def Catenoid(c=1, name="Catenoid"):
r"""
Return a catenoid surface, with parametric representation
.. MATH::
\begin{aligned}
x(u, v) & = c \cosh(v/c) \cos(u); \\
y(u, v) & = c \cosh(v/c) \sin(u); \\
z(u, v) & = v.
\end{aligned}
INPUT:
- ``c`` -- surface parameter.
- ``name`` -- string. Name of the surface.
For more information, see :wikipedia:`Catenoid`.
EXAMPLES::
sage: cat = surfaces.Catenoid(); cat
Parametrized surface ('Catenoid') with equation (cos(u)*cosh(v), cosh(v)*sin(u), v)
sage: cat.plot()
Graphics3d Object
"""
u, v = var('u, v')
catenoid_eq = [c * cosh(v / c) * cos(u), c * cosh(v / c) * sin(u), v]
coords = ((u, 0, 2 * pi), (v, -1, 1))
return ParametrizedSurface3D(catenoid_eq, coords, name)
def Catenoid(c=1, name="Catenoid"):
r"""
Returns a catenoid surface, with parametric representation
.. MATH::
\begin{aligned}
x(u, v) & = c \cosh(v/c) \cos(u); \\
y(u, v) & = c \cosh(v/c) \sin(u); \\
z(u, v) & = v.
\end{aligned}
INPUT:
- ``c`` -- surface parameter.
- ``name`` -- string. Name of the surface.
EXAMPLES::
sage: cat = surfaces.Catenoid(); cat
Parametrized surface ('Catenoid') with equation (cos(u)*cosh(v), cosh(v)*sin(u), v)
sage: cat.plot()
Graphics3d Object
"""
u, v = var("u, v")
catenoid_eq = [c * cosh(v / c) * cos(u), c * cosh(v / c) * sin(u), v]
coords = ((u, 0, 2 * pi), (v, -1, 1))
return ParametrizedSurface3D(catenoid_eq, coords, name)
def show(self, show_hyperboloid=True, **graphics_options):
r"""
Plot ``self``.
EXAMPLES::
sage: from sage.geometry.hyperbolic_space.hyperbolic_geodesic import *
sage: g = HyperbolicPlane().HM().random_geodesic()
sage: g.show()
Graphics3d Object
"""
x = SR.var('x')
opts = self.graphics_options()
opts.update(graphics_options)
v1, u2 = [vector(k.coordinates()) for k in self.endpoints()]
# Lorentzian Gram Shmidt. The original vectors will be
# u1, u2 and the orthogonal ones will be v1, v2. Except
# v1 = u1, and I don't want to declare another variable,
# hence the odd naming convention above.
# We need the Lorentz dot product of v1 and u2.
v1_ldot_u2 = u2[0] * v1[0] + u2[1] * v1[1] - u2[2] * v1[2]
v2 = u2 + v1_ldot_u2 * v1
v2_norm = sqrt(v2[0]**2 + v2[1]**2 - v2[2]**2)
v2 = v2 / v2_norm
v2_ldot_u2 = u2[0] * v2[0] + u2[1] * v2[1] - u2[2] * v2[2]
# Now v1 and v2 are Lorentz orthogonal, and |v1| = -1, |v2|=1
# That is, v1 is unit timelike and v2 is unit spacelike.
# This means that cosh(x)*v1 + sinh(x)*v2 is unit timelike.
hyperbola = cosh(x) * v1 + sinh(x) * v2
endtime = arcsinh(v2_ldot_u2)
from sage.plot.plot3d.all import parametric_plot3d
pic = parametric_plot3d(hyperbola, (x, 0, endtime), **graphics_options)
if show_hyperboloid:
pic += self._model.get_background_graphic()
return pic
def show(self, show_hyperboloid=True, **graphics_options):
r"""
Plot ``self``.
EXAMPLES::
sage: from sage.geometry.hyperbolic_space.hyperbolic_geodesic import *
sage: g = HyperbolicPlane().HM().random_geodesic()
sage: g.show()
Graphics3d Object
"""
x = SR.var('x')
opts = self.graphics_options()
opts.update(graphics_options)
v1, u2 = [vector(k.coordinates()) for k in self.endpoints()]
# Lorentzian Gram Shmidt. The original vectors will be
# u1, u2 and the orthogonal ones will be v1, v2. Except
# v1 = u1, and I don't want to declare another variable,
# hence the odd naming convention above.
# We need the Lorentz dot product of v1 and u2.
v1_ldot_u2 = u2[0]*v1[0] + u2[1]*v1[1] - u2[2]*v1[2]
v2 = u2 + v1_ldot_u2 * v1
v2_norm = sqrt(v2[0]**2 + v2[1]**2 - v2[2]**2)
v2 = v2 / v2_norm
v2_ldot_u2 = u2[0]*v2[0] + u2[1]*v2[1] - u2[2]*v2[2]
# Now v1 and v2 are Lorentz orthogonal, and |v1| = -1, |v2|=1
# That is, v1 is unit timelike and v2 is unit spacelike.
# This means that cosh(x)*v1 + sinh(x)*v2 is unit timelike.
hyperbola = cosh(x)*v1 + sinh(x)*v2
endtime = arcsinh(v2_ldot_u2)
from sage.plot.plot3d.all import parametric_plot3d
pic = parametric_plot3d(hyperbola, (x, 0, endtime), **graphics_options)
if show_hyperboloid:
pic += self._model.get_background_graphic()
return pic
def _eval_(self, n, x):
"""
EXAMPLES::
sage: y=var('y')
sage: bessel_I(y,x)
bessel_I(y, x)
sage: bessel_I(0.0, 1.0)
1.26606587775201
sage: bessel_I(1/2, 1)
sqrt(2)*sinh(1)/sqrt(pi)
sage: bessel_I(-1/2, pi)
sqrt(2)*cosh(pi)/pi
"""
if (not isinstance(n, Expression) and not isinstance(x, Expression)
and (is_inexact(n) or is_inexact(x))):
coercion_model = get_coercion_model()
n, x = coercion_model.canonical_coercion(n, x)
return self._evalf_(n, x, parent(n))
# special identities
if n == Integer(1) / Integer(2):
return sqrt(2 / (pi * x)) * sinh(x)
elif n == -Integer(1) / Integer(2):
return sqrt(2 / (pi * x)) * cosh(x)
return None # leaves the expression unevaluated
def _eval_(self, a, z):
"""
EXAMPLES::
sage: struve_L(-2,0)
struve_L(-2, 0)
sage: struve_L(-1,0)
0
sage: struve_L(pi,0)
0
sage: struve_L(-1/2,x)
sqrt(2)*sqrt(1/(pi*x))*sinh(x)
sage: struve_L(1/2,1)
sqrt(2)*(cosh(1) - 1)/sqrt(pi)
sage: struve_L(2,x)
struve_L(2, x)
sage: struve_L(-3/2,x)
-bessel_I(3/2, x)
"""
from sage.symbolic.ring import SR
if z.is_zero() \
and (SR(a).is_numeric() or SR(a).is_constant()) \
and a.real() >= -1:
return ZZ(0)
if a == -Integer(1)/2:
from sage.functions.hyperbolic import sinh
return sqrt(2/(pi*z)) * sinh(z)
if a == Integer(1)/2:
from sage.functions.hyperbolic import cosh
return sqrt(2/(pi*z)) * (cosh(z)-1)
if a < 0 and not SR(a).is_integer() and SR(2*a).is_integer():
from sage.rings.rational_field import QQ
n = (a*(-2) - 1)/2
return Integer(-1)**n * bessel_I(n+QQ(1)/2, z)
def _eval_(self, n, x):
"""
EXAMPLES::
sage: y=var('y')
sage: bessel_I(y,x)
bessel_I(y, x)
sage: bessel_I(0.0, 1.0)
1.26606587775201
sage: bessel_I(1/2, 1)
sqrt(2)*sinh(1)/sqrt(pi)
sage: bessel_I(-1/2, pi)
sqrt(2)*cosh(pi)/pi
"""
if (not isinstance(n, Expression) and not isinstance(x, Expression) and
(is_inexact(n) or is_inexact(x))):
coercion_model = get_coercion_model()
n, x = coercion_model.canonical_coercion(n, x)
return self._evalf_(n, x, parent(n))
# special identities
if n == Integer(1) / Integer(2):
return sqrt(2 / (pi * x)) * sinh(x)
elif n == -Integer(1) / Integer(2):
return sqrt(2 / (pi * x)) * cosh(x)
return None # leaves the expression unevaluated
def RQ(delta):
# this is the quotient R(F_0,z)/R(F_0,z(F)) for a generic z
# at distance delta from j. See Lemma 4.2 in [HS2018].
cd = cosh(delta).n(prec=prec)
sd = sinh(delta).n(prec=prec)
return prod(
[cd + (cost * phi[0] + sint * phi[1]) * sd for phi in phis])
def get_bound_poly(F, prec=53, norm_type='norm', emb=None):
"""
The hyperbolic distance from `j` which must contain the smallest poly.
This defines the maximum possible distance from `j` to the `z_0` covariant
in the hyperbolic 3-space for which the associated `F` could have smaller
coefficients.
INPUT:
- ``F`` -- binary form of degree at least 3 with no multiple roots
- ``prec``-- positive integer. precision to use in CC
- ``norm_type`` -- string, either norm or height
- ``emb`` -- embedding into CC
OUTPUT: a positive real number
EXAMPLES::
sage: from sage.rings.polynomial.binary_form_reduce import get_bound_poly
sage: R.<x,y> = QQ[]
sage: F = -2*x^3 + 2*x^2*y + 3*x*y^2 + 127*y^3
sage: get_bound_poly(F)
28.0049336543295
sage: get_bound_poly(F, norm_type='height')
111.890642019092
"""
def coshdelta(z):
#The cosh of the hyperbolic distance from z = t+uj to j
return (z.norm() + 1) / (2 * z.imag())
if F.base_ring() != ComplexField(prec=prec):
if emb is None:
compF = F.change_ring(ComplexField(prec=prec))
else:
compF = F.change_ring(emb)
else:
compF = F
n = F.degree()
assert (n > 2), "degree 2 polynomial"
z0F, thetaF = covariant_z0(compF, prec=prec, emb=emb)
cosh_delta = coshdelta(z0F)
if norm_type == 'norm':
#euclidean norm squared
normF = (sum([abs(i)**2 for i in compF.coefficients()]))
target = (2**(n - 1)) * normF / thetaF
elif norm_type == 'height':
hF = e**max([c.global_height(prec=prec)
for c in F.coefficients()]) #height
target = (2**(n - 1)) * (n + 1) * (hF**2) / thetaF
else:
raise ValueError('type must be norm or height')
return cosh(epsinv(F, target, prec=prec))
def _0f1(b, z):
F12 = cosh(2 * sqrt(z))
F32 = sinh(2 * sqrt(z)) / (2 * sqrt(z))
if 2 * b == 1:
return F12
if 2 * b == 3:
return F32
if 2 * b > 3:
return ((b - 2) * (b - 1) / z *
(_0f1(b - 2, z) - _0f1(b - 1, z)))
if 2 * b < 1:
return (_0f1(b + 1, z) + z / (b *
(b + 1)) * _0f1(b + 2, z))
raise ValueError
def _0f1(b, z):
F12 = cosh(2 * sqrt(z))
F32 = sinh(2 * sqrt(z)) / (2 * sqrt(z))
if 2 * b == 1:
return F12
if 2 * b == 3:
return F32
if 2 * b > 3:
return ((b - 2) * (b - 1) / z * (_0f1(b - 2, z) -
_0f1(b - 1, z)))
if 2 * b < 1:
return (_0f1(b + 1, z) + z / (b * (b + 1)) *
_0f1(b + 2, z))
raise ValueError
def _derivative_(self, z, diff_param=None):
"""
The derivative of `\operatorname{Chi}(z)` is `\cosh(z)/z`.
EXAMPLES::
sage: x = var('x')
sage: f = cosh_integral(x)
sage: f.diff(x)
cosh(x)/x
sage: f = cosh_integral(ln(x))
sage: f.diff(x)
cosh(log(x))/(x*log(x))
"""
return cosh(z)/z
def _derivative_(self, z, diff_param=None):
"""
The derivative of `\operatorname{Chi}(z)` is `\cosh(z)/z`.
EXAMPLES::
sage: x = var('x')
sage: f = cosh_integral(x)
sage: f.diff(x)
cosh(x)/x
sage: f = cosh_integral(ln(x))
sage: f.diff(x)
cosh(log(x))/(x*log(x))
"""
return cosh(z) / z
def _eval_(self, n, x):
"""
EXAMPLES::
sage: y=var('y')
sage: bessel_I(y,x)
bessel_I(y, x)
sage: bessel_I(0.0, 1.0)
1.26606587775201
sage: bessel_I(1/2, 1)
sqrt(2)*sinh(1)/sqrt(pi)
sage: bessel_I(-1/2, pi)
sqrt(2)*cosh(pi)/pi
"""
# special identities
if n == Integer(1) / Integer(2):
return sqrt(2 / (pi * x)) * sinh(x)
elif n == -Integer(1) / Integer(2):
return sqrt(2 / (pi * x)) * cosh(x)
def _eval_(self, n, x):
"""
EXAMPLES::
sage: y=var('y')
sage: bessel_I(y,x)
bessel_I(y, x)
sage: bessel_I(0.0, 1.0)
1.26606587775201
sage: bessel_I(1/2, 1)
sqrt(2)*sinh(1)/sqrt(pi)
sage: bessel_I(-1/2, pi)
sqrt(2)*cosh(pi)/pi
"""
# special identities
if n == Integer(1) / Integer(2):
return sqrt(2 / (pi * x)) * sinh(x)
elif n == -Integer(1) / Integer(2):
return sqrt(2 / (pi * x)) * cosh(x)
def get_bound_dynamical(F, f, m=1, dynatomic=True, prec=53, emb=None):
"""
The hyperbolic distance from `j` which must contain the smallest map.
This defines the maximum possible distance from `j` to the `z_0` covariant
of the associated binary form `F` in the hyperbolic 3-space
for which the map `f`` could have smaller coefficients.
INPUT:
- ``F`` -- binary form of degree at least 3 with no multiple roots associated
to ``f``
- ``f`` -- a dynamical system on `P^1`
- ``m`` - positive integer. the period used to create ``F``
- ``dynatomic`` -- boolean. whether ``F`` is the periodic points or the
formal periodic points of period ``m`` for ``f``
- ``prec``-- positive integer. precision to use in CC
- ``emb`` -- embedding into CC
OUTPUT: a positive real number
EXAMPLES::
sage: from sage.dynamics.arithmetic_dynamics.endPN_minimal_model import get_bound_dynamical
sage: P.<x,y> = ProjectiveSpace(QQ,1)
sage: f = DynamicalSystem([50*x^2 + 795*x*y + 2120*y^2, 265*x^2 + 106*y^2])
sage: get_bound_dynamical(f.dynatomic_polynomial(1), f)
35.5546923182219
"""
def coshdelta(z):
#The cosh of the hyperbolic distance from z = t+uj to j
return (z.norm() + 1) / (2 * z.imag())
if F.base_ring() != ComplexField(prec=prec):
if emb is None:
compF = F.change_ring(ComplexField(prec=prec))
else:
compF = F.change_ring(emb)
else:
compF = F
n = F.degree()
z0F, thetaF = covariant_z0(compF, prec=prec, emb=emb)
cosh_delta = coshdelta(z0F)
d = f.degree()
hF = e**f.global_height(prec=prec)
#get precomputed constants C,k
if m == 1:
C = 4 * d + 2
k = 2
else:
Ck_values = {(False, 2, 2): (322, 6), (False, 2, 3): (385034, 14),\
(False, 2, 4): (4088003923454, 30), (False, 3, 2): (18044, 8),\
(False, 4, 2): (1761410, 10), (False, 5, 2): (269283820, 12),\
(True, 2, 2): (43, 4), (True, 2, 3): (106459, 12),\
(True, 2, 4): (39216735905, 24), (True, 3, 2): (1604, 6),\
(True, 4, 2): (114675, 8), (True, 5, 2): (14158456, 10)}
try:
C, k = Ck_values[(dynatomic, d, m)]
except KeyError:
raise ValueError("constants not computed for this (m,d) pair")
if n == 2 and d == 2:
#bound with epsilonF = 1
bound = 2 * ((2 * C * (hF**k)) / (thetaF))
else:
bound = cosh(epsinv(F, (2**(n - 1)) * C * (hF**k) / thetaF, prec=prec))
return bound
def get_bound_dynamical(F, f, m=1, dynatomic=True, prec=53, emb=None):
"""
The hyperbolic distance from `j` which must contain the smallest map.
This defines the maximum possible distance from `j` to the `z_0` covariant
of the assocaited binary form `F` in the hyperbolic 3-space
for which the map `f`` could have smaller coefficients.
INPUT:
- ``F`` -- binary form of degree at least 3 with no multiple roots associated
to ``f``
- ``f`` -- a dynamical system on `P^1`
- ``m`` - positive integer. the period used to create ``F``
- ``dyantomic`` -- boolean. whether ``F`` is the periodic points or the
formal periodic points of period ``m`` for ``f``
- ``prec``-- positive integer. precision to use in CC
- ``emb`` -- embedding into CC
OUTPUT: a positive real number
EXAMPLES::
sage: from sage.dynamics.arithmetic_dynamics.endPN_minimal_model import get_bound_dynamical
sage: P.<x,y> = ProjectiveSpace(QQ,1)
sage: f = DynamicalSystem([50*x^2 + 795*x*y + 2120*y^2, 265*x^2 + 106*y^2])
sage: get_bound_dynamical(f.dynatomic_polynomial(1), f)
35.5546923182219
"""
def coshdelta(z):
#The cosh of the hyperbolic distance from z = t+uj to j
return (z.norm() + 1)/(2*z.imag())
if F.base_ring() != ComplexField(prec=prec):
if emb is None:
compF = F.change_ring(ComplexField(prec=prec))
else:
compF = F.change_ring(emb)
else:
compF = F
n = F.degree()
z0F, thetaF = covariant_z0(compF, prec=prec, emb=emb)
cosh_delta = coshdelta(z0F)
d = f.degree()
hF = e**f.global_height(prec=prec)
#get precomputed constants C,k
if m == 1:
C = 4*d+2;k=2
else:
Ck_values = {(False, 2, 2): (322, 6), (False, 2, 3): (385034, 14),\
(False, 2, 4): (4088003923454, 30), (False, 3, 2): (18044, 8),\
(False, 4, 2): (1761410, 10), (False, 5, 2): (269283820, 12),\
(True, 2, 2): (43, 4), (True, 2, 3): (106459, 12),\
(True, 2, 4): (39216735905, 24), (True, 3, 2): (1604, 6),\
(True, 4, 2): (114675, 8), (True, 5, 2): (14158456, 10)}
try:
C,k = Ck_values[(dynatomic,d,m)]
except KeyError:
raise ValueError("constants not computed for this (m,d) pair")
if n == 2 and d == 2:
#bound with epsilonF = 1
bound = 2*((2*C*(hF**k))/(thetaF))
else:
bound = cosh(epsinv(F, (2**(n-1))*C*(hF**k)/thetaF, prec=prec))
return bound