def Torus(r=2, R=3, name="Torus"):
r"""
Returns a torus obtained by revolving a circle of radius ``r`` around
a coplanar axis ``R`` units away from the center of the circle. The
parametrization used is
.. MATH::
\begin{aligned}
x(u, v) & = (R + r \cos(v)) \cos(u); \\
y(u, v) & = (R + r \cos(v)) \sin(u); \\
z(u, v) & = r \sin(v).
\end{aligned}
INPUT:
- ``r``, ``R`` -- Minor and major radius of the torus.
- ``name`` -- string. Name of the surface.
EXAMPLES::
sage: torus = surfaces.Torus(); torus
Parametrized surface ('Torus') with equation ((2*cos(v) + 3)*cos(u), (2*cos(v) + 3)*sin(u), 2*sin(v))
sage: torus.plot()
Graphics3d Object
"""
u, v = var("u, v")
torus_eq = [(R + r * cos(v)) * cos(u), (R + r * cos(v)) * sin(u), r * sin(v)]
coords = ((u, 0, 2 * pi), (v, 0, 2 * pi))
return ParametrizedSurface3D(torus_eq, coords, name)
def Klein(r=1, name="Klein bottle"):
r"""
Returns the Klein bottle, in the figure-8 parametrization given by
.. MATH::
\begin{aligned}
x(u, v) & = (r + \cos(u/2)\cos(v) - \sin(u/2)\sin(2v)) \cos(u); \\
y(u, v) & = (r + \cos(u/2)\cos(v) - \sin(u/2)\sin(2v)) \sin(u); \\
z(u, v) & = \sin(u/2)\cos(v) + \cos(u/2)\sin(2v).
\end{aligned}
INPUT:
- ``r`` -- radius of the "figure-8" circle.
- ``name`` -- string. Name of the surface.
EXAMPLES::
sage: klein = surfaces.Klein(); klein
Parametrized surface ('Klein bottle') with equation (-(sin(1/2*u)*sin(2*v) - cos(1/2*u)*sin(v) - 1)*cos(u), -(sin(1/2*u)*sin(2*v) - cos(1/2*u)*sin(v) - 1)*sin(u), cos(1/2*u)*sin(2*v) + sin(1/2*u)*sin(v))
sage: klein.plot()
Graphics3d Object
"""
u, v = var("u, v")
x = (r + cos(u / 2) * sin(v) - sin(u / 2) * sin(2 * v)) * cos(u)
y = (r + cos(u / 2) * sin(v) - sin(u / 2) * sin(2 * v)) * sin(u)
z = sin(u / 2) * sin(v) + cos(u / 2) * sin(2 * v)
klein_eq = [x, y, z]
coords = ((u, 0, 2 * pi), (v, 0, 2 * pi))
return ParametrizedSurface3D(klein_eq, coords, name)
def Dini(a=1, b=1, name="Dini's surface"):
r"""
Returns Dini's surface, with parametrization
.. MATH::
\begin{aligned}
x(u, v) & = a \cos(u)\sin(v); \\
y(u, v) & = a \sin(u)\sin(v); \\
z(u, v) & = u + \log(\tan(v/2)) + \cos(v).
\end{aligned}
INPUT:
- ``a, b`` -- surface parameters.
- ``name`` -- string. Name of the surface.
EXAMPLES::
sage: dini = surfaces.Dini(a=3, b=4); dini
Parametrized surface ('Dini's surface') with equation (3*cos(u)*sin(v), 3*sin(u)*sin(v), 4*u + 3*cos(v) + 3*log(tan(1/2*v)))
sage: dini.plot() # not tested -- known bug (see #10132)
"""
u, v = var("u, v")
dini_eq = [a * cos(u) * sin(v), a * sin(u) * sin(v), a * (cos(v) + log(tan(v / 2))) + b * u]
coords = ((u, 0, 2 * pi), (v, 0, 2 * pi))
return ParametrizedSurface3D(dini_eq, coords, name)
def Crosscap(r=1, name="Crosscap"):
r"""
Returns a crosscap surface, with parametrization
.. MATH::
\begin{aligned}
x(u, v) & = r(1 + \cos(v)) \cos(u); \\
y(u, v) & = r(1 + \cos(v)) \sin(u); \\
z(u, v) & = - r\tanh(u - \pi) \sin(v).
\end{aligned}
INPUT:
- ``r`` -- surface parameter.
- ``name`` -- string. Name of the surface.
EXAMPLES::
sage: crosscap = surfaces.Crosscap(); crosscap
Parametrized surface ('Crosscap') with equation ((cos(v) + 1)*cos(u), (cos(v) + 1)*sin(u), -sin(v)*tanh(-pi + u))
sage: crosscap.plot()
Graphics3d Object
"""
u, v = var("u, v")
crosscap_eq = [r * (1 + cos(v)) * cos(u), r * (1 + cos(v)) * sin(u), -tanh(u - pi) * r * sin(v)]
coords = ((u, 0, 2 * pi), (v, 0, 2 * pi))
return ParametrizedSurface3D(crosscap_eq, coords, name)
def show(self, boundary=True, **options):
r"""
Plot ``self``.
EXAMPLES::
sage: HyperbolicPlane().PD().get_geodesic(0, 1).show()
Graphics object consisting of 2 graphics primitives
"""
opts = dict([('axes', False), ('aspect_ratio', 1)])
opts.update(self.graphics_options())
opts.update(options)
end_1, end_2 = [CC(k.coordinates()) for k in self.endpoints()]
bd_1, bd_2 = [CC(k.coordinates()) for k in self.ideal_endpoints()]
# Check to see if it's a line
if bool(real(bd_1)*imag(bd_2) - real(bd_2)*imag(bd_1))**2 < EPSILON:
from sage.plot.line import line
pic = line([(real(bd_1),imag(bd_1)),(real(bd_2),imag(bd_2))],
**opts)
else:
# If we are here, we know it's not a line
# So we compute the center and radius of the circle
center = (1/(real(bd_1)*imag(bd_2) - real(bd_2)*imag(bd_1)) *
((imag(bd_2)-imag(bd_1)) + (real(bd_1)-real(bd_2))*I))
radius = RR(abs(bd_1 - center)) # abs is Euclidean distance
# Now we calculate the angles for the parametric plot
theta1 = CC(end_1 - center).arg()
theta2 = CC(end_2 - center).arg()
if theta2 < theta1:
theta1, theta2 = theta2, theta1
from sage.calculus.var import var
from sage.plot.plot import parametric_plot
x = var('x')
mid = (theta1 + theta2)/2.0
if (radius*cos(mid) + real(center))**2 + \
(radius*sin(mid) + imag(center))**2 > 1.0:
# Swap theta1 and theta2
tmp = theta1 + 2*pi
theta1 = theta2
theta2 = tmp
pic = parametric_plot((radius*cos(x) + real(center),
radius*sin(x) + imag(center)),
(x, theta1, theta2), **opts)
else:
pic = parametric_plot((radius*cos(x) + real(center),
radius*sin(x) + imag(center)),
(x, theta1, theta2), **opts)
if boundary:
bd_pic = self._model.get_background_graphic()
pic = bd_pic + pic
return pic
def transform(self, radius=None, azimuth=None, elevation=None):
"""
A spherical elevation coordinates transform.
EXAMPLE::
sage: T = SphericalElevation('radius', ['azimuth', 'elevation'])
sage: T.transform(radius=var('r'), azimuth=var('theta'), elevation=var('phi'))
(r*cos(phi)*cos(theta), r*cos(phi)*sin(theta), r*sin(phi))
"""
return (radius * cos(elevation) * cos(azimuth),
radius * cos(elevation) * sin(azimuth),
radius * sin(elevation))
def plot_arc(radius, p, q, **opts):
# TODO: THIS SHOULD USE THE EXISTING PLOT OF ARCS!
# plot the arc from p to q differently depending on the type of self
p = ZZ(p)
q = ZZ(q)
t = var('t')
if p - q in [1, -1]:
def f(t):
return (radius * cos(t), radius * sin(t))
(p, q) = sorted([p, q])
angle_p = vertex_to_angle(p)
angle_q = vertex_to_angle(q)
return parametric_plot(f(t), (t, angle_q, angle_p), **opts)
if self.type() == 'A':
angle_p = vertex_to_angle(p)
angle_q = vertex_to_angle(q)
if angle_p < angle_q:
angle_p += 2 * pi
internal_angle = angle_p - angle_q
if internal_angle > pi:
(angle_p, angle_q) = (angle_q + 2 * pi, angle_p)
internal_angle = angle_p - angle_q
angle_center = (angle_p+angle_q) / 2
hypotenuse = radius / cos(internal_angle / 2)
radius_arc = hypotenuse * sin(internal_angle / 2)
center = (hypotenuse * cos(angle_center),
hypotenuse * sin(angle_center))
center_angle_p = angle_p + pi / 2
center_angle_q = angle_q + 3 * pi / 2
def f(t):
return (radius_arc * cos(t) + center[0],
radius_arc * sin(t) + center[1])
return parametric_plot(f(t), (t, center_angle_p,
center_angle_q), **opts)
elif self.type() == 'D':
if p >= q:
q += self.r()
px = -2 * pi * p / self.r() + pi / 2
qx = -2 * pi * q / self.r() + pi / 2
arc_radius = (px - qx) / 2
arc_center = qx + arc_radius
def f(t):
return exp(I * ((cos(t) + I * sin(t)) *
arc_radius + arc_center)) * radius
return parametric_plot((real_part(f(t)), imag_part(f(t))),
(t, 0, pi), **opts)
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 Helicoid(h=1, name="Helicoid"):
r"""
Returns a helicoid surface, with parametrization
.. MATH::
\begin{aligned}
x(\rho, \theta) & = \rho \cos(\theta); \\
y(\rho, \theta) & = \rho \sin(\theta); \\
z(\rho, \theta) & = h\theta/(2\pi).
\end{aligned}
INPUT:
- ``h`` -- distance along the z-axis between two
successive turns of the helicoid.
- ``name`` -- string. Name of the surface.
EXAMPLES::
sage: helicoid = surfaces.Helicoid(h=2); helicoid
Parametrized surface ('Helicoid') with equation (rho*cos(theta), rho*sin(theta), theta/pi)
sage: helicoid.plot()
Graphics3d Object
"""
rho, theta = var("rho, theta")
helicoid_eq = [rho * cos(theta), rho * sin(theta), h * theta / (2 * pi)]
coords = ((rho, -2, 2), (theta, 0, 2 * pi))
return ParametrizedSurface3D(helicoid_eq, [rho, theta], name)
def _eval_(self, a, z):
"""
EXAMPLES::
sage: struve_H(0,0)
0
sage: struve_H(pi,0)
0
sage: struve_H(-1/2,x)
sqrt(2)*sqrt(1/(pi*x))*sin(x)
sage: struve_H(1/2,-1)
-sqrt(2)*sqrt(-1/pi)*(cos(1) - 1)
sage: struve_H(1/2,pi)
2*sqrt(2)/pi
sage: struve_H(2,x)
struve_H(2, x)
sage: struve_H(-3/2,x)
-bessel_J(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.trig import sin
return sqrt(2/(pi*z)) * sin(z)
if a == Integer(1)/2:
from sage.functions.trig import cos
return sqrt(2/(pi*z)) * (1-cos(z))
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_J(n+QQ(1)/2, z)
def _eval_(self, n, m, theta, phi, **kwargs):
r"""
TESTS::
sage: x, y = var('x y')
sage: spherical_harmonic(1, 2, x, y)
0
sage: spherical_harmonic(1, -2, x, y)
0
sage: spherical_harmonic(1/2, 2, x, y)
spherical_harmonic(1/2, 2, x, y)
sage: spherical_harmonic(3, 2, x, y)
1/8*sqrt(30)*sqrt(7)*cos(x)*e^(2*I*y)*sin(x)^2/sqrt(pi)
sage: spherical_harmonic(3, 2, 1, 2)
1/8*sqrt(30)*sqrt(7)*cos(1)*e^(4*I)*sin(1)^2/sqrt(pi)
sage: spherical_harmonic(3 + I, 2., 1, 2)
-0.351154337307488 - 0.415562233975369*I
Check that :trac:`20939` is fixed::
sage: ex = spherical_harmonic(3,2,1,2*pi/3)
sage: QQbar(ex * sqrt(pi)/cos(1)/sin(1)^2).minpoly()
x^4 + 105/32*x^2 + 11025/1024
"""
if n in ZZ and m in ZZ and n > -1:
if abs(m) > n:
return ZZ(0)
if m == 0 and theta.is_zero():
return sqrt((2*n+1)/4/pi)
from sage.arith.misc import factorial
from sage.functions.trig import cos
from sage.functions.orthogonal_polys import gen_legendre_P
return (sqrt(factorial(n-m) * (2*n+1) / (4*pi * factorial(n+m))) *
exp(I*m*phi) * gen_legendre_P(n, m, cos(theta)) *
(-1)**m).simplify_trig()
def _i_rotation(self, z, alpha):
r"""
Return the resulting point after applying a hyperbolic
rotation centered at `0 + i` and angle ``alpha`` to ``z``.
INPUT:
- ``z``-- point in the upper complex halfplane to which
apply the isometry
- ``alpha``-- angle of rotation (radians,counterwise)
OUTPUT:
- rotated point in the upper complex halfplane
TESTS::
sage: from sage.plot.hyperbolic_regular_polygon import HyperbolicRegularPolygon
sage: P = HyperbolicRegularPolygon(4, pi/4, 1+I, {})
sage: P._i_rotation(2+I, pi/2)
I - 2
"""
_a = alpha / 2
_c = cos(_a)
_s = sin(_a)
G = matrix([[_c, _s], [-_s, _c]])
return (G[0][0] * z + G[0][1]) / (G[1][0] * z + G[1][1])
def __init__(self, coxeter_matrix, base_ring, index_set):
"""
Initialize ``self``.
EXAMPLES::
sage: W = CoxeterGroup([[1,3,2],[3,1,3],[2,3,1]])
sage: TestSuite(W).run() # long time
sage: W = CoxeterGroup([[1,3,2],[3,1,4],[2,4,1]], base_ring=QQbar)
sage: TestSuite(W).run() # long time
sage: W = CoxeterGroup([[1,3,2],[3,1,6],[2,6,1]])
sage: TestSuite(W).run(max_runs=30) # long time
sage: W = CoxeterGroup([[1,3,2],[3,1,-1],[2,-1,1]])
sage: TestSuite(W).run(max_runs=30) # long time
"""
self._matrix = coxeter_matrix
self._index_set = index_set
n = ZZ(coxeter_matrix.nrows())
MS = MatrixSpace(base_ring, n, sparse=True)
# FIXME: Hack because there is no ZZ \cup \{ \infty \}: -1 represents \infty
if base_ring is UniversalCyclotomicField():
val = lambda x: base_ring.gen(2*x) + ~base_ring.gen(2*x) if x != -1 else base_ring(2)
else:
from sage.functions.trig import cos
from sage.symbolic.constants import pi
val = lambda x: base_ring(2*cos(pi / x)) if x != -1 else base_ring(2)
gens = [MS.one() + MS({(i, j): val(coxeter_matrix[i, j])
for j in range(n)})
for i in range(n)]
FinitelyGeneratedMatrixGroup_generic.__init__(self, n, base_ring,
gens,
category=CoxeterGroups())
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 show(self, boundary=True, **options):
r"""
Plot ``self``.
EXAMPLES::
sage: HyperbolicPlane().UHP().get_geodesic(0, 1).show()
Graphics object consisting of 2 graphics primitives
"""
opts = {'axes': False, 'aspect_ratio': 1}
opts.update(self.graphics_options())
opts.update(options)
end_1, end_2 = [CC(k.coordinates()) for k in self.endpoints()]
bd_1, bd_2 = [CC(k.coordinates()) for k in self.ideal_endpoints()]
if (abs(real(end_1) - real(end_2)) < EPSILON) \
or CC(infinity) in [end_1, end_2]: #on same vertical line
# If one of the endpoints is infinity, we replace it with a
# large finite point
if end_1 == CC(infinity):
end_1 = (real(end_2), (imag(end_2) + 10))
end_2 = (real(end_2), imag(end_2))
elif end_2 == CC(infinity):
end_2 = (real(end_1), (imag(end_1) + 10))
end_1 = (real(end_1), imag(end_1))
from sage.plot.line import line
pic = line((end_1, end_2), **opts)
if boundary:
cent = min(bd_1, bd_2)
bd_dict = {'bd_min': cent - 3, 'bd_max': cent + 3}
bd_pic = self._model.get_background_graphic(**bd_dict)
pic = bd_pic + pic
return pic
else:
center = (bd_1 + bd_2)/2 # Circle center
radius = abs(bd_1 - bd_2)/2
theta1 = CC(end_1 - center).arg()
theta2 = CC(end_2 - center).arg()
if abs(theta1 - theta2) < EPSILON:
theta2 += pi
[theta1, theta2] = sorted([theta1, theta2])
from sage.calculus.var import var
from sage.plot.plot import parametric_plot
x = var('x')
pic = parametric_plot((radius*cos(x) + real(center),
radius*sin(x) + imag(center)),
(x, theta1, theta2), **opts)
if boundary:
# We want to draw a segment of the real line. The
# computations below compute the projection of the
# geodesic to the real line, and then draw a little
# to the left and right of the projection.
shadow_1, shadow_2 = [real(k) for k in [end_1, end_2]]
midpoint = (shadow_1 + shadow_2)/2
length = abs(shadow_1 - shadow_2)
bd_dict = {'bd_min': midpoint - length, 'bd_max': midpoint +
length}
bd_pic = self._model.get_background_graphic(**bd_dict)
pic = bd_pic + pic
return pic
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 plot(self):
from sage.functions.trig import sin, cos
from sage.plot.circle import circle
from sage.plot.point import point
from sage.plot.text import text
p = circle((0,0),1)
for i in range(self._dimension):
a = self._alpha[i]
p += point([cos(2*pi*a),sin(2*pi*a)], color='blue', size=100)
p += text(r"$\alpha_%i$"%(self._i_alpha[i]+1),
[1.2*cos(2*pi*a),1.2*sin(2*pi*a)],fontsize=40)
for i in range(self._dimension):
b = self._beta[i]
p += point([cos(2*pi*b),sin(2*pi*b)], color='red', size=100)
p += text(r"$\beta_%i$"%(self._i_beta[i]+1),
[1.2*cos(2*pi*b),1.2*sin(2*pi*b)],fontsize=40)
p.show(axes=False, xmin=-1, xmax=1, ymin=-1, ymax=1)
def plot(self, show_box=False, colors=["white","lightgray","darkgray"]):
r"""
Return a plot of ``self``.
INPUT:
- ``show_box`` -- boolean (default: ``False``); if ``True``,
also shows the visible tiles on the `xy`-, `yz`-, `zx`-planes
- ``colors`` -- (default: ``["white", "lightgray", "darkgray"]``)
list ``[A, B, C]`` of 3 strings representing colors
EXAMPLES::
sage: PP = PlanePartition([[4,3,3,1],[2,1,1],[1,1]])
sage: PP.plot()
Graphics object consisting of 27 graphics primitives
"""
x = self._max_x
y = self._max_y
z = self._max_z
from sage.functions.trig import cos, sin
from sage.plot.polygon import polygon
from sage.symbolic.constants import pi
from sage.plot.plot import plot
Uside = [[0,0], [cos(-pi/6),sin(-pi/6)], [0,-1], [cos(7*pi/6),sin(7*pi/6)]]
Lside = [[0,0], [cos(-pi/6),sin(-pi/6)], [cos(pi/6),sin(pi/6)], [0,1]]
Rside = [[0,0], [0,1], [cos(5*pi/6),sin(5*pi/6)], [cos(7*pi/6),sin(7*pi/6)]]
Xdir = [cos(7*pi/6), sin(7*pi/6)]
Ydir = [cos(-pi/6), sin(-pi/6)]
Zdir = [0, 1]
def move(side, i, j, k):
return [[P[0]+i*Xdir[0]+j*Ydir[0]+k*Zdir[0],
P[1]+i*Xdir[1]+j*Ydir[1]+k*Zdir[1]]
for P in side]
def add_topside(i, j, k):
return polygon(move(Uside,i,j,k), edgecolor="black", color=colors[0])
def add_leftside(i, j, k):
return polygon(move(Lside,i,j,k), edgecolor="black", color=colors[1])
def add_rightside(i, j, k):
return polygon(move(Rside,i,j,k), edgecolor="black", color=colors[2])
TP = plot([])
for r in range(len(self.z_tableau())):
for c in range(len(self.z_tableau()[r])):
if self.z_tableau()[r][c] > 0 or show_box:
TP += add_topside(r, c, self.z_tableau()[r][c])
for r in range(len(self.y_tableau())):
for c in range(len(self.y_tableau()[r])):
if self.y_tableau()[r][c] > 0 or show_box:
TP += add_rightside(c, self.y_tableau()[r][c], r)
for r in range(len(self.x_tableau())):
for c in range(len(self.x_tableau()[r])):
if self.x_tableau()[r][c] > 0 or show_box:
TP += add_leftside(self.x_tableau()[r][c], r, c)
TP.axes(show=False)
return TP
def __init__(self, sides, i_angle, center, options):
"""
Initialize HyperbolicRegularPolygon.
EXAMPLES::
sage: from sage.plot.hyperbolic_regular_polygon import HyperbolicRegularPolygon
sage: print(HyperbolicRegularPolygon(5,pi/2,I, {}))
Hyperbolic regular polygon (sides=5, i_angle=1/2*pi, center=1.00000000000000*I)
"""
self.center = CC(center)
if self.center.imag() <= 0 :
raise ValueError("center: %s is not a valid point in the upper half plane model of the hyperbolic plane"%(self.center))
if sides < 3 :
raise ValueError("degenerated polygons (sides<=2) are not supported")
if i_angle <=0 or i_angle >= pi:
raise ValueError("interior angle %s must be in (0, pi) interval"%(i_angle))
if pi*(sides-2) - sides*i_angle <= 0 :
raise ValueError("there exists no hyperbolic regular compact polygon, for sides=%s the interior angle must be less than %s"%(sides, pi * (sides-2) / sides))
self.sides = sides
self.i_angle = i_angle
beta = 2 * pi / self.sides # compute the rotation angle to be used ahead
alpha = self.i_angle / Integer(2)
I = CC(0, 1)
# compute using cosine theorem the radius of the circumscribed circle
# using the triangle formed by the radius and the three known angles
r = arccosh(cot(alpha) * (1 + cos(beta)) / sin(beta))
# The first point will be always on the imaginary axis limited
# to 8 digits for efficiency in the subsequent calculations.
z_0 = [I*(e**r).n(digits=8)]
# Compute the dilation isometry used to move the center
# from I to the imaginary part of the given center.
scale = self.center.imag()
# Compute the parabolic isometry to move the center to the
# real part of the given center.
h_disp = self.center.real()
d_z_k = [z_0[0]*scale + h_disp] #d_k has the points for the polygon in the given center
z_k = z_0 #z_k has the Re(z)>0 vertices for the I centered polygon
r_z_k = [] #r_z_k has the Re(z)<0 vertices
if is_odd(self.sides):
vert = (self.sides - 1) / 2
else:
vert = self.sides / 2 - 1
for k in range(0, vert):
# Compute with 8 digits to accelerate calculations
new_z_k = self._i_rotation(z_k[-1], beta).n(digits=8)
z_k = z_k + [new_z_k]
d_z_k = d_z_k + [new_z_k * scale + h_disp]
r_z_k=[-(new_z_k).conjugate() * scale + h_disp] + r_z_k
if is_odd(self.sides):
HyperbolicPolygon.__init__(self, d_z_k + r_z_k, options)
else:
z_opo = [I * (e**(-r)).n(digits=8) * scale + h_disp]
HyperbolicPolygon.__init__(self, d_z_k + z_opo + r_z_k, options)
def bilinear_form(self, R=None):
"""
Return the bilinear form over ``R`` associated to ``self``.
INPUT:
- ``R`` -- (default: universal cyclotomic field) a ring used to
compute the bilinear form
EXAMPLES::
sage: CoxeterType(['A', 2, 1]).bilinear_form()
[ 1 -1/2 -1/2]
[-1/2 1 -1/2]
[-1/2 -1/2 1]
sage: CoxeterType(['H', 3]).bilinear_form()
[ 1 -1/2 0]
[ -1/2 1 1/2*E(5)^2 + 1/2*E(5)^3]
[ 0 1/2*E(5)^2 + 1/2*E(5)^3 1]
sage: C = CoxeterMatrix([[1,-1,-1],[-1,1,-1],[-1,-1,1]])
sage: C.bilinear_form()
[ 1 -1 -1]
[-1 1 -1]
[-1 -1 1]
"""
n = self.rank()
mat = self.coxeter_matrix()._matrix
base_ring = mat.base_ring()
from sage.rings.universal_cyclotomic_field import UniversalCyclotomicField
UCF = UniversalCyclotomicField()
if UCF.has_coerce_map_from(base_ring):
R = UCF
else:
R = base_ring
# Compute the matrix with entries `- \cos( \pi / m_{ij} )`.
if R is UCF:
val = lambda x: (R.gen(2 * x) + ~R.gen(2 * x)) / R(-2) if x > -1 else R.one() * x
else:
from sage.functions.trig import cos
from sage.symbolic.constants import pi
val = lambda x: -R(cos(pi / SR(x))) if x > -1 else x
MS = MatrixSpace(R, n, sparse=True)
MC = MS._get_matrix_class()
bilinear = MC(
MS,
entries={(i, j): val(mat[i, j]) for i in range(n) for j in range(n) if mat[i, j] != 2},
coerce=True,
copy=True,
)
bilinear.set_immutable()
return bilinear
def plot(self, **kwargs):
"""
Return a graphics object representing the Kontsevich graph.
INPUT:
- ``edge_labels`` (boolean, default True) -- if True, show edge labels.
- ``indices`` (boolean, default False) -- if True, show indices as
edge labels instead of L and R; see :meth:`._latex_`.
- ``upright`` (boolean, default False) -- if True, try to plot the
graph with the ground vertices on the bottom and the rest on top.
"""
if not 'edge_labels' in kwargs:
kwargs['edge_labels'] = True # show edge labels by default
if 'indices' in kwargs:
del kwargs['indices']
KG = DiGraph(self)
for (k,e) in enumerate(self.edges()):
KG.delete_edge(e)
KG.add_edge((e[0], e[1], chr(97 + k)))
return KG.plot(**kwargs)
if len(self.ground_vertices()) == 2 and 'upright' in kwargs:
del kwargs['upright']
kwargs['save_pos'] = True
DiGraph.plot(self, **kwargs)
positions = self.get_pos()
# translate F to origin:
F_pos = vector(positions[self.ground_vertices()[0]])
for p in positions:
positions[p] = list(vector(positions[p]) - F_pos)
# scale F - G distance to 1:
G_len = abs(vector(positions[self.ground_vertices()[1]]))
for p in positions:
positions[p] = list(vector(positions[p])/G_len)
# rotate the vector F - G to (1,0)
from math import atan2
theta = -atan2(positions[self.ground_vertices()[1]][1], positions[self.ground_vertices()[1]][0])
for p in positions:
positions[p] = list(matrix([[cos(theta),-(sin(theta))],[sin(theta),cos(theta)]]) * vector(positions[p]))
# flip if most things are below the x-axis:
if len([(x,y) for (x,y) in positions.values() if y < 0])/len(self.internal_vertices()) > 0.5:
for p in positions:
positions[p] = [positions[p][0], -positions[p][1]]
return DiGraph.plot(self, **kwargs)
def _derivative_(self, x, diff_param=None):
"""
EXAMPLES::
sage: x = var('x')
sage: fresnel_cos(x).diff(x)
cos(1/2*pi*x^2)
"""
from sage.functions.trig import cos
return cos(pi*x**2/2)
def Ellipsoid(center=(0,0,0), axes=(1,1,1), name="Ellipsoid"):
r"""
Returns an ellipsoid centered at ``center`` whose semi-principal axes
have lengths given by the components of ``axes``. The
parametrization of the ellipsoid is given by
.. MATH::
\begin{aligned}
x(u, v) & = x_0 + a \cos(u) \cos(v); \\
y(u, v) & = y_0 + b \sin(u) \cos(v); \\
z(u, v) & = z_0 + c \sin(v).
\end{aligned}
INPUT:
- ``center`` -- 3-tuple. Coordinates of the center of the ellipsoid.
- ``axes`` -- 3-tuple. Lengths of the semi-principal axes.
- ``name`` -- string. Name of the ellipsoid.
EXAMPLES::
sage: ell = surfaces.Ellipsoid(axes=(1, 2, 3)); ell
Parametrized surface ('Ellipsoid') with equation (cos(u)*cos(v), 2*cos(v)*sin(u), 3*sin(v))
sage: ell.plot()
"""
u, v = var ('u, v')
x, y, z = center
a, b, c = axes
ellipsoid_parametric_eq = [x + a*cos(u)*cos(v),
y + b*sin(u)*cos(v),
z + c*sin(v)]
coords = ((u, 0, 2*pi), (v, -pi/2, pi/2))
return ParametrizedSurface3D(ellipsoid_parametric_eq, coords, name)
def transform(self, radius=None, azimuth=None, height=None):
"""
A cylindrical coordinates transform.
EXAMPLE::
sage: T = Cylindrical('height', ['azimuth', 'radius'])
sage: T.transform(radius=var('r'), azimuth=var('theta'), height=var('z'))
(r*cos(theta), r*sin(theta), z)
"""
return (radius * cos(azimuth),
radius * sin(azimuth),
height)
def _derivative_(self, z, diff_param=None):
r"""
The derivative of `\operatorname{Ci}(z)` is `\cos(z)/z` if `z` is not zero.
EXAMPLES::
sage: x = var('x')
sage: f = cos_integral(x)
sage: f.diff(x)
cos(x)/x
sage: f = cos_integral(x^2)
sage: f.diff(x)
2*cos(x^2)/x
"""
return cos(z)/z
def photon_orbit_radius(self, retrograde=False):
r"""
Return the Boyer-Lindquist radial coordinate of the circular orbit
of photons in the equatorial plane.
INPUT:
- ``retrograde`` -- (default: ``False``) boolean determining whether
retrograde or prograde (direct) orbits are considered
OUTPUT:
- Boyer-Lindquist radial coordinate `r` of the circular orbit of
photons in the equatorial plane
EXAMPLES::
sage: from kerrgeodesic_gw import KerrBH
sage: a, m = var('a m')
sage: BH = KerrBH(a, m)
sage: BH.photon_orbit_radius()
2*m*(cos(2/3*arccos(-a/m)) + 1)
sage: BH.photon_orbit_radius(retrograde=True)
2*m*(cos(2/3*arccos(a/m)) + 1)
Photon orbit in Schwarzschild spacetime::
sage: KerrBH(0, m).photon_orbit_radius()
3*m
Photon orbits in extreme Kerr spacetime (`a=m`)::
sage: KerrBH(m, m).photon_orbit_radius()
m
sage: KerrBH(m, m).photon_orbit_radius(retrograde=True)
4*m
"""
m = self._m
a = self._a
eps = -1 if not retrograde else 1
# Eq. (2.18) in Bardeen, Press & Teukolsky, ApJ 178, 347 (1972)
return 2 * m * (1 + cos(2 * acos(eps * a / m) / 3))
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
from sage.functions.trig import (cos, sin)
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)]
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 __init__(self, coxeter_matrix, base_ring, index_set):
"""
Initialize ``self``.
EXAMPLES::
sage: W = CoxeterGroup([[1,3,2],[3,1,3],[2,3,1]])
sage: TestSuite(W).run() # long time
sage: W = CoxeterGroup([[1,3,2],[3,1,4],[2,4,1]], base_ring=QQbar)
sage: TestSuite(W).run() # long time
sage: W = CoxeterGroup([[1,3,2],[3,1,6],[2,6,1]])
sage: TestSuite(W).run(max_runs=30) # long time
sage: W = CoxeterGroup([[1,3,2],[3,1,-1],[2,-1,1]])
sage: TestSuite(W).run(max_runs=30) # long time
"""
self._matrix = coxeter_matrix
self._index_set = index_set
n = ZZ(coxeter_matrix.nrows())
MS = MatrixSpace(base_ring, n, sparse=True)
# FIXME: Hack because there is no ZZ \cup \{ \infty \}: -1 represents \infty
if base_ring is UniversalCyclotomicField():
val = lambda x: base_ring.gen(2 * x) + ~base_ring.gen(
2 * x) if x != -1 else base_ring(2)
else:
from sage.functions.trig import cos
from sage.symbolic.constants import pi
val = lambda x: base_ring(2 * cos(pi / x)
) if x != -1 else base_ring(2)
gens = [
MS.one() + MS({(i, j): val(coxeter_matrix[i, j])
for j in range(n)}) for i in range(n)
]
FinitelyGeneratedMatrixGroup_generic.__init__(self,
n,
base_ring,
gens,
category=CoxeterGroups())
def regular_polygon(self, n, base_ring=QQ):
"""
Return a regular polygon with n vertices. Over the rational
field the vertices may not be exact.
INPUT:
- ``n`` -- a positive integer, the number of vertices.
- ``field`` -- either ``QQ`` or ``RDF``.
EXAMPLES::
sage: octagon = polytopes.regular_polygon(8)
sage: len(octagon.vertices())
8
"""
npi = 3.14159265359
verts = []
for i in range(n):
t = 2*npi*i/n
verts.append([sin(t),cos(t)])
verts = [[base_ring(RDF(x)) for x in y] for y in verts]
return Polyhedron(vertices=verts, base_ring=base_ring)
def _eval_(self, n, m, theta, phi, **kwargs):
r"""
TESTS::
sage: x, y = var('x y')
sage: spherical_harmonic(1, 2, x, y)
0
sage: spherical_harmonic(1, -2, x, y)
0
sage: spherical_harmonic(1/2, 2, x, y)
spherical_harmonic(1/2, 2, x, y)
sage: spherical_harmonic(3, 2, x, y)
1/8*sqrt(30)*sqrt(7)*cos(x)*e^(2*I*y)*sin(x)^2/sqrt(pi)
sage: spherical_harmonic(3, 2, 1, 2)
1/8*sqrt(30)*sqrt(7)*cos(1)*e^(4*I)*sin(1)^2/sqrt(pi)
sage: spherical_harmonic(3 + I, 2., 1, 2)
-0.351154337307488 - 0.415562233975369*I
Check that :trac:`20939` is fixed::
sage: ex = spherical_harmonic(3,2,1,2*pi/3)
sage: QQbar(ex * sqrt(pi)/cos(1)/sin(1)^2).minpoly()
x^4 + 105/32*x^2 + 11025/1024
"""
if n in ZZ and m in ZZ and n > -1:
if abs(m) > n:
return ZZ(0)
if m == 0 and theta.is_zero():
return sqrt((2 * n + 1) / 4 / pi)
from sage.arith.misc import factorial
from sage.functions.trig import cos
from sage.functions.orthogonal_polys import gen_legendre_P
return (sqrt(
factorial(n - m) * (2 * n + 1) / (4 * pi * factorial(n + m))) *
exp(I * m * phi) * gen_legendre_P(n, m, cos(theta)) *
(-1)**m).simplify_trig()
def regular_polygon(self, n, base_ring=QQ):
"""
Return a regular polygon with n vertices. Over the rational
field the vertices may not be exact.
INPUT:
- ``n`` -- a positive integer, the number of vertices.
- ``field`` -- either ``QQ`` or ``RDF``.
EXAMPLES::
sage: octagon = polytopes.regular_polygon(8)
sage: len(octagon.vertices())
8
"""
npi = 3.14159265359
verts = []
for i in range(n):
t = 2 * npi * i / n
verts.append([sin(t), cos(t)])
verts = [[base_ring(RDF(x)) for x in y] for y in verts]
return Polyhedron(vertices=verts, base_ring=base_ring)
def f(t):
return exp(I * ((cos(t) + I * sin(t)) *
arc_radius + arc_center)) * radius
raise ValueError("Ellipse not implemented")
first_unit_vector = first_vector / radius
second_unit_vector = second_vector / radius
normal_vector = second_vector - (second_vector *
first_unit_vector) * first_unit_vector
if norm(normal_vector) == 0:
print(first_point, second_point)
return
normal_unit_vector = normal_vector / norm(normal_vector)
scalar_product = first_unit_vector * second_unit_vector
if abs(scalar_product) == 1:
raise ValueError("The points are alligned")
angle = arccos(scalar_product)
var('t')
return parametric_plot3d(
center + first_vector * cos(t) + radius * normal_unit_vector * sin(t),
(0, angle), **kwds)
def _arc(p, q, s, **kwds):
#rewrite this to use polar_plot and get points to do filled triangles
from sage.misc.functional import det
from sage.plot.line import line
from sage.misc.functional import norm
from sage.symbolic.all import pi
from sage.plot.arc import arc
p, q, s = map(lambda x: vector(x), [p, q, s])
# to avoid running into division by 0 we set to be colinear vectors that are
# almost colinear
def surface_density_gaussian(r, phi, param):
r"""
Surface density of a matter blob with a Gaussian profile
INPUT:
- ``r`` -- Boyer-Lindquist radial coordinate `\bar{r}` in the matter blob
- ``phi`` -- Boyer-Lindquist azimuthal coordinate `\bar{\phi}` in the
matter blob
- ``param`` -- list of parameters defining the position and width of
matter blob:
- ``param[0]``: mean radius `r_0` (Boyer-Lindquist coordinate)
- ``param[1]``: mean azimuthal angle `\phi_0` (Boyer-Lindquist
coordinate)
- ``param[2]``: width `\lambda` of the Gaussian profile
- ``param[3]`` (optional): amplitude `\Sigma_0`; if not provided,
then `\Sigma_0=1` is used
OUTPUT:
- surface density `\Sigma(\bar{r}, \bar{\phi})`
EXAMPLES::
sage: from kerrgeodesic_gw import surface_density_gaussian
sage: param = [6.5, 0., 0.3]
sage: surface_density_gaussian(6.5, 0, param)
1.0
sage: surface_density_gaussian(8., 0, param) # tol 1.0e-13
1.3887943864964021e-11
sage: surface_density_gaussian(6.5, pi/16, param) # tol 1.0e-13
1.4901161193847656e-08
3D representation: `z=\Sigma(\bar{r}, \bar{\phi})` in terms of
`x:=\bar{r}\cos\bar\phi` and `y:=\bar{r}\sin\bar\phi`::
sage: s_plot = lambda r, phi: surface_density_gaussian(r, phi, param)
sage: r, phi, z = var('r phi z')
sage: plot3d(s_plot, (r, 6, 8), (phi, -0.4, 0.4),
....: transformation=(r*cos(phi), r*sin(phi), z))
Graphics3d Object
.. PLOT::
from kerrgeodesic_gw import surface_density_gaussian
param = param = [6.5, 0., 0.3]
s_plot = lambda r, phi: surface_density_gaussian(r, phi, param)
r, phi, z = var('r phi z')
g = plot3d(s_plot, (r, 6, 8), (phi, -0.4, 0.4), \
transformation=(r*cos(phi), r*sin(phi), z))
sphinx_plot(g)
Use with a non-default amplitude (`\Sigma_0=10^{-5}`)::
sage: sigma0 = 1.e-5
sage: param = [6.5, 0., 0.3, sigma0]
sage: surface_density_gaussian(6.5, 0, param)
1e-05
"""
r0, phi0, lam = param[0], param[1], param[2]
Sigma0 = param[3] if len(param) == 4 else float(1)
return float(Sigma0*exp(-((r - r0*cos(phi-phi0))**2 +
(r0*sin(phi-phi0))**2)/lam**2))
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 revolution_plot3d(curve,
trange,
phirange=None,
parallel_axis='z',
axis=(0, 0),
print_vector=False,
show_curve=False,
**kwds):
r"""
Return a plot of a revolved curve.
There are three ways to call this function:
- ``revolution_plot3d(f,trange)`` where `f` is a function located in the `x z` plane.
- ``revolution_plot3d((f_x,f_z),trange)`` where `(f_x,f_z)` is a parametric curve on the `x z` plane.
- ``revolution_plot3d((f_x,f_y,f_z),trange)`` where `(f_x,f_y,f_z)` can be any parametric curve.
INPUT:
- ``curve`` - A curve to be revolved, specified as a function, a 2-tuple or a 3-tuple.
- ``trange`` - A 3-tuple `(t,t_{\min},t_{\max})` where t is the independent variable of the curve.
- ``phirange`` - A 2-tuple of the form `(\phi_{\min},\phi_{\max})`, (default `(0,\pi)`) that specifies the angle in which the curve is to be revolved.
- ``parallel_axis`` - A string (Either 'x', 'y', or 'z') that specifies the coordinate axis parallel to the revolution axis.
- ``axis`` - A 2-tuple that specifies the position of the revolution axis. If parallel is:
- 'z' - then axis is the point in which the revolution axis intersects the `x y` plane.
- 'x' - then axis is the point in which the revolution axis intersects the `y z` plane.
- 'y' - then axis is the point in which the revolution axis intersects the `x z` plane.
- ``print_vector`` - If True, the parametrization of the surface of revolution will be printed.
- ``show_curve`` - If True, the curve will be displayed.
EXAMPLES:
Let's revolve a simple function around different axes::
sage: u = var('u')
sage: f = u^2
sage: revolution_plot3d(f, (u,0,2), show_curve=True, opacity=0.7).show(aspect_ratio=(1,1,1))
.. PLOT::
u = var('u')
f = u**2
P = revolution_plot3d(f, (u,0,2), show_curve=True, opacity=0.7).plot()
sphinx_plot(P)
If we move slightly the axis, we get a goblet-like surface::
sage: revolution_plot3d(f, (u,0,2), axis=(1,0.2), show_curve=True, opacity=0.5).show(aspect_ratio=(1,1,1))
.. PLOT::
u = var('u')
f = u**2
P = revolution_plot3d(f, (u,0,2), axis=(1,0.2), show_curve=True, opacity=0.5).plot()
sphinx_plot(P)
A common problem in calculus books, find the volume within the following revolution solid::
sage: line = u
sage: parabola = u^2
sage: sur1 = revolution_plot3d(line, (u,0,1), opacity=0.5, rgbcolor=(1,0.5,0), show_curve=True, parallel_axis='x')
sage: sur2 = revolution_plot3d(parabola, (u,0,1), opacity=0.5, rgbcolor=(0,1,0), show_curve=True, parallel_axis='x')
sage: (sur1+sur2).show()
.. PLOT::
u = var('u')
line = u
parabola = u**2
sur1 = revolution_plot3d(line, (u,0,1), opacity=0.5, rgbcolor=(1,0.5,0), show_curve=True, parallel_axis='x')
sur2 = revolution_plot3d(parabola, (u,0,1), opacity=0.5, rgbcolor=(0,1,0), show_curve=True, parallel_axis='x')
P = sur1 + sur2
sphinx_plot(P)
Now let's revolve a parametrically defined circle. We can play with the topology of the surface by changing the axis,
an axis in `(0,0)` (as the previous one) will produce a sphere-like surface::
sage: u = var('u')
sage: circle = (cos(u), sin(u))
sage: revolution_plot3d(circle, (u,0,2*pi), axis=(0,0), show_curve=True, opacity=0.5).show(aspect_ratio=(1,1,1))
.. PLOT::
u = var('u')
circle = (cos(u), sin(u))
P = revolution_plot3d(circle, (u,0,2*pi), axis=(0,0), show_curve=True, opacity=0.5)
sphinx_plot(P)
An axis on `(0,y)` will produce a cylinder-like surface::
sage: revolution_plot3d(circle, (u,0,2*pi), axis=(0,2), show_curve=True, opacity=0.5).show(aspect_ratio=(1,1,1))
.. PLOT::
u = var('u')
circle = (cos(u), sin(u))
P = revolution_plot3d(circle, (u,0,2*pi), axis=(0,2), show_curve=True, opacity=0.5)
sphinx_plot(P)
And any other axis will produce a torus-like surface::
sage: revolution_plot3d(circle, (u,0,2*pi), axis=(2,0), show_curve=True, opacity=0.5).show(aspect_ratio=(1,1,1))
.. PLOT::
u = var('u')
circle = (cos(u), sin(u))
P = revolution_plot3d(circle, (u,0,2*pi), axis=(2,0), show_curve=True, opacity=0.5)
sphinx_plot(P)
Now, we can get another goblet-like surface by revolving a curve in 3d::
sage: u = var('u')
sage: curve = (u, cos(4*u), u^2)
sage: P = revolution_plot3d(curve, (u,0,2), show_curve=True, parallel_axis='z',axis=(1,.2), opacity=0.5)
sage: P.show(aspect_ratio=(1,1,1))
.. PLOT::
u = var('u')
curve = (u, cos(4*u), u**2)
P = revolution_plot3d(curve, (u,0,2), show_curve=True, parallel_axis='z', axis=(1,.2), opacity=0.5)
sphinx_plot(P)
A curvy curve with only a quarter turn::
sage: u = var('u')
sage: curve = (sin(3*u), .8*cos(4*u), cos(u))
sage: revolution_plot3d(curve, (u,0,pi), (0,pi/2), show_curve=True, parallel_axis='z', opacity=0.5).show(aspect_ratio=(1,1,1),frame=False)
.. PLOT::
u = var('u')
curve = (sin(3*u), .8*cos(4*u), cos(u))
P = revolution_plot3d(curve, (u,0,pi), (0,pi/2), show_curve=True, parallel_axis='z', opacity=0.5)
sphinx_plot(P)
One can also color the surface using a coloring function of two
parameters and a colormap as follows. Note that the coloring
function must take values in the interval [0,1]. ::
sage: u, phi = var('u,phi')
sage: def cf(u,phi): return sin(phi+u) ^ 2
sage: curve = (1+u^2/4, 0, u)
sage: revolution_plot3d(curve, (u,-2,2), (0,2*pi), parallel_axis='z', color=(cf, colormaps.PiYG)).show(aspect_ratio=(1,1,1))
.. PLOT::
u, phi = var('u,phi')
def cf(u, phi): return sin(phi+u) ** 2
curve = (1+u**2/4, 0, u)
P = revolution_plot3d(curve, (u,-2,2), (0,2*pi), parallel_axis='z', color=(cf, colormaps.PiYG))
sphinx_plot(P)
The first parameter of the coloring function will be identified with the
parameter of the curve, and the second with the angle parameter.
.. WARNING::
This kind of coloring using a colormap can be visualized using
Jmol, Tachyon (option ``viewer='tachyon'``) and Canvas3D
(option ``viewer='canvas3d'`` in the notebook).
Another colored example, illustrating that one can use (colormap, color function) instead of (color function, colormap)::
sage: u, phi = var('u,phi')
sage: def cf(u, phi): return float(2 * u / pi) % 1
sage: curve = (sin(u), 0, u)
sage: revolution_plot3d(curve, (u,0,pi), (0,2*pi), parallel_axis
....: ='z', color=(colormaps.brg, cf)).show(aspect_ratio=1)
.. PLOT::
u, phi = var('u,phi')
def cf(u, phi): return float(2 * u / pi) % 1
curve = (sin(u), 0, u)
P = revolution_plot3d(curve, (u,0,pi), (0,2*pi), parallel_axis='z', color=(colormaps.brg, cf))
sphinx_plot(P)
"""
from sage.symbolic.ring import SR
from sage.symbolic.constants import pi
from sage.misc.functional import sqrt
from sage.functions.trig import sin
from sage.functions.trig import cos
from sage.functions.trig import atan2
if parallel_axis not in ['x', 'y', 'z']:
raise ValueError("parallel_axis must be either 'x', 'y', or 'z'.")
vart = trange[0]
if str(vart) == 'phi':
phi = SR.var('fi')
else:
phi = SR.var('phi')
if phirange is None: # this if-else provides a phirange
phirange = (phi, 0, 2 * pi)
elif len(phirange) == 3:
phi = phirange[0]
else:
phirange = (phi, phirange[0], phirange[1])
if isinstance(curve, (tuple, list)):
#this if-else provides a vector v to be plotted
#if curve is a tuple or a list of length 2, it is interpreted as a parametric curve
#in the x-z plane.
#if it is of length 3 it is interpreted as a parametric curve in 3d space
if len(curve) == 2:
x = curve[0]
y = 0
z = curve[1]
elif len(curve) == 3:
x = curve[0]
y = curve[1]
z = curve[2]
else:
x = vart
y = 0
z = curve
phase = 0
if parallel_axis == 'z':
x0 = axis[0]
y0 = axis[1]
# (0,0) must be handled separately for the phase value
if x0 != 0 or y0 != 0:
phase = atan2(y - y0, x - x0)
R = sqrt((x - x0)**2 + (y - y0)**2)
v = (R * cos(phi + phase) + x0, R * sin(phi + phase) + y0, z)
elif parallel_axis == 'x':
y0 = axis[0]
z0 = axis[1]
# (0,0) must be handled separately for the phase value
if z0 != 0 or y0 != 0:
phase = atan2(z - z0, y - y0)
R = sqrt((y - y0)**2 + (z - z0)**2)
v = (x, R * cos(phi + phase) + y0, R * sin(phi + phase) + z0)
elif parallel_axis == 'y':
x0 = axis[0]
z0 = axis[1]
# (0,0) must be handled separately for the phase value
if z0 != 0 or x0 != 0:
phase = atan2(z - z0, x - x0)
R = sqrt((x - x0)**2 + (z - z0)**2)
v = (R * cos(phi + phase) + x0, y, R * sin(phi + phase) + z0)
if print_vector:
print(v)
if show_curve:
curveplot = parametric_plot3d((x, y, z),
trange,
thickness=2,
rgbcolor=(1, 0, 0))
return parametric_plot3d(v, trange, phirange, **kwds) + curveplot
return parametric_plot3d(v, trange, phirange, **kwds)
def Sphere(dim=None, radius=1, names=None, stereo2d=False, stereo_lim=None):
"""
Generate a sphere embedded in Euclidean space.
The shortcut operator ``.<,>`` can be used to specify the coordinates.
INPUT:
- ``dim`` -- (optional) the dimension of the sphere; if not specified,
equals to the number of coordinate names
- ``radius`` -- (default: ``1``) radius of the sphere
- ``names`` -- (default: ``None``) name of the coordinates,
automatically set by the shortcut operator
- ``stereo2d`` -- (default: ``False``) if ``True``, defines only the
stereographic charts, only implemented in 2d
- ``stereo_lim`` -- (default: ``None``) parameter used to restrict the
span of the stereographic charts, so that they don't cover the whole
sphere; valid domain will be ``x**2 + y**2 < stereo_lim**2``
OUTPUT:
- Riemannian manifold
EXAMPLES::
sage: S.<th, ph> = manifolds.Sphere()
sage: S
2-dimensional Riemannian submanifold S embedded in the Euclidean
space E^3
sage: S.atlas()
[Chart (S, (th, ph))]
sage: S.metric().display()
gamma = dth*dth + sin(th)^2 dph*dph
sage: S = manifolds.Sphere(2, stereo2d=True) # long time
sage: S # long time
2-dimensional Riemannian submanifold S embedded in the Euclidean
space E^3
sage: S.metric().display() # long time
gamma = 4/(x^4 + y^4 + 2*(x^2 + 1)*y^2 + 2*x^2 + 1) dx*dx
+ 4/(x^4 + y^4 + 2*(x^2 + 1)*y^2 + 2*x^2 + 1) dy*dy
"""
from sage.functions.trig import cos, sin, atan, atan2
from sage.functions.other import sqrt
from sage.symbolic.constants import pi
from sage.misc.misc_c import prod
from sage.manifolds.manifold import Manifold
from sage.manifolds.differentiable.euclidean import EuclideanSpace
if dim is None:
if names is None:
raise ValueError("either the names or the dimension must be specified")
dim = len(names)
else:
if names is not None and dim != len(names):
raise ValueError("the number of coordinates does not match the dimension")
if stereo2d:
if dim != 2:
raise NotImplementedError("stereographic charts only "
"implemented for 2d spheres")
E = EuclideanSpace(3, names=("X", "Y", "Z"))
S2 = Manifold(dim, 'S', ambient=E, structure='Riemannian')
U = S2.open_subset('U')
V = S2.open_subset('V')
stereoN = U.chart(names=("x", "y"))
x, y = stereoN[:]
stereoS = V.chart(names=("xp", "yp"))
xp, yp = stereoS[:]
if stereo_lim is not None:
stereoN.add_restrictions(x**2+y**2 < stereo_lim**2)
stereoS.add_restrictions(xp**2+yp**2 < stereo_lim**2)
stereoN_to_S = stereoN.transition_map(stereoS,
(x / (x**2 + y**2), y / (x**2 + y**2)), intersection_name='W',
restrictions1=x**2 + y**2 != 0, restrictions2=xp**2+yp**2!=0)
stereoN_to_S.set_inverse(xp / (xp**2 + yp**2), yp / (xp**2 + yp**2),
check=False)
W = U.intersection(V)
stereoN_W = stereoN.restrict(W)
stereoS_W = stereoS.restrict(W)
A = W.open_subset('A', coord_def={stereoN_W: (y != 0, x < 0),
stereoS_W: (yp != 0, xp < 0)})
stereoN_A = stereoN_W.restrict(A)
if names is None:
names = tuple(["phi_{}:(0,pi)".format(i) for i in range(dim - 1)] +
["phi_{}:(-pi,pi):periodic".format(dim - 1)])
else:
names = tuple([names[i] + ":(0,pi)" for i in range(dim - 1)] +
[names[dim - 1] + ":(-pi,pi):periodic"])
spher = A.chart(names=names)
th, ph = spher[:]
spher_to_stereoN = spher.transition_map(stereoN_A, (sin(th)*cos(ph) / (1-cos(th)),
sin(th)*sin(ph) / (1-cos(th))))
spher_to_stereoN.set_inverse(2*atan(1/sqrt(x**2+y**2)), atan2(-y, -x)+pi,
check=False)
stereoN_to_S_A = stereoN_to_S.restrict(A)
stereoN_to_S_A * spher_to_stereoN # generates spher_to_stereoS
stereoS_to_N_A = stereoN_to_S.inverse().restrict(A)
spher_to_stereoN.inverse() * stereoS_to_N_A # generates stereoS_to_spher
coordfunc1 = [sin(th)*cos(ph), sin(th)*sin(ph), cos(th)]
coordfunc2 = [2*x/(1+x**2+y**2), 2*y/(1+x**2+y**2), (x**2+y**2-1)/(1+x**2+y**2)]
coordfunc3 = [2*xp/(1+xp**2+yp**2), 2*yp/(1+xp**2+yp**2),(1-xp**2-yp**2)/(1+xp**2+yp**2)]
imm = S2.diff_map(E, {(spher, E.default_chart()): coordfunc1,
(stereoN, E.default_chart()): coordfunc2,
(stereoS, E.default_chart()): coordfunc3})
S2.set_embedding(imm)
S2.induced_metric()
return S2
if dim != 2:
raise NotImplementedError("only implemented for 2 dimensional spheres")
E = EuclideanSpace(3, symbols='X Y Z')
M = Manifold(dim, 'S', ambient=E, structure='Riemannian')
if names is None:
names = tuple(["phi_{}:(0,pi)".format(i) for i in range(dim-1)] +
["phi_{}:(-pi,pi):periodic".format(dim-1)])
else:
names = tuple([names[i]+":(0,pi)"for i in range(dim - 1)] +
[names[dim-1]+":(-pi,pi):periodic"])
C = M.chart(names=names)
M._first_ngens = C._first_ngens
phi = M._first_ngens(dim)[:]
coordfunc = ([radius * prod(sin(phi[j]) for j in range(dim))] +
[radius * cos(phi[i]) * prod(sin(phi[j]) for j in range(i))
for i in range(dim)])
imm = M.diff_map(E, coordfunc)
M.set_embedding(imm)
M.induced_metric()
return M
def val(x):
if x == -1:
return 2
else:
return base_ring(2 * cos(pi / x))
def barycentric_projection_matrix(n, angle=0):
r"""
Returns a family of `n+1` vectors evenly spaced in a real vector space of dimension `n`
Those vectors are of norm `1`, the scalar product between any two
vector is `1/n`, thus the distance between two tips is constant.
The family is built recursively and uniquely determined by the
following property: the last vector is `(0,\dots,0,-1)`, and the
projection of the first `n` vectors in dimension `n-1`, after
appropriate rescaling to norm `1`, retrieves the family for `n-1`.
OUTPUT:
A matrix with `n+1` columns of height `n` with rational or
symbolic coefficients.
EXAMPLES:
One vector in dimension `0`::
sage: from sage.combinat.root_system.root_lattice_realizations import barycentric_projection_matrix
sage: m = barycentric_projection_matrix(0); m
[]
sage: matrix(QQ,0,1).nrows()
0
sage: matrix(QQ,0,1).ncols()
1
Two vectors in dimension 1::
sage: barycentric_projection_matrix(1)
[ 1 -1]
Three vectors in dimension 2::
sage: barycentric_projection_matrix(2)
[ 1/2*sqrt(3) -1/2*sqrt(3) 0]
[ 1/2 1/2 -1]
Four vectors in dimension 3::
sage: m = barycentric_projection_matrix(3); m
[ 1/3*sqrt(3)*sqrt(2) -1/3*sqrt(3)*sqrt(2) 0 0]
[ 1/3*sqrt(2) 1/3*sqrt(2) -2/3*sqrt(2) 0]
[ 1/3 1/3 1/3 -1]
The columns give four vectors that sum up to zero::
sage: sum(m.columns())
(0, 0, 0)
and have regular mutual angles::
sage: m.transpose()*m
[ 1 -1/3 -1/3 -1/3]
[-1/3 1 -1/3 -1/3]
[-1/3 -1/3 1 -1/3]
[-1/3 -1/3 -1/3 1]
Here is a plot of them::
sage: sum(arrow((0,0,0),x) for x in m.columns())
For 2D drawings of root systems, it is desirable to rotate the
result to match with the usual conventions::
sage: barycentric_projection_matrix(2, angle=2*pi/3)
[ 1/2 -1 1/2]
[ 1/2*sqrt(3) 0 -1/2*sqrt(3)]
TESTS::
sage: for n in range(1, 7):
... m = barycentric_projection_matrix(n)
... assert sum(m.columns()).is_zero()
... assert matrix(QQ, n+1,n+1, lambda i,j: 1 if i==j else -1/n) == m.transpose()*m
"""
from sage.matrix.constructor import matrix
from sage.functions.other import sqrt
n = ZZ(n)
if n == 0:
return matrix(QQ, 0, 1)
a = 1 / n
b = sqrt(1 - a**2)
result = b * barycentric_projection_matrix(n - 1)
result = result.augment(vector([0] * (n - 1)))
result = result.stack(matrix([[a] * n + [-1]]))
assert sum(result.columns()).is_zero()
if angle and n == 2:
from sage.functions.trig import sin
from sage.functions.trig import cos
rotation = matrix([[sin(angle), cos(angle)], [-cos(angle),
sin(angle)]])
result = rotation * result
result.set_immutable()
return result
def _compute_sw_spherical_harm(s,
l,
m,
theta,
phi,
condon_shortley=True,
numerical=None):
r"""
Compute the spin-weighted spherical harmonic of spin weight ``s`` and
indices ``(l,m)`` as a callable symbolic expression in (theta,phi)
INPUT:
- ``s`` -- integer; the spin weight
- ``l`` -- non-negative integer; the harmonic degree
- ``m`` -- integer within the range ``[-l, l]``; the azimuthal number
- ``theta`` -- colatitude angle
- ``phi`` -- azimuthal angle
- ``condon_shortley`` -- (default: ``True``) determines whether the
Condon-Shortley phase of `(-1)^m` is taken into account (see below)
- ``numerical`` -- (default: ``None``) determines whether a symbolic or
a numerical computation of a given type is performed; allowed values are
- ``None``: a symbolic computation is performed
- ``RDF``: Sage's machine double precision floating-point numbers
(``RealDoubleField``)
- ``RealField(n)``, where ``n`` is a number of bits: Sage's
floating-point numbers with an arbitrary precision; note that ``RR`` is
a shortcut for ``RealField(53)``.
- ``float``: Python's floating-point numbers
OUTPUT:
- `{}_s Y_l^m(\theta,\phi)` either
- as a symbolic expression if ``numerical`` is ``None``
- or a pair of floating-point numbers, each of them being of the type
corresponding to ``numerical`` and representing respectively the
real and imaginary parts of `{}_s Y_l^m(\theta,\phi)`
ALGORITHM:
The spin-weighted spherical harmonic is evaluated according to Eq. (3.1)
of J. N. Golberg et al., J. Math. Phys. **8**, 2155 (1967)
[:doi:`10.1063/1.1705135`], with an extra `(-1)^m` factor (the so-called
*Condon-Shortley phase*) if ``condon_shortley`` is ``True``, the actual
formula being then the one given in
:wikipedia:`Spin-weighted_spherical_harmonics#Calculating`
TESTS::
sage: from kerrgeodesic_gw.spin_weighted_spherical_harm import _compute_sw_spherical_harm
sage: theta, phi = var("theta phi")
sage: _compute_sw_spherical_harm(-2, 2, 1, theta, phi)
1/4*(sqrt(5)*cos(theta) + sqrt(5))*e^(I*phi)*sin(theta)/sqrt(pi)
"""
if abs(s) > l:
return ZZ(0)
if abs(theta) < 1.e-6: # TODO: fix the treatment of small theta values
if theta < 0: # possibly with exact formula for theta=0
theta = -1.e-6 #
else: #
theta = 1.e-6 #
cott2 = cos(theta / 2) / sin(theta / 2)
res = 0
for r in range(l - s + 1):
res += (-1)**(l - r - s) * (binomial(l - s, r) * binomial(
l + s, r + s - m) * cott2**(2 * r + s - m))
res *= sin(theta / 2)**(2 * l)
ff = factorial(l + m) * factorial(l - m) * (2 * l + 1) / (
factorial(l + s) * factorial(l - s))
if numerical:
pre = sqrt(numerical(ff) / numerical(pi)) / 2
else:
pre = sqrt(ff) / (2 * sqrt(pi))
res *= pre
if condon_shortley:
res *= (-1)**m
if numerical:
return (numerical(res * cos(m * phi)), numerical(res * sin(m * phi)))
# Symbolic case:
res = res.simplify_full()
res = res.reduce_trig() # get rid of cos(theta/2) and sin(theta/2)
res = res.simplify_trig() # further trigonometric simplifications
res *= exp(I * m * phi)
return res
def plot(self, **kwds):
r"""
Plot the initial triangulation associated to ``self``.
INPUT:
- ``radius`` - the radius of the disk; by default the length of
the circle is the number of vertices
- ``points_color`` - the color of the vertices; default 'black'
- ``points_size`` - the size of the vertices; default 7
- ``triangulation_color`` - the color of the arcs; default 'black'
- ``triangulation_thickness`` - the thickness of the arcs; default 0.5
- ``shading_color`` - the color of the shading used on neuter
intervals; default 'lightgray'
- ``reflections_color`` - the color of the reflection axes; default
'blue'
- ``reflections_thickness`` - the thickness of the reflection axes;
default 1
EXAMPLES::
sage: Y = SineGordonYsystem('A',(6,4,3));
sage: Y.plot() # not tested
"""
# Set up plotting options
if 'radius' in kwds:
radius = kwds['radius']
else:
radius = ceil(self.r() / (2 * pi))
points_opts = {}
if 'points_color' in kwds:
points_opts['color'] = kwds['points_color']
else:
points_opts['color'] = 'black'
if 'points_size' in kwds:
points_opts['size'] = kwds['points_size']
else:
points_opts['size'] = 7
triangulation_opts = {}
if 'triangulation_color' in kwds:
triangulation_opts['color'] = kwds['triangulation_color']
else:
triangulation_opts['color'] = 'black'
if 'triangulation_thickness' in kwds:
triangulation_opts['thickness'] = kwds['triangulation_thickness']
else:
triangulation_opts['thickness'] = 0.5
shading_opts = {}
if 'shading_color' in kwds:
shading_opts['color'] = kwds['shading_color']
else:
shading_opts['color'] = 'lightgray'
reflections_opts = {}
if 'reflections_color' in kwds:
reflections_opts['color'] = kwds['reflections_color']
else:
reflections_opts['color'] = 'blue'
if 'reflections_thickness' in kwds:
reflections_opts['thickness'] = kwds['reflections_thickness']
else:
reflections_opts['thickness'] = 1
# Helper functions
def triangle(x):
(a, b) = sorted(x[:2])
for p in self.vertices():
if (p, a) in self.triangulation() or (a, p) in self.triangulation():
if (p, b) in self.triangulation() or (b, p) in self.triangulation():
if p < a or p > b:
return sorted((a, b, p))
def plot_arc(radius, p, q, **opts):
# plot the arc from p to q differently depending on the type of self
p = ZZ(p)
q = ZZ(q)
t = var('t')
if p - q in [1, -1]:
def f(t):
return (radius * cos(t), radius * sin(t))
(p, q) = sorted([p, q])
angle_p = vertex_to_angle(p)
angle_q = vertex_to_angle(q)
return parametric_plot(f(t), (t, angle_q, angle_p), **opts)
if self.type() == 'A':
angle_p = vertex_to_angle(p)
angle_q = vertex_to_angle(q)
if angle_p < angle_q:
angle_p += 2 * pi
internal_angle = angle_p - angle_q
if internal_angle > pi:
(angle_p, angle_q) = (angle_q + 2 * pi, angle_p)
internal_angle = angle_p - angle_q
angle_center = (angle_p+angle_q) / 2
hypotenuse = radius / cos(internal_angle / 2)
radius_arc = hypotenuse * sin(internal_angle / 2)
center = (hypotenuse * cos(angle_center),
hypotenuse * sin(angle_center))
center_angle_p = angle_p + pi / 2
center_angle_q = angle_q + 3 * pi / 2
def f(t):
return (radius_arc * cos(t) + center[0],
radius_arc * sin(t) + center[1])
return parametric_plot(f(t), (t, center_angle_p,
center_angle_q), **opts)
elif self.type() == 'D':
if p >= q:
q += self.r()
px = -2 * pi * p / self.r() + pi / 2
qx = -2 * pi * q / self.r() + pi / 2
arc_radius = (px - qx) / 2
arc_center = qx + arc_radius
def f(t):
return exp(I * ((cos(t) + I * sin(t)) *
arc_radius + arc_center)) * radius
return parametric_plot((real_part(f(t)), imag_part(f(t))),
(t, 0, pi), **opts)
def vertex_to_angle(v):
# v==0 corresponds to pi/2
return -2 * pi * RR(v) / self.r() + 5 * pi / 2
# Begin plotting
P = Graphics()
# Shade neuter intervals
neuter_intervals = [x for x in flatten(self.intervals()[:-1],
max_level=1)
if x[2] in ["NR", "NL"]]
shaded_triangles = map(triangle, neuter_intervals)
for (p, q, r) in shaded_triangles:
points = list(plot_arc(radius, p, q)[0])
points += list(plot_arc(radius, q, r)[0])
points += list(reversed(plot_arc(radius, p, r)[0]))
P += polygon2d(points, **shading_opts)
# Disk boundary
P += circle((0, 0), radius, **triangulation_opts)
# Triangulation
for (p, q) in self.triangulation():
P += plot_arc(radius, p, q, **triangulation_opts)
if self.type() == 'D':
s = radius / 50.0
P += polygon2d([(s, 5 * s), (s, 7 * s),
(3 * s, 5 * s), (3 * s, 7 * s)],
color=triangulation_opts['color'])
P += bezier_path([[(0, 0), (2 * s, 1 * s), (2 * s, 6 * s)],
[(2 * s, 10 * s), (s, 20 * s)],
[(0, 30 * s), (0, radius)]],
**triangulation_opts)
P += bezier_path([[(0, 0), (-2 * s, 1 * s), (-2 * s, 6 * s)],
[(-2 * s, 10 * s), (-s, 20 * s)],
[(0, 30 * s), (0, radius)]],
**triangulation_opts)
P += point((0, 0), zorder=len(P), **points_opts)
# Vertices
v_points = {x: (radius * cos(vertex_to_angle(x)),
radius * sin(vertex_to_angle(x)))
for x in self.vertices()}
for v in v_points:
P += point(v_points[v], zorder=len(P), **points_opts)
# Reflection axes
P += line([(0, 1.1 * radius), (0, -1.1 * radius)],
zorder=len(P), **reflections_opts)
axis_angle = vertex_to_angle(-0.5 * (self.rk() + (1, 1))[1])
(a, b) = (1.1 * radius * cos(axis_angle),
1.1 * radius * sin(axis_angle))
P += line([(a, b), (-a, -b)], zorder=len(P), **reflections_opts)
# Wrap up
P.set_aspect_ratio(1)
P.axes(False)
return P
def f(t):
return (radius * cos(t), radius * sin(t))
def f(t):
return (radius_arc * cos(t) + center[0],
radius_arc * sin(t) + center[1])
def val(x):
if x > -1:
return -R(cos(pi / SR(x)))
else:
return R(x)
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 __init__(self, coxeter_matrix, base_ring, index_set):
"""
Initialize ``self``.
EXAMPLES::
sage: W = CoxeterGroup([[1,3,2],[3,1,3],[2,3,1]])
sage: TestSuite(W).run() # long time
sage: W = CoxeterGroup([[1,3,2],[3,1,4],[2,4,1]], base_ring=QQbar)
sage: TestSuite(W).run() # long time
sage: W = CoxeterGroup([[1,3,2],[3,1,6],[2,6,1]])
sage: TestSuite(W).run(max_runs=30) # long time
sage: W = CoxeterGroup([[1,3,2],[3,1,-1],[2,-1,1]])
sage: TestSuite(W).run(max_runs=30) # long time
We check that :trac:`16630` is fixed::
sage: CoxeterGroup(['D',4], base_ring=QQ).category()
Category of finite coxeter groups
sage: CoxeterGroup(['H',4], base_ring=QQbar).category()
Category of finite coxeter groups
sage: F = CoxeterGroups().Finite()
sage: all(CoxeterGroup([letter,i]) in F
....: for i in range(2,5) for letter in ['A','B','D'])
True
sage: all(CoxeterGroup(['E',i]) in F for i in range(6,9))
True
sage: CoxeterGroup(['F',4]).category()
Category of finite coxeter groups
sage: CoxeterGroup(['G',2]).category()
Category of finite coxeter groups
sage: all(CoxeterGroup(['H',i]) in F for i in range(3,5))
True
sage: all(CoxeterGroup(['I',i]) in F for i in range(2,5))
True
"""
self._matrix = coxeter_matrix
n = coxeter_matrix.rank()
# Compute the matrix with entries `2 \cos( \pi / m_{ij} )`.
MS = MatrixSpace(base_ring, n, sparse=True)
MC = MS._get_matrix_class()
# FIXME: Hack because there is no ZZ \cup \{ \infty \}: -1 represents \infty
if base_ring is UniversalCyclotomicField():
val = lambda x: base_ring.gen(2*x) + ~base_ring.gen(2*x) if x != -1 else base_ring(2)
else:
from sage.functions.trig import cos
from sage.symbolic.constants import pi
val = lambda x: base_ring(2*cos(pi / x)) if x != -1 else base_ring(2)
gens = [MS.one() + MC(MS, entries={(i, j): val(coxeter_matrix[index_set[i], index_set[j]])
for j in range(n)},
coerce=True, copy=True)
for i in range(n)]
category = CoxeterGroups()
# Now we shall see if the group is finite, and, if so, refine
# the category to ``category.Finite()``. Otherwise the group is
# infinite and we refine the category to ``category.Infinite()``.
if self._matrix.is_finite():
category = category.Finite()
else:
category = category.Infinite()
FinitelyGeneratedMatrixGroup_generic.__init__(self, ZZ(n), base_ring,
gens, category=category)
def signal_to_noise_particle(a,
r0,
theta,
psd,
t_obs,
BH_time_scale,
m_min=1,
m_max=None,
scale=1,
approximation=None):
r"""
Evaluate the signal-to-noise ratio of gravitational radiation emitted
by a single orbiting particle observed in a detector of a given power
spectral density.
INPUT:
- ``a`` -- BH angular momentum parameter (in units of `M`, the BH mass)
- ``r0`` -- Boyer-Lindquist radius of the orbit (in units of `M`)
- ``theta`` -- Boyer-Lindquist colatitute `\theta` of the observer
- ``psd`` -- function with a single argument (`f`) representing the
detector's one-sided noise power spectral density `S_n(f)` (see e.g. :func:`.lisa_detector.power_spectral_density`)
- ``t_obs`` -- observation period, in the same time unit as `S_n(f)`
- ``BH_time_scale`` -- value of `M` in the same time unit as `S_n(f)`; if
`S_n(f)` is provided in `\mathrm{Hz}^{-1}`, then ``BH_time_scale`` must
be `M` expressed in seconds.
- ``m_min`` -- (default: 1) lower bound in the summation over the Fourier
mode `m`
- ``m_max`` -- (default: ``None``) upper bound in the summation over the
Fourier mode `m`; if ``None``, ``m_max`` is set to 10 for `r_0 \leq 20 M`
and to 5 for `r_0 > 20 M`
- ``scale`` -- (default: ``1``) scale factor by which `h(t)` must be
multiplied to get the actual signal; this should by `\mu/r`, where `\mu`
is the particle mass and `r` the radial coordinate of the detector
- ``approximation`` -- (default: ``None``) string describing the
computational method for the signal; allowed values are
- ``None``: exact computation
- ``'quadrupole'``: quadrupole approximation; see
:func:`.gw_particle.h_particle_quadrupole`
- ``'1.5PN'`` (only for ``a=0``): 1.5-post-Newtonian expansion following
E. Poisson, Phys. Rev. D **47**, 1497 (1993)
[:doi:`10.1103/PhysRevD.47.1497`]
OUTPUT:
- the signal-to-noise ratio `\rho`
EXAMPLES:
Let us evaluate the SNR of the gravitational signal generated by a 1-solar
mass object orbiting at the ISCO of Sgr A* observed by LISA during 1 day::
sage: from kerrgeodesic_gw import (signal_to_noise_particle,
....: lisa_detector, astro_data)
sage: a, r0 = 0., 6.
sage: theta = pi/2
sage: t_obs = 24*3600 # 1 day in seconds
sage: BH_time_scale = astro_data.SgrA_mass_s # Sgr A* mass in seconds
sage: psd = lisa_detector.power_spectral_density_RCLfit
sage: mu_ov_r = astro_data.Msol_m / astro_data.dSgrA # mu/r
sage: signal_to_noise_particle(a, r0, theta, psd, t_obs, # tol 1.0e-13
....: BH_time_scale, scale=mu_ov_r)
7565.6612762972445
Using the quadrupole approximation::
sage: signal_to_noise_particle(a, r0, theta, psd, t_obs, # tol 1.0e-13
....: BH_time_scale, scale=mu_ov_r,
....: approximation='quadrupole')
5230.403692883996
Using the 1.5-PN approximation (``m_max`` has to be at most 5)::
sage: signal_to_noise_particle(a, r0, theta, psd, t_obs, # tol 1.0e-13
....: BH_time_scale, scale=mu_ov_r,
....: approximation='1.5PN', m_max=5)
7601.344521598601
For large values of `r_0`, the 1.5-PN approximation and the quadrupole one
converge::
sage: r0 = 100
sage: signal_to_noise_particle(a, r0, theta, psd, t_obs, # tol 1.0e-13
....: BH_time_scale, scale=mu_ov_r,
....: approximation='quadrupole')
0.0030532227165507805
sage: signal_to_noise_particle(a, r0, theta, psd, t_obs, # tol 1.0e-13
....: BH_time_scale, scale=mu_ov_r,
....: approximation='1.5PN')
0.0031442135473417616
::
sage: r0 = 1000
sage: signal_to_noise_particle(a, r0, theta, psd, t_obs, # tol 1.0e-13
....: BH_time_scale, scale=mu_ov_r,
....: approximation='quadrupole')
9.663790254603111e-09
sage: signal_to_noise_particle(a, r0, theta, psd, t_obs, # tol 1.0e-13
....: BH_time_scale, scale=mu_ov_r,
....: approximation='1.5PN')
9.687469292984858e-09
"""
from .gw_particle import h_amplitude_particle_fourier
from .zinf import _lmax
if approximation == 'quadrupole':
fm2 = RDF(1. / (pi * r0**1.5) / BH_time_scale)
return RDF(2. * scale / r0 *
sqrt(t_obs / psd(fm2) *
(1 + 6 * cos(theta)**2 + cos(theta)**4)))
if m_max is None:
m_max = _lmax(a, r0)
# Orbital frequency in the same time units as S_n(f) (generally seconds):
f0 = RDF(1. / (2 * pi * (r0**1.5 + a)) / BH_time_scale)
rho2 = 0
for m in range(m_min, m_max + 1):
hmp, hmc = h_amplitude_particle_fourier(m,
a,
r0,
theta,
l_max=m_max,
algorithm_Zinf=approximation)
rho2 += (hmp**2 + hmc**2) / psd(m * f0)
return sqrt(rho2 * t_obs) * scale
def plot(self, show_box=False, colors=["white", "lightgray", "darkgray"]):
r"""
Return a plot of ``self``.
INPUT:
- ``show_box`` -- boolean (default: ``False``); if ``True``,
also shows the visible tiles on the `xy`-, `yz`-, `zx`-planes
- ``colors`` -- (default: ``["white", "lightgray", "darkgray"]``)
list ``[A, B, C]`` of 3 strings representing colors
EXAMPLES::
sage: PP = PlanePartition([[4,3,3,1],[2,1,1],[1,1]])
sage: PP.plot()
Graphics object consisting of 27 graphics primitives
"""
from sage.functions.trig import cos, sin
from sage.plot.polygon import polygon
from sage.symbolic.constants import pi
from sage.plot.plot import plot
Uside = [[0, 0], [cos(-pi / 6), sin(-pi / 6)], [0, -1],
[cos(7 * pi / 6), sin(7 * pi / 6)]]
Lside = [[0, 0], [cos(-pi / 6), sin(-pi / 6)],
[cos(pi / 6), sin(pi / 6)], [0, 1]]
Rside = [[0, 0], [0, 1], [cos(5 * pi / 6),
sin(5 * pi / 6)],
[cos(7 * pi / 6), sin(7 * pi / 6)]]
Xdir = [cos(7 * pi / 6), sin(7 * pi / 6)]
Ydir = [cos(-pi / 6), sin(-pi / 6)]
Zdir = [0, 1]
def move(side, i, j, k):
return [[
P[0] + i * Xdir[0] + j * Ydir[0] + k * Zdir[0],
P[1] + i * Xdir[1] + j * Ydir[1] + k * Zdir[1]
] for P in side]
def add_topside(i, j, k):
return polygon(move(Uside, i, j, k),
edgecolor="black",
color=colors[0])
def add_leftside(i, j, k):
return polygon(move(Lside, i, j, k),
edgecolor="black",
color=colors[1])
def add_rightside(i, j, k):
return polygon(move(Rside, i, j, k),
edgecolor="black",
color=colors[2])
TP = plot([])
for r in range(len(self.z_tableau())):
for c in range(len(self.z_tableau()[r])):
if self.z_tableau()[r][c] > 0 or show_box:
TP += add_topside(r, c, self.z_tableau()[r][c])
for r in range(len(self.y_tableau())):
for c in range(len(self.y_tableau()[r])):
if self.y_tableau()[r][c] > 0 or show_box:
TP += add_rightside(c, self.y_tableau()[r][c], r)
for r in range(len(self.x_tableau())):
for c in range(len(self.x_tableau()[r])):
if self.x_tableau()[r][c] > 0 or show_box:
TP += add_leftside(self.x_tableau()[r][c], r, c)
TP.axes(show=False)
return TP
def __init__(self, sides, i_angle, center, options):
"""
Initialize HyperbolicRegularPolygon.
EXAMPLES::
sage: from sage.plot.hyperbolic_regular_polygon import HyperbolicRegularPolygon
sage: print(HyperbolicRegularPolygon(5,pi/2,I, {}))
Hyperbolic regular polygon (sides=5, i_angle=1/2*pi, center=1.00000000000000*I)
"""
self.center = CC(center)
if self.center.imag() <= 0:
raise ValueError(
"center: %s is not a valid point in the upper half plane model of the hyperbolic plane"
% (self.center))
if sides < 3:
raise ValueError(
"degenerated polygons (sides<=2) are not supported")
if i_angle <= 0 or i_angle >= pi:
raise ValueError("interior angle %s must be in (0, pi) interval" %
(i_angle))
if pi * (sides - 2) - sides * i_angle <= 0:
raise ValueError(
"there exists no hyperbolic regular compact polygon, for sides=%s the interior angle must be less than %s"
% (sides, pi * (sides - 2) / sides))
self.sides = sides
self.i_angle = i_angle
beta = 2 * pi / self.sides # compute the rotation angle to be used ahead
alpha = self.i_angle / Integer(2)
I = CC(0, 1)
# compute using cosine theorem the radius of the circumscribed circle
# using the triangle formed by the radius and the three known angles
r = arccosh(cot(alpha) * (1 + cos(beta)) / sin(beta))
# The first point will be always on the imaginary axis limited
# to 8 digits for efficiency in the subsequent calculations.
z_0 = [I * (e**r).n(digits=8)]
# Compute the dilation isometry used to move the center
# from I to the imaginary part of the given center.
scale = self.center.imag()
# Compute the parabolic isometry to move the center to the
# real part of the given center.
h_disp = self.center.real()
d_z_k = [z_0[0] * scale + h_disp
] #d_k has the points for the polygon in the given center
z_k = z_0 #z_k has the Re(z)>0 vertices for the I centered polygon
r_z_k = [] #r_z_k has the Re(z)<0 vertices
if is_odd(self.sides):
vert = (self.sides - 1) // 2
else:
vert = self.sides // 2 - 1
for k in range(vert):
# Compute with 8 digits to accelerate calculations
new_z_k = self._i_rotation(z_k[-1], beta).n(digits=8)
z_k = z_k + [new_z_k]
d_z_k = d_z_k + [new_z_k * scale + h_disp]
r_z_k = [-(new_z_k).conjugate() * scale + h_disp] + r_z_k
if is_odd(self.sides):
HyperbolicPolygon.__init__(self, d_z_k + r_z_k, options)
else:
z_opo = [I * (e**(-r)).n(digits=8) * scale + h_disp]
HyperbolicPolygon.__init__(self, d_z_k + z_opo + r_z_k, options)
def barycentric_projection_matrix(n, angle=0):
r"""
Returns a family of `n+1` vectors evenly spaced in a real vector space of dimension `n`
Those vectors are of norm `1`, the scalar product between any two
vector is `1/n`, thus the distance between two tips is constant.
The family is built recursively and uniquely determined by the
following property: the last vector is `(0,\dots,0,-1)`, and the
projection of the first `n` vectors in dimension `n-1`, after
appropriate rescaling to norm `1`, retrieves the family for `n-1`.
OUTPUT:
A matrix with `n+1` columns of height `n` with rational or
symbolic coefficients.
EXAMPLES:
One vector in dimension `0`::
sage: from sage.combinat.root_system.root_lattice_realizations import barycentric_projection_matrix
sage: m = barycentric_projection_matrix(0); m
[]
sage: matrix(QQ,0,1).nrows()
0
sage: matrix(QQ,0,1).ncols()
1
Two vectors in dimension 1::
sage: barycentric_projection_matrix(1)
[ 1 -1]
Three vectors in dimension 2::
sage: barycentric_projection_matrix(2)
[ 1/2*sqrt(3) -1/2*sqrt(3) 0]
[ 1/2 1/2 -1]
Four vectors in dimension 3::
sage: m = barycentric_projection_matrix(3); m
[ 1/3*sqrt(3)*sqrt(2) -1/3*sqrt(3)*sqrt(2) 0 0]
[ 1/3*sqrt(2) 1/3*sqrt(2) -2/3*sqrt(2) 0]
[ 1/3 1/3 1/3 -1]
The columns give four vectors that sum up to zero::
sage: sum(m.columns())
(0, 0, 0)
and have regular mutual angles::
sage: m.transpose()*m
[ 1 -1/3 -1/3 -1/3]
[-1/3 1 -1/3 -1/3]
[-1/3 -1/3 1 -1/3]
[-1/3 -1/3 -1/3 1]
Here is a plot of them::
sage: sum(arrow((0,0,0),x) for x in m.columns())
For 2D drawings of root systems, it is desirable to rotate the
result to match with the usual conventions::
sage: barycentric_projection_matrix(2, angle=2*pi/3)
[ 1/2 -1 1/2]
[ 1/2*sqrt(3) 0 -1/2*sqrt(3)]
TESTS::
sage: for n in range(1, 7):
... m = barycentric_projection_matrix(n)
... assert sum(m.columns()).is_zero()
... assert matrix(QQ, n+1,n+1, lambda i,j: 1 if i==j else -1/n) == m.transpose()*m
"""
from sage.matrix.constructor import matrix
from sage.functions.other import sqrt
n = ZZ(n)
if n == 0:
return matrix(QQ, 0, 1)
a = 1/n
b = sqrt(1-a**2)
result = b * barycentric_projection_matrix(n-1)
result = result.augment(vector([0]*(n-1)))
result = result.stack(matrix([[a]*n+[-1]]))
assert sum(result.columns()).is_zero()
if angle and n == 2:
from sage.functions.trig import sin
from sage.functions.trig import cos
rotation = matrix([[sin(angle), cos(angle)],[-cos(angle), sin(angle)]])
result = rotation * result
result.set_immutable()
return result
def subexpressions_list(f, pars=None):
"""
Construct the lists with the intermediate steps on the evaluation of the
function.
INPUT:
- ``f`` -- a symbolic function of several components.
- ``pars`` -- a list of the parameters that appear in the function
this should be the symbolic constants that appear in f but are not
arguments.
OUTPUT:
- a list of the intermediate subexpressions that appear in the evaluation
of f.
- a list with the operations used to construct each of the subexpressions.
each element of this list is a tuple, formed by a string describing the
operation made, and the operands.
For the trigonometric functions, some extra expressions will be added.
These extra expressions will be used later to compute their derivatives.
EXAMPLES::
sage: from sage.interfaces.tides import subexpressions_list
sage: var('x,y')
(x, y)
sage: f(x,y) = [x^2+y, cos(x)/log(y)]
sage: subexpressions_list(f)
([x^2, x^2 + y, sin(x), cos(x), log(y), cos(x)/log(y)],
[('mul', x, x),
('add', y, x^2),
('sin', x),
('cos', x),
('log', y),
('div', log(y), cos(x))])
::
sage: f(a)=[cos(a), arctan(a)]
sage: from sage.interfaces.tides import subexpressions_list
sage: subexpressions_list(f)
([sin(a), cos(a), a^2, a^2 + 1, arctan(a)],
[('sin', a), ('cos', a), ('mul', a, a), ('add', 1, a^2), ('atan', a)])
::
sage: from sage.interfaces.tides import subexpressions_list
sage: var('s,b,r')
(s, b, r)
sage: f(t,x,y,z)= [s*(y-x),x*(r-z)-y,x*y-b*z]
sage: subexpressions_list(f,[s,b,r])
([-y,
x - y,
s*(x - y),
-s*(x - y),
-z,
r - z,
(r - z)*x,
-y,
(r - z)*x - y,
x*y,
b*z,
-b*z,
x*y - b*z],
[('mul', -1, y),
('add', -y, x),
('mul', x - y, s),
('mul', -1, s*(x - y)),
('mul', -1, z),
('add', -z, r),
('mul', x, r - z),
('mul', -1, y),
('add', -y, (r - z)*x),
('mul', y, x),
('mul', z, b),
('mul', -1, b*z),
('add', -b*z, x*y)])
::
sage: var('x, y')
(x, y)
sage: f(x,y)=[exp(x^2+sin(y))]
sage: from sage.interfaces.tides import *
sage: subexpressions_list(f)
([x^2, sin(y), cos(y), x^2 + sin(y), e^(x^2 + sin(y))],
[('mul', x, x),
('sin', y),
('cos', y),
('add', sin(y), x^2),
('exp', x^2 + sin(y))])
"""
from sage.functions.trig import sin, cos, arcsin, arctan, arccos
variables = f[0].arguments()
if not pars:
parameters = []
else:
parameters = pars
varpar = list(parameters) + list(variables)
F = symbolic_expression([i(*variables) for i in f]).function(*varpar)
lis = flatten([fast_callable(i,vars=varpar).op_list() for i in F], max_level=1)
stack = []
const =[]
stackcomp=[]
detail=[]
for i in lis:
if i[0] == 'load_arg':
stack.append(varpar[i[1]])
elif i[0] == 'ipow':
if i[1] in NN:
basis = stack[-1]
for j in range(i[1]-1):
a=stack.pop(-1)
detail.append(('mul', a, basis))
stack.append(a*basis)
stackcomp.append(stack[-1])
else:
detail.append(('pow',stack[-1],i[1]))
stack[-1]=stack[-1]**i[1]
stackcomp.append(stack[-1])
elif i[0] == 'load_const':
const.append(i[1])
stack.append(i[1])
elif i == 'mul':
a=stack.pop(-1)
b=stack.pop(-1)
detail.append(('mul', a, b))
stack.append(a*b)
stackcomp.append(stack[-1])
elif i == 'div':
a=stack.pop(-1)
b=stack.pop(-1)
detail.append(('div', a, b))
stack.append(b/a)
stackcomp.append(stack[-1])
elif i == 'add':
a=stack.pop(-1)
b=stack.pop(-1)
detail.append(('add',a,b))
stack.append(a+b)
stackcomp.append(stack[-1])
elif i == 'pow':
a=stack.pop(-1)
b=stack.pop(-1)
detail.append(('pow', b, a))
stack.append(b**a)
stackcomp.append(stack[-1])
elif i[0] == 'py_call' and str(i[1])=='log':
a=stack.pop(-1)
detail.append(('log', a))
stack.append(log(a))
stackcomp.append(stack[-1])
elif i[0] == 'py_call' and str(i[1])=='exp':
a=stack.pop(-1)
detail.append(('exp', a))
stack.append(exp(a))
stackcomp.append(stack[-1])
elif i[0] == 'py_call' and str(i[1])=='sin':
a=stack.pop(-1)
detail.append(('sin', a))
detail.append(('cos', a))
stackcomp.append(sin(a))
stackcomp.append(cos(a))
stack.append(sin(a))
elif i[0] == 'py_call' and str(i[1])=='cos':
a=stack.pop(-1)
detail.append(('sin', a))
detail.append(('cos', a))
stackcomp.append(sin(a))
stackcomp.append(cos(a))
stack.append(cos(a))
elif i[0] == 'py_call' and str(i[1])=='tan':
a=stack.pop(-1)
b = sin(a)
c = cos(a)
detail.append(('sin', a))
detail.append(('cos', a))
detail.append(('div', b, c))
stackcomp.append(b)
stackcomp.append(c)
stackcomp.append(b/c)
stack.append(b/c)
elif i[0] == 'py_call' and str(i[1])=='arctan':
a=stack.pop(-1)
detail.append(('mul', a, a))
detail.append(('add', 1, a*a))
detail.append(('atan', a))
stackcomp.append(a*a)
stackcomp.append(1+a*a)
stackcomp.append(arctan(a))
stack.append(arctan(a))
elif i[0] == 'py_call' and str(i[1])=='arcsin':
a=stack.pop(-1)
detail.append(('mul', a, a))
detail.append(('mul', -1, a*a))
detail.append(('add', 1, -a*a))
detail.append(('pow', 1- a*a, 0.5))
detail.append(('asin', a))
stackcomp.append(a*a)
stackcomp.append(-a*a)
stackcomp.append(1-a*a)
stackcomp.append(sqrt(1-a*a))
stackcomp.append(arcsin(a))
stack.append(arcsin(a))
elif i[0] == 'py_call' and str(i[1])=='arccos':
a=stack.pop(-1)
detail.append(('mul', a, a))
detail.append(('mul', -1, a*a))
detail.append(('add', 1, -a*a))
detail.append(('pow', 1- a*a, 0.5))
detail.append(('mul', -1, sqrt(1-a*a)))
detail.append(('acos', a))
stackcomp.append(a*a)
stackcomp.append(-a*a)
stackcomp.append(1-a*a)
stackcomp.append(sqrt(1-a*a))
stackcomp.append(-sqrt(1-a*a))
stackcomp.append(arccos(a))
stack.append(arccos(a))
elif i[0] == 'py_call' and 'sqrt' in str(i[1]):
a=stack.pop(-1)
detail.append(('pow', a, 0.5))
stackcomp.append(sqrt(a))
stack.append(sqrt(a))
elif i == 'neg':
a = stack.pop(-1)
detail.append(('mul', -1, a))
stack.append(-a)
stackcomp.append(-a)
return stackcomp,detail
def Kerr(m=1, a=0, coordinates="BL", names=None):
"""
Generate a Kerr spacetime.
A Kerr spacetime is a 4 dimensional manifold describing a rotating black
hole. Two coordinate systems are implemented: Boyer-Lindquist and Kerr
(3+1 version).
The shortcut operator ``.<,>`` can be used to specify the coordinates.
INPUT:
- ``m`` -- (default: ``1``) mass of the black hole in natural units
(`c=1`, `G=1`)
- ``a`` -- (default: ``0``) angular momentum in natural units; if set to
``0``, the resulting spacetime corresponds to a Schwarzschild black hole
- ``coordinates`` -- (default: ``"BL"``) either ``"BL"`` for
Boyer-Lindquist coordinates or ``"Kerr"`` for Kerr coordinates (3+1
version)
- ``names`` -- (default: ``None``) name of the coordinates,
automatically set by the shortcut operator
OUTPUT:
- Lorentzian manifold
EXAMPLES::
sage: m, a = var('m, a')
sage: K = manifolds.Kerr(m, a)
sage: K
4-dimensional Lorentzian manifold M
sage: K.atlas()
[Chart (M, (t, r, th, ph))]
sage: K.metric().display()
g = (2*m*r/(a^2*cos(th)^2 + r^2) - 1) dt⊗dt
+ 2*a*m*r*sin(th)^2/(a^2*cos(th)^2 + r^2) dt⊗dph
+ (a^2*cos(th)^2 + r^2)/(a^2 - 2*m*r + r^2) dr⊗dr
+ (a^2*cos(th)^2 + r^2) dth⊗dth
+ 2*a*m*r*sin(th)^2/(a^2*cos(th)^2 + r^2) dph⊗dt
+ (2*a^2*m*r*sin(th)^2/(a^2*cos(th)^2 + r^2) + a^2 + r^2)*sin(th)^2 dph⊗dph
sage: K.<t, r, th, ph> = manifolds.Kerr()
sage: K
4-dimensional Lorentzian manifold M
sage: K.metric().display()
g = (2/r - 1) dt⊗dt + r^2/(r^2 - 2*r) dr⊗dr
+ r^2 dth⊗dth + r^2*sin(th)^2 dph⊗dph
sage: K.default_chart().coord_range()
t: (-oo, +oo); r: (0, +oo); th: (0, pi); ph: [-pi, pi] (periodic)
sage: m, a = var('m, a')
sage: K.<t, r, th, ph> = manifolds.Kerr(m, a, coordinates="Kerr")
sage: K
4-dimensional Lorentzian manifold M
sage: K.atlas()
[Chart (M, (t, r, th, ph))]
sage: K.metric().display()
g = (2*m*r/(a^2*cos(th)^2 + r^2) - 1) dt⊗dt
+ 2*m*r/(a^2*cos(th)^2 + r^2) dt⊗dr
- 2*a*m*r*sin(th)^2/(a^2*cos(th)^2 + r^2) dt⊗dph
+ 2*m*r/(a^2*cos(th)^2 + r^2) dr⊗dt
+ (2*m*r/(a^2*cos(th)^2 + r^2) + 1) dr⊗dr
- a*(2*m*r/(a^2*cos(th)^2 + r^2) + 1)*sin(th)^2 dr⊗dph
+ (a^2*cos(th)^2 + r^2) dth⊗dth
- 2*a*m*r*sin(th)^2/(a^2*cos(th)^2 + r^2) dph⊗dt
- a*(2*m*r/(a^2*cos(th)^2 + r^2) + 1)*sin(th)^2 dph⊗dr
+ (2*a^2*m*r*sin(th)^2/(a^2*cos(th)^2 + r^2)
+ a^2 + r^2)*sin(th)^2 dph⊗dph
sage: K.default_chart().coord_range()
t: (-oo, +oo); r: (0, +oo); th: (0, pi); ph: [-pi, pi] (periodic)
"""
from sage.misc.functional import sqrt
from sage.functions.trig import cos, sin
from sage.manifolds.manifold import Manifold
M = Manifold(4, 'M', structure="Lorentzian")
if coordinates == "Kerr":
if names is None:
names = (r't:(-oo,+oo)', r'r:(0,+oo)', r'th:(0,pi):\theta',
r'ph:(-pi,pi):periodic:\phi')
else:
names = (names[0] + r':(-oo,+oo)', names[1] + r':(0,+oo)',
names[2] + r':(0,pi):\theta',
names[3] + r':(-pi,pi):periodic:\phi')
C = M.chart(names=names)
M._first_ngens = C._first_ngens
g = M.metric('g')
t, r, th, ph = C[:]
rho = sqrt(r**2 + a**2 * cos(th)**2)
g[0, 0], g[1, 1], g[2, 2], g[3, 3] = -(1-2*m*r/rho**2), 1+2*m*r/rho**2,\
rho**2, (r**2+a**2+2*a**2*m*r*sin(th)**2/rho**2)*sin(th)**2
g[0, 1] = 2 * m * r / rho**2
g[0, 3] = -2 * a * m * r / rho**2 * sin(th)**2
g[1, 3] = -a * sin(th)**2 * (1 + 2 * m * r / rho**2)
return M
if coordinates == "BL":
if names is None:
names = (r't:(-oo,+oo)', r'r:(0,+oo)', r'th:(0,pi):\theta',
r'ph:(-pi,pi):periodic:\phi')
else:
names = (names[0] + r':(-oo,+oo)', names[1] + r':(0,+oo)',
names[2] + r':(0,pi):\theta',
names[3] + r':(-pi,pi):periodic:\phi')
C = M.chart(names=names)
M._first_ngens = C._first_ngens
g = M.metric('g')
t, r, th, ph = C[:]
rho = sqrt(r**2 + a**2 * cos(th)**2)
g[0, 0], g[1, 1], g[2, 2], g[3, 3] = -(1-2*m*r/rho**2), \
rho**2/(r**2-2*m*r+a**2), rho**2, \
(r**2+a**2+2*m*r*a**2/rho**2*sin(th)**2)*sin(th)**2
g[0, 3] = 2 * m * r * a * sin(th)**2 / rho**2
return M
raise NotImplementedError("coordinates system not implemented, see help"
" for details")
def _compute_init_vector(self, point, pt0, pr0, pth0, pph0, r_increase,
th_increase, verbose):
r"""
Computes the initial 4-momentum vector `p` from the constants of motion
"""
BLchart = self._spacetime.boyer_lindquist_coordinates()
basis = BLchart.frame().at(point)
r, th = BLchart(point)[1:3]
a, m = self._a, self._m
r2 = r**2
a2 = a**2
rho2 = r2 + (a * cos(th))**2
Delta = r2 - 2 * m * r + a2
if pt0 is None:
if (self._mu is not None and pr0 is not None and pth0 is not None
and pph0 is not None):
xxx = SR.var('xxx')
v = self._spacetime.tangent_space(point)(
(xxx, pr0, pth0, pph0), basis=basis)
muv2 = -self._spacetime.metric().at(point)(v, v)
muv2 = muv2.substitute(self._numerical_substitutions())
solutions = solve(muv2 == self._mu**2, xxx, solution_dict=True)
if verbose:
print("Solutions for p^t:")
pretty_print(solutions)
for sol in solutions:
if sol[xxx] > 0:
pt0 = sol[xxx]
break
else: # pt0 <= 0 might occur in the ergoregion
pt0 = solutions[0][xxx]
try:
pt0 = RR(pt0)
except TypeError: # pt0 contains some symbolic expression
pass
else:
if self._E is None:
raise ValueError("the constant E must be provided")
if self._L is None:
raise ValueError("the constant L must be provided")
E, L = self._E, self._L
pt0 = ((r2 + a2) / Delta * ((r2 + a2) * E - a * L) + a *
(L - a * E * sin(th)**2)) / rho2
if pph0 is None:
if self._E is None:
raise ValueError("the constant E must be provided")
if self._L is None:
raise ValueError("the constant L must be provided")
E, L = self._E, self._L
pph0 = (L / sin(th)**2 - a * E + a / Delta *
((r2 + a2) * E - a * L)) / rho2
if pr0 is None:
if self._E is None:
raise ValueError("the constant E must be provided")
if self._L is None:
raise ValueError("the constant L must be provided")
if self._mu is None:
raise ValueError("the constant mu must be provided")
if self._Q is None:
raise ValueError("the constant Q must be provided")
E, L, Q = self._E, self._L, self._Q
mu2 = self._mu**2
E2_mu2 = E**2 - mu2
pr0 = sqrt((E2_mu2) * r**4 + 2 * m * mu2 * r**3 +
(a2 * E2_mu2 - L**2 - Q) * r**2 + 2 * m *
(Q + (L - a * E)**2) * r - a2 * Q) / rho2
if not r_increase:
pr0 = -pr0
if pth0 is None:
if self._E is None:
raise ValueError("the constant E must be provided")
if self._L is None:
raise ValueError("the constant L must be provided")
if self._mu is None:
raise ValueError("the constant mu must be provided")
if self._Q is None:
raise ValueError("the constant Q must be provided")
E2 = self._E**2
L2 = self._L**2
mu2 = self._mu**2
Q = self._Q
pth0 = sqrt(Q + cos(th)**2 * (a2 *
(E2 - mu2) - L2 / sin(th)**2)) / rho2
if not th_increase:
pth0 = -pth0
return self._spacetime.tangent_space(point)((pt0, pr0, pth0, pph0),
basis=basis,
name='p')
def _init_spherical(self, names=None):
r"""
Construct the chart of spherical coordinates.
TESTS:
Spherical coordinates on a 2-sphere::
sage: S2 = manifolds.Sphere(2)
sage: S2.spherical_coordinates()
Chart (A, (theta, phi))
Spherical coordinates on a 1-sphere::
sage: S1 = manifolds.Sphere(1, coordinates='stereographic')
sage: spher = S1.spherical_coordinates(); spher # create coords
Chart (A, (phi,))
sage: S1.atlas()
[Chart (S^1-{NP}, (y1,)),
Chart (S^1-{SP}, (yp1,)),
Chart (S^1-{NP,SP}, (y1,)),
Chart (S^1-{NP,SP}, (yp1,)),
Chart (A, (phi,)),
Chart (A, (y1,)),
Chart (A, (yp1,))]
"""
# speed-up via simplification method...
self.set_simplify_function(lambda expr: expr.simplify_trig())
# get domain...
A = self._spher_dom
# initialize coordinates...
n = self._dim
if names:
# add interval:
names = tuple([x + ':(0,pi)' for x in names[:-1]] +
[names[-1] + ':(-pi,pi):periodic'])
else:
if n == 1:
names = ('phi:(-pi,pi):periodic', )
elif n == 2:
names = ('theta:(0,pi)', 'phi:(-pi,pi):periodic')
elif n == 3:
names = ('chi:(0,pi)', 'theta:(0,pi)', 'phi:(-pi,pi):periodic')
else:
names = tuple(["phi_{}:(0,pi)".format(i)
for i in range(1, n)] +
["phi_{}:(-pi,pi):periodic".format(n)])
spher = A.chart(names=names)
coord = spher[:]
# manage embedding...
from sage.misc.misc_c import prod
from sage.functions.trig import cos, sin
R = self._radius
coordfunc = [
R * cos(coord[n - 1]) * prod(sin(coord[i]) for i in range(n - 1))
]
coordfunc += [R * prod(sin(coord[i]) for i in range(n))]
for k in reversed(range(n - 1)):
c = R * cos(coord[k]) * prod(sin(coord[i]) for i in range(k))
coordfunc.append(c)
cart = self._ambient.cartesian_coordinates()
# shift coordinates to barycenter:
coordfunc = self._shift_coords(coordfunc, s='+')
# add expression to embedding:
self._immersion.add_expr(spher, cart, coordfunc)
self.clear_cache() # clear cache of extrinsic information
# finish process...
self._coordinates['spherical'] = [spher]
# adapt other coordinates...
if 'stereographic' in self._coordinates:
self._transition_spher_stereo()
# reset simplification method...
self.set_simplify_function('default')
def revolution_plot3d(curve,
trange,
phirange=None,
parallel_axis='z',
axis=(0, 0),
print_vector=False,
show_curve=False,
**kwds):
"""
Return a plot of a revolved curve.
There are three ways to call this function:
- ``revolution_plot3d(f,trange)`` where `f` is a function located in the `x z` plane.
- ``revolution_plot3d((f_x,f_z),trange)`` where `(f_x,f_z)` is a parametric curve on the `x z` plane.
- ``revolution_plot3d((f_x,f_y,f_z),trange)`` where `(f_x,f_y,f_z)` can be any parametric curve.
INPUT:
- ``curve`` - A curve to be revolved, specified as a function, a 2-tuple or a 3-tuple.
- ``trange`` - A 3-tuple `(t,t_{\min},t_{\max})` where t is the independent variable of the curve.
- ``phirange`` - A 2-tuple of the form `(\phi_{\min},\phi_{\max})`, (default `(0,\pi)`) that specifies the angle in which the curve is to be revolved.
- ``parallel_axis`` - A string (Either 'x', 'y', or 'z') that specifies the coordinate axis parallel to the revolution axis.
- ``axis`` - A 2-tuple that specifies the position of the revolution axis. If parallel is:
- 'z' - then axis is the point in which the revolution axis intersects the `x y` plane.
- 'x' - then axis is the point in which the revolution axis intersects the `y z` plane.
- 'y' - then axis is the point in which the revolution axis intersects the `x z` plane.
- ``print_vector`` - If True, the parametrization of the surface of revolution will be printed.
- ``show_curve`` - If True, the curve will be displayed.
EXAMPLES:
Let's revolve a simple function around different axes::
sage: u = var('u')
sage: f=u^2
sage: revolution_plot3d(f,(u,0,2),show_curve=True,opacity=0.7).show(aspect_ratio=(1,1,1))
If we move slightly the axis, we get a goblet-like surface::
sage: revolution_plot3d(f,(u,0,2),axis=(1,0.2),show_curve=True,opacity=0.5).show(aspect_ratio=(1,1,1))
A common problem in calculus books, find the volume within the following revolution solid::
sage: line=u
sage: parabola=u^2
sage: sur1=revolution_plot3d(line,(u,0,1),opacity=0.5,rgbcolor=(1,0.5,0),show_curve=True,parallel_axis='x')
sage: sur2=revolution_plot3d(parabola,(u,0,1),opacity=0.5,rgbcolor=(0,1,0),show_curve=True,parallel_axis='x')
sage: (sur1+sur2).show()
Now let's revolve a parametrically defined circle. We can play with the topology of the surface by changing the axis, an axis in `(0,0)` (as the previous one) will produce a sphere-like surface::
sage: u = var('u')
sage: circle=(cos(u),sin(u))
sage: revolution_plot3d(circle,(u,0,2*pi),axis=(0,0),show_curve=True,opacity=0.5).show(aspect_ratio=(1,1,1))
An axis on `(0,y)` will produce a cylinder-like surface::
sage: revolution_plot3d(circle,(u,0,2*pi),axis=(0,2),show_curve=True,opacity=0.5).show(aspect_ratio=(1,1,1))
And any other axis will produce a torus-like surface::
sage: revolution_plot3d(circle,(u,0,2*pi),axis=(2,0),show_curve=True,opacity=0.5).show(aspect_ratio=(1,1,1))
Now, we can get another goblet-like surface by revolving a curve in 3d::
sage: u = var('u')
sage: curve=(u,cos(4*u),u^2)
sage: revolution_plot3d(curve,(u,0,2),show_curve=True,parallel_axis='z',axis=(1,.2),opacity=0.5).show(aspect_ratio=(1,1,1))
A curvy curve with only a quarter turn::
sage: u = var('u')
sage: curve=(sin(3*u),.8*cos(4*u),cos(u))
sage: revolution_plot3d(curve,(u,0,pi),(0,pi/2),show_curve=True,parallel_axis='z',opacity=0.5).show(aspect_ratio=(1,1,1),frame=False)
"""
from sage.symbolic.ring import SR
from sage.symbolic.constants import pi
from sage.functions.other import sqrt
from sage.functions.trig import sin
from sage.functions.trig import cos
from sage.functions.trig import atan2
if parallel_axis not in ['x', 'y', 'z']:
raise ValueError("parallel_axis must be either 'x', 'y', or 'z'.")
vart = trange[0]
if str(vart) == 'phi':
phi = SR.var('fi')
else:
phi = SR.var('phi')
if phirange is None: #this if-else provides a phirange
phirange = (phi, 0, 2 * pi)
elif len(phirange) == 3:
phi = phirange[0]
pass
else:
phirange = (phi, phirange[0], phirange[1])
if isinstance(curve, tuple) or isinstance(curve, list):
#this if-else provides a vector v to be plotted
#if curve is a tuple or a list of length 2, it is interpreted as a parametric curve
#in the x-z plane.
#if it is of length 3 it is interpreted as a parametric curve in 3d space
if len(curve) == 2:
x = curve[0]
y = 0
z = curve[1]
elif len(curve) == 3:
x = curve[0]
y = curve[1]
z = curve[2]
else:
x = vart
y = 0
z = curve
if parallel_axis == 'z':
x0 = axis[0]
y0 = axis[1]
# (0,0) must be handled separately for the phase value
phase = 0
if x0 != 0 or y0 != 0:
phase = atan2((y - y0), (x - x0))
R = sqrt((x - x0)**2 + (y - y0)**2)
v = (R * cos(phi + phase) + x0, R * sin(phi + phase) + y0, z)
elif parallel_axis == 'x':
y0 = axis[0]
z0 = axis[1]
# (0,0) must be handled separately for the phase value
phase = 0
if z0 != 0 or y0 != 0:
phase = atan2((z - z0), (y - y0))
R = sqrt((y - y0)**2 + (z - z0)**2)
v = (x, R * cos(phi + phase) + y0, R * sin(phi + phase) + z0)
elif parallel_axis == 'y':
x0 = axis[0]
z0 = axis[1]
# (0,0) must be handled separately for the phase value
phase = 0
if z0 != 0 or x0 != 0:
phase = atan2((z - z0), (x - x0))
R = sqrt((x - x0)**2 + (z - z0)**2)
v = (R * cos(phi + phase) + x0, y, R * sin(phi + phase) + z0)
if print_vector:
print(v)
if show_curve:
curveplot = parametric_plot3d((x, y, z),
trange,
thickness=2,
rgbcolor=(1, 0, 0))
return parametric_plot3d(v, trange, phirange, **kwds) + curveplot
return parametric_plot3d(v, trange, phirange, **kwds)
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 simplify_abs_trig(expr):
r"""
Simplify ``abs(sin(...))`` and ``abs(cos(...))`` in symbolic expressions.
EXAMPLES::
sage: M = Manifold(3, 'M', structure='topological')
sage: X.<x,y,z> = M.chart(r'x y:(0,pi) z:(-pi/3,0)')
sage: X.coord_range()
x: (-oo, +oo); y: (0, pi); z: (-1/3*pi, 0)
Since `x` spans all `\RR`, no simplification of ``abs(sin(x))``
occurs, while ``abs(sin(y))`` and ``abs(sin(3*z))`` are correctly
simplified, given that `y \in (0,\pi)` and `z \in (-\pi/3,0)`::
sage: from sage.manifolds.utilities import simplify_abs_trig
sage: simplify_abs_trig( abs(sin(x)) + abs(sin(y)) + abs(sin(3*z)) )
abs(sin(x)) + sin(y) + sin(-3*z)
Note that neither
:meth:`~sage.symbolic.expression.Expression.simplify_trig` nor
:meth:`~sage.symbolic.expression.Expression.simplify_full`
works in this case::
sage: s = abs(sin(x)) + abs(sin(y)) + abs(sin(3*z))
sage: s.simplify_trig()
abs(4*cos(-z)^2 - 1)*abs(sin(-z)) + abs(sin(x)) + abs(sin(y))
sage: s.simplify_full()
abs(4*cos(-z)^2 - 1)*abs(sin(-z)) + abs(sin(x)) + abs(sin(y))
despite the following assumptions hold::
sage: assumptions()
[x is real, y is real, y > 0, y < pi, z is real, z > -1/3*pi, z < 0]
Additional checks are::
sage: simplify_abs_trig( abs(sin(y/2)) ) # shall simplify
sin(1/2*y)
sage: simplify_abs_trig( abs(sin(2*y)) ) # must not simplify
abs(sin(2*y))
sage: simplify_abs_trig( abs(sin(z/2)) ) # shall simplify
sin(-1/2*z)
sage: simplify_abs_trig( abs(sin(4*z)) ) # must not simplify
abs(sin(-4*z))
Simplification of ``abs(cos(...))``::
sage: forget()
sage: M = Manifold(3, 'M', structure='topological')
sage: X.<x,y,z> = M.chart(r'x y:(0,pi/2) z:(pi/4,3*pi/4)')
sage: X.coord_range()
x: (-oo, +oo); y: (0, 1/2*pi); z: (1/4*pi, 3/4*pi)
sage: simplify_abs_trig( abs(cos(x)) + abs(cos(y)) + abs(cos(2*z)) )
abs(cos(x)) + cos(y) - cos(2*z)
Additional tests::
sage: simplify_abs_trig(abs(cos(y-pi/2))) # shall simplify
cos(-1/2*pi + y)
sage: simplify_abs_trig(abs(cos(y+pi/2))) # shall simplify
-cos(1/2*pi + y)
sage: simplify_abs_trig(abs(cos(y-pi))) # shall simplify
-cos(-pi + y)
sage: simplify_abs_trig(abs(cos(2*y))) # must not simplify
abs(cos(2*y))
sage: simplify_abs_trig(abs(cos(y/2)) * abs(sin(z))) # shall simplify
cos(1/2*y)*sin(z)
TESTS:
Simplification of expressions involving some symbolic derivatives::
sage: f = function('f')
sage: s = abs(cos(x)) + abs(cos(y))*diff(f(x),x) + abs(cos(2*z))
sage: simplify_abs_trig(s)
cos(y)*diff(f(x), x) + abs(cos(x)) - cos(2*z)
sage: s = abs(sin(x))*diff(f(x),x).subs(x=y^2) + abs(cos(y))
sage: simplify_abs_trig(s)
abs(sin(x))*D[0](f)(y^2) + cos(y)
sage: forget() # for doctests below
"""
w0 = SR.wild()
if expr.has(abs_symbolic(sin(w0))) or expr.has(abs_symbolic(cos(w0))):
return SimplifyAbsTrig(expr)()
return expr
def composition(self, ex, operator):
r"""
This is the only method of the base class
:class:`~sage.symbolic.expression_conversions.ExpressionTreeWalker`
that is reimplemented, since it manages the composition of
``abs`` with ``cos`` or ``sin``.
INPUT:
- ``ex`` -- a symbolic expression
- ``operator`` -- an operator
OUTPUT:
- a symbolic expression, equivalent to ``ex`` with ``abs(cos(...))``
and ``abs(sin(...))`` simplified, according to the range of their
argument.
EXAMPLES::
sage: from sage.manifolds.utilities import SimplifyAbsTrig
sage: assume(-pi/2 < x, x<0)
sage: a = abs(sin(x))
sage: s = SimplifyAbsTrig(a)
sage: a.operator()
abs
sage: s.composition(a, a.operator())
sin(-x)
::
sage: a = exp(function('f')(x)) # no abs(sin_or_cos(...))
sage: a.operator()
exp
sage: s.composition(a, a.operator())
e^f(x)
::
sage: forget() # no longer any assumption on x
sage: a = abs(cos(sin(x))) # simplifiable since -1 <= sin(x) <= 1
sage: s.composition(a, a.operator())
cos(sin(x))
sage: a = abs(sin(cos(x))) # not simplifiable
sage: s.composition(a, a.operator())
abs(sin(cos(x)))
"""
if operator is abs_symbolic:
argum = ex.operands()[0] # argument of abs
if argum.operator() is sin:
# Case of abs(sin(...))
x = argum.operands()[0] # argument of sin
w0 = SR.wild()
if x.has(abs_symbolic(sin(w0))) or x.has(abs_symbolic(
cos(w0))):
x = self(x) # treatment of nested abs(sin_or_cos(...))
# Simplifications for values of x in the range [-pi, 2*pi]:
if x >= 0 and x <= pi:
ex = sin(x)
elif (x > pi and x <= 2 * pi) or (x >= -pi and x < 0):
ex = -sin(x)
return ex
if argum.operator() is cos:
# Case of abs(cos(...))
x = argum.operands()[0] # argument of cos
w0 = SR.wild()
if x.has(abs_symbolic(sin(w0))) or x.has(abs_symbolic(
cos(w0))):
x = self(x) # treatment of nested abs(sin_or_cos(...))
# Simplifications for values of x in the range [-pi, 2*pi]:
if (x >= -pi / 2 and x <= pi / 2) or (x >= 3 * pi / 2
and x <= 2 * pi):
ex = cos(x)
elif (x > pi / 2 and x <= 3 * pi / 2) or (x >= -pi
and x < -pi / 2):
ex = -cos(x)
return ex
# If no pattern is found, we default to ExpressionTreeWalker:
return super(SimplifyAbsTrig, self).composition(ex, operator)
