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 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()
"""
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 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"""
Return 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.
For more information, see :wikipedia:`Cross-cap`.
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 Dini(a=1, b=1, name="Dini's surface"):
r"""
Return 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.
For more information, see :wikipedia:`Dini%27s_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()
Graphics3d Object
"""
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 evaluate_L(self, affine_parameter, solution_key=None):
r"""
Compute the conserved angular momentum about the rotation axis `L` at
a given value of the affine parameter `\lambda`.
INPUT:
- ``affine_parameter`` -- value of the affine parameter `\lambda`
- ``solution_key`` -- (default: ``None``) string denoting the numerical
solution to use for the evaluation; if ``None``, the latest solution
computed by :meth:`integrate` is used.
OUTPUT:
- value of `L`
"""
p = self.evaluate_tangent_vector(affine_parameter,
solution_key=solution_key)
point = p.parent().base_point()
BLchart = self._spacetime.boyer_lindquist_coordinates()
r, th = BLchart(point)[1:3]
a, m = self._a, self._m
r2 = r**2
a2 = a**2
rho2 = r2 + (a * cos(th))**2
p_comp = p.components(basis=BLchart.frame().at(point))
pt = p_comp[0]
pph = p_comp[3]
bs = 2 * a * m * r * sin(th)**2 / rho2
L = -bs * pt + (r2 + a2 + a * bs) * sin(th)**2 * pph
return L.substitute(self._numerical_substitutions())
def Torus(r=2, R=3, name="Torus"):
r"""
Return 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.
For more information, see :wikipedia:`Torus`.
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 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 Klein(r=1, name="Klein bottle"):
r"""
Return 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.
For more information, see :wikipedia:`Klein_bottle`.
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 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 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.
EXAMPLES::
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 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 Torus(R=2, r=1, names=None):
"""
Generate a 2-dimensional torus embedded in Euclidean space.
The shortcut operator ``.<,>`` can be used to specify the coordinates.
INPUT:
- ``R`` -- (default: ``2``) distance form the center to the
center of the tube
- ``r`` -- (default: ``1``) radius of the tube
- ``names`` -- (default: ``None``) name of the coordinates,
automatically set by the shortcut operator
OUTPUT:
- Riemannian manifold
EXAMPLES::
sage: T.<theta, phi> = manifolds.Torus(3, 1)
sage: T
2-dimensional Riemannian submanifold T embedded in the Euclidean
space E^3
sage: T.atlas()
[Chart (T, (theta, phi))]
sage: T.embedding().display()
T → E^3
(theta, phi) ↦ (X, Y, Z) = ((cos(theta) + 3)*cos(phi),
(cos(theta) + 3)*sin(phi),
sin(theta))
sage: T.metric().display()
gamma = dtheta⊗dtheta + (cos(theta)^2 + 6*cos(theta) + 9) dphi⊗dphi
"""
from sage.functions.trig import cos, sin
from sage.manifolds.manifold import Manifold
from sage.manifolds.differentiable.examples.euclidean import EuclideanSpace
E = EuclideanSpace(3, symbols='X Y Z')
M = Manifold(2, 'T', ambient=E, structure="Riemannian")
if names is None:
names = ("th", "ph")
names = tuple([names[i] + ":(-pi,pi):periodic" for i in range(2)])
C = M.chart(names=names)
M._first_ngens = C._first_ngens
th, ph = C[:]
coordfunc = [(R + r * cos(th)) * cos(ph), (R + r * cos(th)) * sin(ph),
r * sin(th)]
imm = M.diff_map(E, coordfunc)
M.set_embedding(imm)
M.induced_metric()
return M
def metric(self):
r"""
Return the metric tensor.
EXAMPLES::
sage: from kerrgeodesic_gw import KerrBH
sage: a, m = var('a m')
sage: BH = KerrBH(a, m)
sage: BH.metric()
Lorentzian metric g on the Kerr spacetime M
sage: BH.metric().display()
g = -(a^2*cos(th)^2 - 2*m*r + r^2)/(a^2*cos(th)^2 + r^2) 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)^4 + (a^2*r^2 + r^4 + (a^4 + a^2*r^2)*cos(th)^2)*sin(th)^2)/(a^2*cos(th)^2 + r^2) dph*dph
The Schwarzschild metric::
sage: KerrBH(0, m).metric().display()
g = (2*m - r)/r dt*dt - r/(2*m - r) dr*dr + r^2 dth*dth
+ r^2*sin(th)^2 dph*dph
"""
if self._metric is None:
# Initialization of the metric tensor in Boyer-Lindquist coordinates
cBL = self.boyer_lindquist_coordinates()
t, r, th, ph = cBL[:]
g = super(KerrBH, self).metric() # the initialized metric object
m = self._m
a = self._a
r2 = r**2
a2 = a**2
rho2 = r2 + (a * cos(th))**2
Delta = r2 - 2 * m * r + a2
fBL = cBL.frame() # vector frame associated to BL coordinates
g[fBL, 0, 0, cBL] = -1 + 2 * m * r / rho2
g[fBL, 0, 3, cBL] = -2 * a * m * r * sin(th)**2 / rho2
g[fBL, 1, 1, cBL] = rho2 / Delta
g[fBL, 2, 2, cBL] = rho2
g[fBL, 3, 3, cBL] = (r2 + a2 + 2 * m * r *
(a * sin(th))**2 / rho2) * sin(th)**2
for i in self.irange():
g[fBL, i, i, cBL].simplify()
g[fBL, 0, 3, cBL].simplify()
return self._metric
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 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 _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 _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 Helicoid(h=1, name="Helicoid"):
r"""
Return 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.
For more information, see :wikipedia:`Helicoid`.
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, coords, name)
def evaluate_E(self, affine_parameter, solution_key=None):
r"""
Compute the conserved energy `E` at a given value of the affine
parameter `\lambda`.
INPUT:
- ``affine_parameter`` -- value of the affine parameter `\lambda`
- ``solution_key`` -- (default: ``None``) string denoting the numerical
solution to use for the evaluation; if ``None``, the latest solution
computed by :meth:`integrate` is used.
OUTPUT:
- value of `E`
"""
p = self.evaluate_tangent_vector(affine_parameter,
solution_key=solution_key)
point = p.parent().base_point()
BLchart = self._spacetime.boyer_lindquist_coordinates()
r, th = BLchart(point)[1:3]
a, m = self._a, self._m
rho2 = r**2 + (a * cos(th))**2
p_comp = p.components(basis=BLchart.frame().at(point))
pt = p_comp[0]
pph = p_comp[3]
b = 2 * m * r / rho2
E = (1 - b) * pt + b * a * sin(th)**2 * pph
return E.substitute(self._numerical_substitutions())
def Catenoid(c=1, name="Catenoid"):
r"""
Return a catenoid surface, with parametric representation
.. MATH::
\begin{aligned}
x(u, v) & = c \cosh(v/c) \cos(u); \\
y(u, v) & = c \cosh(v/c) \sin(u); \\
z(u, v) & = v.
\end{aligned}
INPUT:
- ``c`` -- surface parameter.
- ``name`` -- string. Name of the surface.
For more information, see :wikipedia:`Catenoid`.
EXAMPLES::
sage: cat = surfaces.Catenoid(); cat
Parametrized surface ('Catenoid') with equation (cos(u)*cosh(v), cosh(v)*sin(u), v)
sage: cat.plot()
Graphics3d Object
"""
u, v = var('u, v')
catenoid_eq = [c * cosh(v / c) * cos(u), c * cosh(v / c) * sin(u), v]
coords = ((u, 0, 2 * pi), (v, -1, 1))
return ParametrizedSurface3D(catenoid_eq, coords, name)
def evaluate_Q(self, affine_parameter, solution_key=None):
r"""
Compute the Carter constant `Q` at a given value of the affine
parameter `\lambda`.
INPUT:
- ``affine_parameter`` -- value of the affine parameter `\lambda`
- ``solution_key`` -- (default: ``None``) string denoting the numerical
solution to use for the evaluation; if ``None``, the latest solution
computed by :meth:`integrate` is used.
OUTPUT:
- value of `Q`
"""
p = self.evaluate_tangent_vector(affine_parameter,
solution_key=solution_key)
point = p.parent().base_point()
BLchart = self._spacetime.boyer_lindquist_coordinates()
r, th = BLchart(point)[1:3]
a = self._a
rho4 = (r**2 + (a * cos(th))**2)**2
mu2 = self.evaluate_mu(affine_parameter)**2
E2 = self.evaluate_E(affine_parameter, solution_key=solution_key)**2
L2 = self.evaluate_L(affine_parameter, solution_key=solution_key)**2
p_comp = p.components(basis=BLchart.frame().at(point))
pth = p_comp[2]
Q = rho4 * pth**2 + cos(th)**2 * (L2 / sin(th)**2 + a**2 * (mu2 - E2))
return Q.substitute(self._numerical_substitutions())
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 _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 _circ_arc(t0, t1, c, r, num_pts=5000):
r""" Circular arc
INPUTS:
- ''t0'' -- starting parameter
- ''t1'' -- ending parameter
- ''c'' -- center point of the circle
- ''r'' -- radius of circle
- ''num_pts'' -- (default 100) number of points on polygon
OUTPUT:
- ''ca'' -- a polygonal approximation of a circular arc centered
at c and radius r, starting at t0 and ending at t1
EXAMPLES::
sage: ca=_circ_arc(0.1,0.2,0.0,1.0,100)
"""
from sage.plot.plot import line, parametric_plot
from sage.functions.trig import (cos, sin)
from sage.all import var
t00 = t0
t11 = t1
## To make sure the line is correct we reduce all arguments to the same branch,
## e.g. [0,2pi]
pi = RR.pi()
while (t00 < 0.0):
t00 = t00 + RR(2.0 * pi)
while (t11 < 0):
t11 = t11 + RR(2.0 * pi)
while (t00 > 2 * pi):
t00 = t00 - RR(2.0 * pi)
while (t11 > 2 * pi):
t11 = t11 - RR(2.0 * pi)
xc = CC(c).real()
yc = CC(c).imag()
num_pts = 3
t = var('t')
if t11 > t00:
ca = parametric_plot((r * cos(t) + xc, r * sin(t) + yc), (t, t00, t11))
else:
ca = parametric_plot((r * cos(t) + xc, r * sin(t) + yc), (t, t11, t00))
#L0 = [[RR(r*cos(t00+i*(t11-t00)/num_pts))+xc,RR(r*sin(t00+i*(t11-t00)/num_pts))+yc] for i in range(0 ,num_pts)]
#ca=line(L0)
return ca
def 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 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 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 _derivative_(self, x, diff_param=None):
"""
EXAMPLES::
sage: x = var('x')
sage: fresnel_sin(x).diff(x)
sin(1/2*pi*x^2)
"""
from sage.functions.trig import sin
return sin(pi*x**2/2)
def map_to_Euclidean(self):
r"""
Map from Kerr spacetime to the Euclidean 4-space, based on
Boyer-Lindquist coordinates
"""
if self._map_to_E4 is None:
E4 = EuclideanSpace(4,
coordinates='Cartesian',
symbols='t x y z',
start_index=0)
X4 = E4.cartesian_coordinates()
BL = self.boyer_lindquist_coordinates()
t, r, th, ph = BL[:]
self._map_to_E4 = self.diff_map(E4, {
(BL, X4):
[t, r * sin(th) * cos(ph), r * sin(th) * sin(ph), r * cos(th)]
},
name='F')
return self._map_to_E4
def _derivative_(self, x, diff_param=None):
"""
EXAMPLES::
sage: x = var('x')
sage: fresnel_sin(x).diff(x)
sin(1/2*pi*x^2)
"""
from sage.functions.trig import sin
return sin(pi * x**2 / 2)
def transform(self, radius=None, azimuth=None, height=None):
"""
A cylindrical coordinates transform.
EXAMPLES::
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 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()
Graphics3d Object
"""
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 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 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 triangulation(self):
r"""
Plot a regular polygon with some diagonals.
If ``self`` is positive and integral then this will be a triangulation.
.. PLOT::
:width: 600 px
G = path_tableaux.FriezePattern([1,2,7,5,3,7,4,1]).triangulation()
p = graphics_array(G, 7, 6)
sphinx_plot(p)
EXAMPLES::
sage: path_tableaux.FriezePattern([1,2,7,5,3,7,4,1]).triangulation()
Graphics object consisting of 25 graphics primitives
sage: path_tableaux.FriezePattern([1,2,1/7,5,3]).triangulation()
Graphics object consisting of 12 graphics primitives
sage: K.<sqrt2> = NumberField(x^2-2)
sage: path_tableaux.FriezePattern([1,sqrt2,1,sqrt2,3,2*sqrt2,5,3*sqrt2,1], field=K).triangulation()
Graphics object consisting of 24 graphics primitives
"""
n = len(self) - 1
cd = CylindricalDiagram(self).diagram
from sage.plot.plot import Graphics
from sage.plot.line import line
from sage.plot.text import text
from sage.functions.trig import sin, cos
from sage.all import pi
G = Graphics()
G.set_aspect_ratio(1.0)
vt = [(cos(2 * theta * pi / (n)), sin(2 * theta * pi / (n)))
for theta in range(n + 1)]
for i, p in enumerate(vt):
G += text(str(i), [1.05 * p[0], 1.05 * p[1]])
for i, r in enumerate(cd):
for j, a in enumerate(r[:n]):
if a == 1:
G += line([vt[i], vt[j]])
G.axes(False)
return G
def _derivative_(self, z, diff_param=None):
r"""
The derivative of `\operatorname{Si}(z)` is `\sin(z)/z` if `z`
is not zero. The derivative at `z = 0` is `1` (but this
exception is not currently implemented).
EXAMPLES::
sage: x = var('x')
sage: f = sin_integral(x)
sage: f.diff(x)
sin(x)/x
sage: f = sin_integral(x^2)
sage: f.diff(x)
2*sin(x^2)/x
"""
return sin(z)/z
def _derivative_(self, z, diff_param=None):
r"""
The derivative of `\operatorname{Si}(z)` is `\sin(z)/z` if `z`
is not zero. The derivative at `z = 0` is `1` (but this
exception is not currently implemented).
EXAMPLES::
sage: x = var('x')
sage: f = sin_integral(x)
sage: f.diff(x)
sin(x)/x
sage: f = sin_integral(x^2)
sage: f.diff(x)
2*sin(x^2)/x
"""
return sin(z) / z
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 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 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 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 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 _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 f(t):
return (radius_arc * cos(t) + center[0],
radius_arc * sin(t) + center[1])
def f(t):
return exp(I * ((cos(t) + I * sin(t)) *
arc_radius + arc_center)) * radius
radius=norm(first_vector)
if norm(second_vector)!=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 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
