),w2.real()],[w1.imag(),w2.imag()],self._options))
#self.path.append([(0,w0.imag()),CC(0,y), (0,w3.imag())])
return
x0 = z0.real()
y0 = z0.imag()
x3 = z3.real()
y3 = z3.imag()
if y0 == 0 and y3 == 0:
p = (z0.real() + z3.real()) / 2
r = abs(z0 - p)
zm = CC(p, r)
self._hyperbolic_arc_d(z0, zm, first)
self._hyperbolic_arc_d(zm, z3)
return
else:
if abs(x0 - x3) < 1e-10: ## on the same vertical line
xmid = (x0 + x3) * 0.5
h = y3 - y0
zm = [CC(xmid, y0 + t * h / (npts - 1)) for t in range(npts)]
else:
p = RR((x0 + x3) * (x3 - x0) + (y0 + y3) *
(y3 - y0)) / (2 * (x3 - x0))
r = RR((p - x0)**2 + y0**2).sqrt() # radius of the circle in H
zm = ((z0 + z3) / 2 - p) / abs(
(z0 + z3) / 2 - p
) * r + p # midpoint (at least approximately) of geodesic between z0 and z3
t0 = CC(z0 - p).argument()
t3 = CC(z3 - p).argument()
if self._verbose > 1:
print "x0,x3=", x0, x3
print "t0,t3=", t0, t3
print "r=", r
print "opt=", self._options
if x0 <= x3:
zm = [
p + r * CC(0, (t0 + t * (t3 - t0) / (npts - 1))).exp()
for t in range(npts)
]
else:
zm = [
p + r * CC(0, (t0 + t * (t3 - t0) / (npts - 1))).exp()
for t in range(npts)
]
#print "zm=",zm
#zm.insert(0,z0)
#zm.append(z3)
pts = [self._cayley_transform(x) for x in zm]
opt = self._options
opt['fill'] = False
#w0 = self._cayley_transform(z0)
#w3 = self._cayley_transform(z3)
if self._verbose > 2:
print "pts=", pts
self._graphics.add_primitive(
BezierPath([[(x.real(), x.imag()) for x in pts]], opt))
return
#print "z0_test=",(p+r*exp(t0*I))
#print "z3_test=",(p+r*exp(t3*I))
#t = (8*zm-4*(z0+z3)).imag()/3/(z3-z0).real()
# I have no idea what these points should represent....
#z1 = z0 + t*CC(z0.imag(), (p-z0.real()))
#z2 = z3 - t*CC(z3.imag(), (p-z3.real()))
wm = [self._cayley_transform(x) for x in zm]
pp = self._cayley_transform(CC(p, 0))
w1 = self._cayley_transform(z0)
w2 = self._cayley_transform(z3)
c = self._cayley_transform(CC(
p, 0)) # center of circle on the unit disk.
if self._verbose > 2:
print "p,r=", p, r
print "zm=", zm
#print "t=",t
#print "tt=",(8*zm-4*(z0+z3)).imag()/3/(z3-z0).real()
print "C(c)=", pp
print "C(zm)=", wm
print "C(z0)=", w1
print "C(z3)=", w2
#print "z2=",z2
r = abs(w1 - c) # radius
t0 = CC(w0 - pp).argument()
t3 = CC(w3 - pp).argument()
t = abs(t0 - t3)
if self._verbose > 0:
print "adding a:rc ", zm.real(), zm.imag(), r, r, t, t0, t3
self._graphics.add_primitive(
Line([w1.real(), w2.real(), wm.real()],
[w1.imag(), w2.imag(), wm.imag()], {
'thickness': 2,
'alpha': 1,
'rgbcolor': 'blue',
'legend_label': ""
}))
self._graphics.add_primitive(
Point(
[w1.real(), w2.real(), wm.real()],
[w1.imag(), w2.imag(), wm.imag()], {
'size': 10,
'alpha': 1,
'faceted': True,
'rgbcolor': 'red',
'legend_label': ""
}))
#self._graphics.add_primitive(Arc(pp.real(),pp.imag(),r,r,t,t0,t3,self._options))
#self._graphics. Arc(zm.real(),zm.imag(),r,r,abs(t),-abs(t),abs(t),self._options)
def __call__(self):
return self._graphics
def build_connected_path(P, **kwds):
from sage.all import deepcopy, hyperbolic_arc
paths = []
ymax = P._axes_range.get('ymax', kwds.get('ymax', 10000))
xmax = P._axes_range.get('xmax', kwds.get('xmax', 0))
xmin = P._axes_range.get('xmin', kwds.get('xmin', 0))
new_paths = []
if len(P) == 1: ### SL2Z
A = P[0].A
B = P[0].B
C = P[0].C
new_paths = [
hyperbolic_arc(CC(A), CC(B))[0],
hyperbolic_arc(CC(B), CC(C))[0],
hyperbolic_arc(CC(C), CC(A))[0]
]
else:
for x in P:
if x.A.imag() >= 10000:
### these have to be treated specially..
### We truncate to the maximum y height
### and set the x-coordinate to the same as the other endpoint.
A = CC(x.B.real(), ymax)
B = CC(x.B.real(), x.B.imag())
paths.append(hyperbolic_arc(A, B)[0])
xmin = x.B.real()
elif x.B.imag() >= 10000:
### these have to be treated specially..
### We truncate to the maximum y height
### and set the x-coordinate to the same as the other endpoint.
A = CC(x.A.real(), x.A.imag())
B = CC(x.A.real(), ymax)
paths.append(hyperbolic_arc(A, B)[0])
xmax = x.A.real()
else:
paths.append(x)
## Add a 'closing' path between the two vertical sides.
paths.append(hyperbolic_arc(CC(xmin, ymax), CC(xmax, ymax))[0])
## first find the left most:
## an arc has a .A and .B
As = [x.A for x in P]
minA = min(As)
mini = As.index(minA)
new_paths = [P[mini]]
paths.remove(P[mini])
eps = 1e-15
current = P[mini].B
while paths != []:
current = new_paths[-1].B
#print "current=",current
try:
for p in paths:
#print "p.A=",p.A
#print "p.B=",p.B
if abs(p.A - current) < eps:
#print "appending p"
new_paths.append(p)
paths.remove(p)
raise StopIteration
elif abs(p.B - current) < eps: ## we reverse it
#print "appending p reversed"
pnew = hyperbolic_arc(p.B, p.A)
new_paths.append(pnew[0])
paths.remove(p)
raise StopIteration
else:
continue
print paths
raise ArithmeticError(
"Could not connect from {0}".format(current))
except StopIteration:
pass
### new paths is now a list of hyperbolic arcs
res = []
import matplotlib.patches as patches
from matplotlib.path import Path
import numpy
i = 0
vertices = []
codes = []
for p in new_paths:
new_codes = p.codes
if i > 0:
new_codes[0] = 2
i += 1
if i == len(new_paths):
new_codes[-1] = 79
for v in p.vertices:
vertices.append(v)
codes = codes + new_codes
#pt = patches.Path(p.vertices,codes)
#res.append(pt)
vertices = numpy.array(vertices, float)
res = path = Path(vertices, codes)
# res = patches.Path.make_compound_path(*res)
return res
def value(self, z, embedding=0):
if self.prec == 0:
return 0
else:
q = exp(2*CC.pi()*CC(0,1)*z)
return sum(self.coefficient_embedding(n,embedding)*q**n for n in range(self.prec))
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 _hyperbolic_arc_d(self, z0, z3, first=False):
"""
Function to construct Bezier path as an approximation to
the hyperbolic arc between the complex numbers z0 and z3 in the
hyperbolic plane.
"""
w0 = self._cayley_transform(z0)
w3 = self._cayley_transform(z3)
if self._verbose > 0:
print "in plane z0,z3=", z0, z3
print "in disc: ", w0, w3
npts = self._npts
if z0 == infinity or z0 == CC(infinity):
zm = [z3 + CC(0, j + 0.5) for j in range(npts - 2)]
wm = [self._cayley_transform(x) for x in zm]
pts = [w3]
pts.extend(wm)
pts.append(w0)
opt = self._options
opt['fill'] = False
self._graphics.add_primitive(
BezierPath([[(x.real(), x.imag()) for x in pts]], opt))
return
if z3 == infinity or z3 == CC(infinity):
zm = [z0 + CC(0, j + 0.5) for j in range(npts - 2)]
wm = [self._cayley_transform(x) for x in zm]
pts = [w0]
pts.extend(wm)
pts.append(w3)
opt = self._options
opt['fill'] = False
self._graphics.add_primitive(
BezierPath([[(x.real(), x.imag()) for x in pts]], opt))
#self._graphics.add_primitive(Line([w1.real(),w2.real()],[w1.imag(),w2.imag()],self._options))
#self.path.append([(0,w0.imag()),CC(0,y), (0,w3.imag())])
return
x0 = z0.real()
y0 = z0.imag()
x3 = z3.real()
y3 = z3.imag()
if y0 == 0 and y3 == 0:
p = (z0.real() + z3.real()) / 2
r = abs(z0 - p)
zm = CC(p, r)
self._hyperbolic_arc_d(z0, zm, first)
self._hyperbolic_arc_d(zm, z3)
return
else:
if abs(x0 - x3) < 1e-10: ## on the same vertical line
xmid = (x0 + x3) * 0.5
h = y3 - y0
zm = [CC(xmid, y0 + t * h / (npts - 1)) for t in range(npts)]
else:
p = RR((x0 + x3) * (x3 - x0) + (y0 + y3) *
(y3 - y0)) / (2 * (x3 - x0))
r = RR((p - x0)**2 + y0**2).sqrt() # radius of the circle in H
zm = ((z0 + z3) / 2 - p) / abs(
(z0 + z3) / 2 - p
) * r + p # midpoint (at least approximately) of geodesic between z0 and z3
t0 = CC(z0 - p).argument()
t3 = CC(z3 - p).argument()
if self._verbose > 1:
print "x0,x3=", x0, x3
print "t0,t3=", t0, t3
print "r=", r
print "opt=", self._options
if x0 <= x3:
zm = [
p + r * CC(0, (t0 + t * (t3 - t0) / (npts - 1))).exp()
for t in range(npts)
]
else:
zm = [
p + r * CC(0, (t0 + t * (t3 - t0) / (npts - 1))).exp()
for t in range(npts)
]
#print "zm=",zm
#zm.insert(0,z0)
#zm.append(z3)
pts = [self._cayley_transform(x) for x in zm]
opt = self._options
opt['fill'] = False
#w0 = self._cayley_transform(z0)
#w3 = self._cayley_transform(z3)
if self._verbose > 2:
print "pts=", pts
self._graphics.add_primitive(
BezierPath([[(x.real(), x.imag()) for x in pts]], opt))
return
#print "z0_test=",(p+r*exp(t0*I))
#print "z3_test=",(p+r*exp(t3*I))
#t = (8*zm-4*(z0+z3)).imag()/3/(z3-z0).real()
# I have no idea what these points should represent....
#z1 = z0 + t*CC(z0.imag(), (p-z0.real()))
#z2 = z3 - t*CC(z3.imag(), (p-z3.real()))
wm = [self._cayley_transform(x) for x in zm]
pp = self._cayley_transform(CC(p, 0))
w1 = self._cayley_transform(z0)
w2 = self._cayley_transform(z3)
c = self._cayley_transform(CC(
p, 0)) # center of circle on the unit disk.
if self._verbose > 2:
print "p,r=", p, r
print "zm=", zm
#print "t=",t
#print "tt=",(8*zm-4*(z0+z3)).imag()/3/(z3-z0).real()
print "C(c)=", pp
print "C(zm)=", wm
print "C(z0)=", w1
print "C(z3)=", w2
#print "z2=",z2
r = abs(w1 - c) # radius
t0 = CC(w0 - pp).argument()
t3 = CC(w3 - pp).argument()
t = abs(t0 - t3)
if self._verbose > 0:
print "adding a:rc ", zm.real(), zm.imag(), r, r, t, t0, t3
self._graphics.add_primitive(
Line([w1.real(), w2.real(), wm.real()],
[w1.imag(), w2.imag(), wm.imag()], {
'thickness': 2,
'alpha': 1,
'rgbcolor': 'blue',
'legend_label': ""
}))
self._graphics.add_primitive(
Point(
[w1.real(), w2.real(), wm.real()],
[w1.imag(), w2.imag(), wm.imag()], {
'size': 10,
'alpha': 1,
'faceted': True,
'rgbcolor': 'red',
'legend_label': ""
}))
def _cayley_transform(self, z):
#print "z=",z,z==infyinity,type(z),type(infinity)
if z == infinity or z == CC(infinity):
return CC(1, 0)
return (CC(z) - self._z0) / (CC(z) - self._z0bar)