def plot_cluster_fan_stereographically(self, northsign=1, north=None, right=None, colors=None):
from sage.plot.graphics import Graphics
from sage.plot.point import point
from sage.misc.flatten import flatten
from sage.plot.line import line
from sage.misc.functional import norm
if self.rk !=3:
raise ValueError("Can only stereographically project fans in 3d.")
if not self.is_finite() and self._depth == infinity:
raise ValueError("For infinite algebras you must specify the depth.")
if north == None:
if self.is_affine():
north = vector(self.delta())
else:
north = vector( (-1,-1,-1) )
if right == None:
if self.is_affine():
right = vector(self.gamma())
else:
right = vector( (1,0,0) )
if colors == None:
colors = dict([(0,'red'),(1,'green'),(2,'blue'),(3,'cyan'),(4,'yellow')])
G = Graphics()
roots = list(self.g_vectors())
compatible = []
while roots:
x = roots.pop()
for y in roots:
if self.compatibility_degree(x,y) == 0:
compatible.append((x,y))
for (u,v) in compatible:
G += _stereo_arc(vector(u),vector(v),vector(u+v),north=northsign*north,right=right,thickness=0.5,color='black')
for i in range(3):
orbit = self.ith_orbit(i)
for j in orbit:
G += point(_stereo_coordinates(vector(orbit[j]),north=northsign*north,right=right),color=colors[i],zorder=len(G))
if self.is_affine():
tube_vectors = map(vector,flatten(self.affine_tubes()))
for v in tube_vectors:
G += point(_stereo_coordinates(v,north=northsign*north,right=right),color=colors[3],zorder=len(G))
if north != vector(self.delta()):
G += _stereo_arc(tube_vectors[0],tube_vectors[1],vector(self.delta()),north=northsign*north,right=right,thickness=2,color=colors[4],zorder=0)
else:
# FIXME: refactor this before publishing
tube_projections = [
_stereo_coordinates(v,north=northsign*north,right=right)
for v in tube_vectors ]
t=min((G.get_minmax_data()['xmax'],G.get_minmax_data()['ymax']))
G += line([tube_projections[0],tube_projections[0]+t*(_normalize(tube_projections[0]-tube_projections[1]))],thickness=2,color=colors[4],zorder=0)
G += line([tube_projections[1],tube_projections[1]+t*(_normalize(tube_projections[1]-tube_projections[0]))],thickness=2,color=colors[4],zorder=0)
G.set_aspect_ratio(1)
G._show_axes = False
return G
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
if abs(det(matrix([p-s,q-s])))<0.01:
return line((p,q),**kwds)
(cx,cy)=var('cx','cy')
equations=[
2*cx*(s[0]-p[0])+2*cy*(s[1]-p[1]) == s[0]**2+s[1]**2-p[0]**2-p[1]**2,
2*cx*(s[0]-q[0])+2*cy*(s[1]-q[1]) == s[0]**2+s[1]**2-q[0]**2-q[1]**2
]
c = vector( [solve( equations, (cx,cy), solution_dict=True )[0][i] for i in [cx,cy]] )
r = norm(p-c)
a_p,a_q,a_s = map( _to_angle, [p-c,q-c,s-c])
angles = [a_p,a_q,a_s]
angles.sort()
if a_s == angles[0]:
return arc( c, r, angle=angles[2], sector=(0,2*pi-angles[2]+angles[1]), **kwds)
if a_s == angles[1]:
return arc( c, r, angle=angles[0], sector=(0,angles[2]-angles[0]), **kwds)
if a_s == angles[2]:
return arc( c, r, angle=angles[1], sector=(0,2*pi-angles[1]+angles[0]), **kwds)
def plot_y(self, plot_points=128, **kwds):
r"""Plot the y-part of the path in the complex y-plane.
Additional arguments and keywords are passed to
``matplotlib.pyplot.plot``.
Parameters
----------
N : int
The number of interpolating points used to plot.
t0 : double
Starting t-value in [0,1].
t1 : double
Ending t-value in [0,1].
Returns
-------
plt : Sage plot.
A plot of the complex y-projection of the path.
"""
s = numpy.linspace(0, 1, plot_points, dtype=double)
vals = numpy.array([self.get_y(si)[0] for si in s], dtype=complex)
pts = [(real_part(y), imag_part(y)) for y in vals]
plt = line(pts, **kwds)
return plt
def plot_n_matrices_eigenvectors(self, n, side='right', color_index=0, draw_line=False):
r"""
INPUT:
- ``n`` -- integer, length
- ``side`` -- ``'left'`` or ``'right'``, drawing left or right
eigenvectors
- ``color_index`` -- 0 for first letter, -1 for last letter
- ``draw_line`` -- boolean
EXAMPLES::
sage: from slabbe.matrix_cocycle import cocycles
sage: ARP = cocycles.ARP()
sage: G = ARP.plot_n_matrices_eigenvectors(2)
"""
from sage.plot.graphics import Graphics
from sage.plot.point import point
from sage.plot.line import line
from sage.plot.text import text
from sage.plot.colors import hue
from sage.modules.free_module_element import vector
from .matrices import M3to2
R = self.n_matrices_eigenvectors(n)
L = [(w, M3to2*(a/sum(a)), M3to2*(b/sum(b))) for (w,a,b) in R]
G = Graphics()
alphabet = self._language._alphabet
color_ = dict( (letter, hue(i/float(len(alphabet)))) for i,letter in
enumerate(alphabet))
for letter in alphabet:
L_filtered = [(w,p1,p2) for (w,p1,p2) in L if w[color_index] == letter]
words,rights,lefts = zip(*L_filtered)
if side == 'right':
G += point(rights, color=color_[letter], legend_label=letter)
elif side == 'left':
G += point(lefts, color=color_[letter], legend_label=letter)
else:
raise ValueError("side(=%s) should be left or right" % side)
if draw_line:
for (a,b) in L:
G += line([a,b], color='black', linestyle=":")
G += line([M3to2*vector(a) for a in [(1,0,0), (0,1,0), (0,0,1), (1,0,0)]])
title = "%s eigenvectors, colored by letter w[%s] of cylinder w" % (side, color_index)
G += text(title, (0.5, 1.05), axis_coords=True)
G.axes(False)
return G
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 piecewise_linear_image(A,B):
# Jumps up and down going around circle, not used
v = circle_drops(A,B)
G = Graphics()
w = [(Rational(i)/len(v), j) for i,j in enumerate(v)]
for pt in w:
G += line([(pt[0],pt[1]),(pt[0]+Rational(1)/len(w),pt[1])])
return G
def legend_3d(hyperplane_arrangement, hyperplane_colors, length):
r"""
Create plot of a 3d legend for an arrangement of planes in 3-space. The
``length`` parameter determines whether short or long labels are used in
the legend.
INPUT:
- ``hyperplane_arrangement`` -- a hyperplane arrangement
- ``hyperplane_colors`` -- list of colors
- ``length`` -- either ``'short'`` or ``'long'``
OUTPUT:
- A graphics object.
EXAMPLES::
sage: a = hyperplane_arrangements.semiorder(3)
sage: from sage.geometry.hyperplane_arrangement.plot import legend_3d
sage: legend_3d(a, list(colors.values())[:6],length='long')
Graphics object consisting of 6 graphics primitives
sage: b = hyperplane_arrangements.semiorder(4)
sage: c = b.essentialization()
sage: legend_3d(c, list(colors.values())[:12], length='long')
Graphics object consisting of 12 graphics primitives
sage: legend_3d(c, list(colors.values())[:12], length='short')
Graphics object consisting of 12 graphics primitives
sage: p = legend_3d(c, list(colors.values())[:12], length='short')
sage: p.set_legend_options(ncol=4)
sage: type(p)
<class 'sage.plot.graphics.Graphics'>
"""
if hyperplane_arrangement.dimension() != 3:
raise ValueError('arrangements must be in 3-space')
hyps = hyperplane_arrangement.hyperplanes()
N = len(hyperplane_arrangement)
if length == 'short':
labels = [' ' + str(i) for i in range(N)]
else:
labels = [' ' + hyps[i]._repr_linear(include_zero=False) for i in
range(N)]
p = Graphics()
for i in range(N):
p += line([(0,0),(0,0)], color=hyperplane_colors[i], thickness=8,
legend_label=labels[i], axes=False)
p.set_legend_options(title='Hyperplanes', loc='center', labelspacing=0.4,
fancybox=True, font_size='x-large', ncol=2)
p.legend(True)
return p
def plot(self, **kwargs):
"""
Plot this Newton polygon.
.. NOTE::
All usual rendering options (color, thickness, etc.) are available.
EXAMPLES:
sage: from sage.geometry.newton_polygon import NewtonPolygon
sage: NP = NewtonPolygon([ (0,0), (1,1), (2,6) ])
sage: polygon = NP.plot()
"""
vertices = self.vertices()
if len(vertices) == 0:
from sage.plot.graphics import Graphics
return Graphics()
else:
from sage.plot.line import line
(xstart,ystart) = vertices[0]
(xend,yend) = vertices[-1]
if self.last_slope() is Infinity:
return line([(xstart, ystart+1), (xstart,ystart+0.5)], linestyle="--", **kwargs) \
+ line([(xstart, ystart+0.5)] + vertices + [(xend, yend+0.5)], **kwargs) \
+ line([(xend, yend+0.5), (xend, yend+1)], linestyle="--", **kwargs)
else:
return line([(xstart, ystart+1), (xstart,ystart+0.5)], linestyle="--", **kwargs) \
+ line([(xstart, ystart+0.5)] + vertices + [(xend+0.5, yend + 0.5*self.last_slope())], **kwargs) \
+ line([(xend+0.5, yend + 0.5*self.last_slope()), (xend+1, yend+self.last_slope())], linestyle="--", **kwargs)
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 get_background_graphic(self, **bdry_options):
r"""
Return a graphic object that makes the model easier to visualize.
For the upper half space, the background object is the ideal boundary.
EXAMPLES::
sage: hp = HyperbolicPlane().UHP().get_background_graphic()
"""
from sage.plot.line import line
bd_min = bdry_options.get('bd_min', -5)
bd_max = bdry_options.get('bd_max', 5)
return line(((bd_min, 0), (bd_max, 0)), color='black')
def show(self, boundary=True, **options):
r"""
Plot ``self``.
EXAMPLES:
First some lines::
sage: PD = HyperbolicPlane().PD()
sage: PD.get_geodesic(0, 1).show()
Graphics object consisting of 2 graphics primitives
sage: PD.get_geodesic(0, 0.3+0.8*I).show()
Graphics object consisting of 2 graphics primitives
Then some generic geodesics::
sage: PD.get_geodesic(-0.5, 0.3+0.4*I).show()
Graphics object consisting of 2 graphics primitives
sage: PD.get_geodesic(-1, exp(3*I*pi/7)).show(linestyle="dashed", color="red")
Graphics object consisting of 2 graphics primitives
sage: PD.get_geodesic(exp(2*I*pi/11), exp(1*I*pi/11)).show(thickness=6, color="orange")
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()]
# Check to see if it's a line
if abs(bd_1 + bd_2) < EPSILON:
pic = line([end_1, end_2], **opts)
else:
# If we are here, we know it's not a line
# So we compute the center and radius of the circle
invdet = RR.one() / (real(bd_1)*imag(bd_2) - real(bd_2)*imag(bd_1))
centerx = (imag(bd_2) - imag(bd_1)) * invdet
centery = (real(bd_1) - real(bd_2)) * invdet
center = centerx + I * centery
radius = RR(abs(bd_1 - center))
# Now we calculate the angles for the arc
theta1 = CC(end_1 - center).arg()
theta2 = CC(end_2 - center).arg()
theta1, theta2 = sorted([theta1, theta2])
# Make sure the sector is inside the disk
if theta2 - theta1 > pi:
theta1 += 2 * pi
pic = arc((centerx, centery), radius,
sector=(theta1, theta2), **opts)
if boundary:
pic += self._model.get_background_graphic()
return pic
def show(self, boundary=True, **options):
r"""
Plot ``self``.
EXAMPLES::
sage: HyperbolicPlane().KM().get_geodesic((0,0), (1,0)).show()
Graphics object consisting of 2 graphics primitives
"""
opts = {'axes': False, 'aspect_ratio': 1}
opts.update(self.graphics_options())
pic = line([k.coordinates() for k in self.endpoints()], **opts)
if boundary:
pic += self._model.get_background_graphic()
return pic
def plot2d(self,depth=None):
# FIXME: refactor this before publishing
from sage.plot.line import line
from sage.plot.graphics import Graphics
if self._n !=2:
raise ValueError("Can only 2d plot fans.")
if depth == None:
depth = self._depth
if not self.is_finite() and depth==infinity:
raise ValueError("For infinite algebras you must specify the depth.")
colors = dict([(0,'red'),(1,'green')])
G = Graphics()
for i in range(2):
orbit = self.ith_orbit(i,depth=depth)
for j in orbit:
G += line([(0,0),vector(orbit[j])],color=colors[i],thickness=0.5, zorder=2*j+1)
G.set_aspect_ratio(1)
G._show_axes = False
return G
def plot(self, m, pointsize=100, thickness=3, axes=False):
r"""
Return 2d graphics object contained in the primal box [-m,m]^d.
INPUT:
- ``pointsize``, integer (default:``100``),
- ``thickness``, integer (default:``3``),
- ``axes``, bool (default:``False``),
EXAMPLES::
sage: from slabbe import BondPercolationSample
sage: S = BondPercolationSample(0.5,2)
sage: S.plot(2) # optional long
It works in 3d!!::
sage: S = BondPercolationSample(0.5,3)
sage: S.plot(3, pointsize=10, thickness=1) # optional long
Graphics3d Object
"""
s = ""
s += "\\begin{tikzpicture}\n"
s += "[inner sep=0pt,thick,\n"
s += "reddot/.style={fill=red,draw=red,circle,minimum size=5pt}]\n"
s += "\\clip %s rectangle %s;\n" % ((-m-.4,-m-.4), (m+.4,m+.4))
G = Graphics()
for u in self.cluster_in_box(m+1):
G += point(u, color='blue', size=pointsize)
for (u,v) in self.edges_in_box(m+1):
G += line((u,v), thickness=thickness, alpha=0.8)
G += text("p=%.3f" % self._p, (0.5,1.03), axis_coords=True, color='black')
G += circle((0,0), 0.5, color='red', thickness=thickness)
if self._dimension == 2:
G.axes(axes)
return G
def show(self, **args):
r"""
Displays the pseudoline arrangement as a wiring diagram.
INPUT:
- ``**args`` -- any arguments to be forwarded to the ``show`` method. In
particular, to tune the dimensions, use the ``figsize`` argument
(example below).
EXAMPLES::
sage: from sage.geometry.pseudolines import PseudolineArrangement
sage: permutations = [[3, 2, 1], [3, 2, 0], [3, 1, 0], [2, 1, 0]]
sage: p = PseudolineArrangement(permutations)
sage: p.show(figsize=[7,5])
TESTS::
sage: from sage.geometry.pseudolines import PseudolineArrangement
sage: permutations = [[3, 2, 1], [3, 2, 0], [3, 0, 1], [2, 0, 1]]
sage: p = PseudolineArrangement(permutations)
sage: p.show()
Traceback (most recent call last):
...
ValueError: There has been a problem while plotting the figure...
"""
x = 1
from sage.plot.line import line
from sage.plot.text import text
lines = [[(0,self._n-1-i)] for i in range(self._n)]
for i,j in self.transpositions():
iy = lines[i][-1][1]
jy = lines[j][-1][1]
lines[i].append((x, iy))
lines[j].append((x, jy))
if abs(iy-jy) != 1:
raise ValueError(
"There has been a problem while plotting the figure. It "+
"seems that the lines are not correctly ordered. Please "+
"check the pseudolines modules documentation, there is a "
+"warning about that. ")
lines[i].append((x+2,jy))
lines[j].append((x+2,iy))
x += 2
L = line([(1,1)])
for i, l in enumerate(lines):
l.append((x+2, l[-1][1]))
L += line(l)
L += text(str(i), (0, l[0][1]+.3), horizontal_alignment="right")
L += text(str(i), (x+2, l[-1][1]+.3), horizontal_alignment="left")
return L.show(axes = False, **args)
def list_plot(cls, points, **kwargs):
r"""
Returns a sage figure containing a list plot.
INPUT:
- ``cls`` -- class on which the function is invoked.
- ``points`` -- list or tuple of 2D or 3D points to be plotted.
- ``joined`` -- boolean (default: False) flag triggering the production
of a graph whose points are joined instead of a scatter plot.
- ``alpha`` -- number (default: not used) opacity value of the points
(or lines) in the produced graph.
- ``size`` -- integer (default: not used) size of the points (or lines)
in the produced graph.
Other named arguments affecting the graphic style are forwarded to
matplotlib's ``plot`` or ``scatter``.
OUTPUT:
figure containing a list plot.
EXAMPLES:
The following instructions generate and show a figure showing three
points:
::
>>> points = ((1, 1), (3, -1), (7, 2))
>>> from yaplf.graph import SagePlotter
>>> SagePlotter.list_plot(points)
The same graph can be obtained joining the single points:
::
>>> SagePlotter.list_plot(points, joined = True)
When ``joined`` is set to ``True``, the ``size``, ``color``, and
``alpha`` arguments affect respectively the line size, color, and
opacity:
::
>>> SagePlotter.list_plot(points, joined = True, size = 3,
... alpha = .2)
When the first argument of ``list_plot`` is a list or tuple of
three-sized list or tuples, the result is a 3D graph:
::
>>> points = ((1, 3, -4), (2, 1, 2), (1, 6, 5))
>>> SagePlotter.list_plot(points)
AUTHORS:
- Dario Malchiodi (2010-02-22)
"""
try:
joined = kwargs['joined']
del kwargs['joined']
except KeyError:
joined = False
if len(shape(points)) == 1:
points = zip(range(len(points)), points)
if len(points[0]) == 2:
try:
size = kwargs['size']
del kwargs['size']
if joined:
kwargs['thickness'] = size
else:
kwargs['pointsize'] = size
except KeyError:
pass
elif len(points[0]) == 3:
try:
alpha = kwargs['alpha']
del kwargs['alpha']
kwargs['opacity'] = alpha
if joined:
size = kwargs['size']
del kwargs['size']
kwargs['thickness'] = size
except KeyError:
pass
else:
raise ValueError('scatter() only available for 2D and 3D points')
if joined:
return line(points, **kwargs)
else:
return point(points, **kwargs)
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() # long time 2s
Graphics object consisting of 219 graphics primitives
"""
# 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):
# 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 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 plot(self, chart=None, ambient_coords=None, mapping=None, prange=None,
include_end_point=(True, True), end_point_offset=(0.001, 0.001),
max_value=8, parameters=None, color='red', style='-',
thickness=1, plot_points=75, label_axes=True,
aspect_ratio='automatic'):
r"""
Plot the current curve (``self``) in a Cartesian graph based on the
coordinates of some ambient chart.
The curve is drawn in terms of two (2D graphics) or three (3D graphics)
coordinates of a given chart, called hereafter the *ambient chart*.
The ambient chart's domain must overlap with the curve's codomain or
with the codomain of the composite curve `\Phi\circ c`, where `c` is
``self`` and `\Phi` some manifold differential mapping (argument
``mapping`` below).
INPUT:
- ``chart`` -- (default: ``None``) the ambient chart (see above);
if ``None``, the default chart of the codomain of the curve (or of
the curve composed with `\Phi`) is used
- ``ambient_coords`` -- (default: ``None``) tuple containing the 2 or 3
coordinates of the ambient chart in terms of which the plot is
performed; if ``None``, all the coordinates of the ambient chart are
considered
- ``mapping`` -- (default: ``None``) differentiable mapping `\Phi`
(instance of
:class:`~sage.geometry.manifolds.diffmapping.DiffMapping`)
providing the link between ``self`` and the ambient chart ``chart``
(cf. above); if ``None``, the ambient chart is supposed to be defined
on the codomain of the curve ``self``.
- ``prange`` -- (default: ``None``) range of the curve parameter for
the plot; if ``None``, the entire parameter range declared during the
curve construction is considered (with -Infinity
replaced by ``-max_value`` and +Infinity by ``max_value``)
- ``include_end_point`` -- (default: ``(True, True)``) determines
whether the end points of ``prange`` are included in the plot
- ``end_point_offset`` -- (default: ``(0.001, 0.001)``) offsets from
the end points when they are not included in the plot: if
``include_end_point[0] == False``, the minimal value of the curve
parameter used for the plot is ``prange[0] + end_point_offset[0]``,
while if ``include_end_point[1] == False``, the maximal value is
``prange[1] - end_point_offset[1]``.
- ``max_value`` -- (default: 8) numerical value substituted to
+Infinity if the latter is the upper bound of the parameter range;
similarly ``-max_value`` is the numerical valued substituted for
-Infinity
- ``parameters`` -- (default: ``None``) dictionary giving the numerical
values of the parameters that may appear in the coordinate expression
of ``self``
- ``color`` -- (default: 'red') color of the drawn curve
- ``style`` -- (default: '-') color of the drawn curve; NB: ``style``
is effective only for 2D plots
- ``thickness`` -- (default: 1) thickness of the drawn curve
- ``plot_points`` -- (default: 75) number of points to plot the curve
- ``label_axes`` -- (default: ``True``) boolean determining whether the
labels of the coordinate axes of ``chart`` shall be added to the
graph; can be set to ``False`` if the graph is 3D and must be
superposed with another graph.
- ``aspect_ratio`` -- (default: 'automatic') aspect ratio of the plot;
the default value ('automatic') applies only for 2D plots; for
3D plots, the default value is ``1`` instead.
OUTPUT:
- a graphic object, either an instance of
:class:`~sage.plot.graphics.Graphics` for a 2D plot (i.e. based on
2 coordinates of ``chart``) or an instance of
:class:`~sage.plot.plot3d.base.Graphics3d` for a 3D plot (i.e.
based on 3 coordinates of ``chart``)
EXAMPLES:
Plot of the lemniscate of Gerono::
sage: R2 = Manifold(2, 'R^2')
sage: X.<x,y> = R2.chart()
sage: R.<t> = RealLine()
sage: c = R2.curve([sin(t), sin(2*t)/2], (t, 0, 2*pi), name='c')
sage: c.plot() # 2D plot
Graphics object consisting of 1 graphics primitive
Plot for a subinterval of the curve's domain::
sage: c.plot(prange=(0,pi))
Graphics object consisting of 1 graphics primitive
Plot with various options::
sage: c.plot(color='green', style=':', thickness=3, aspect_ratio=1)
Graphics object consisting of 1 graphics primitive
Plot via a mapping to another manifold: loxodrome of a sphere viewed
in `\RR^3`::
sage: S2 = Manifold(2, 'S^2')
sage: U = S2.open_subset('U')
sage: XS.<th,ph> = U.chart(r'th:(0,pi):\theta ph:(0,2*pi):\phi')
sage: R3 = Manifold(3, 'R^3')
sage: X3.<x,y,z> = R3.chart()
sage: F = S2.diff_mapping(R3, {(XS, X3): [sin(th)*cos(ph),
....: sin(th)*sin(ph), cos(th)]}, name='F')
sage: F.display()
F: S^2 --> R^3
on U: (th, ph) |--> (x, y, z) = (cos(ph)*sin(th), sin(ph)*sin(th), cos(th))
sage: c = S2.curve([2*atan(exp(-t/10)), t], (t, -oo, +oo), name='c')
sage: graph_c = c.plot(mapping=F, max_value=40,
....: plot_points=200, thickness=2, label_axes=False) # 3D plot
sage: graph_S2 = XS.plot(X3, mapping=F, nb_values=11, color='black') # plot of the sphere
sage: show(graph_c + graph_S2) # the loxodrome + the sphere
Example of use of the argument ``parameters``: we define a curve with
some symbolic parameters ``a`` and ``b``::
sage: a, b = var('a b')
sage: c = R2.curve([a*cos(t) + b, a*sin(t)], (t, 0, 2*pi), name='c')
To make a plot, we set spectific values for ``a`` and ``b`` by means
of the Python dictionary ``parameters``::
sage: c.plot(parameters={a: 2, b: -3}, aspect_ratio=1)
Graphics object consisting of 1 graphics primitive
"""
from sage.rings.infinity import Infinity
from sage.misc.functional import numerical_approx
from sage.plot.graphics import Graphics
from sage.plot.line import line
from sage.geometry.manifolds.chart import Chart
from sage.geometry.manifolds.utilities import set_axes_labels
#
# The "effective" curve to be plotted
#
if mapping is None:
eff_curve = self
else:
eff_curve = mapping.restrict(self.codomain()) * self
#
# The chart w.r.t. which the curve is plotted
#
if chart is None:
chart = eff_curve._codomain.default_chart()
elif not isinstance(chart, Chart):
raise TypeError("{} is not a chart".format(chart))
#
# Coordinates of the above chart w.r.t. which the curve is plotted
#
if ambient_coords is None:
ambient_coords = chart[:] # all chart coordinates are used
n_pc = len(ambient_coords)
if n_pc != 2 and n_pc !=3:
raise ValueError("The number of coordinates involved in the " +
"plot must be either 2 or 3, not {}".format(n_pc))
ind_pc = [chart[:].index(pc) for pc in ambient_coords] # indices of plot
# coordinates
#
# Parameter range for the plot
#
if prange is None:
prange = (self._domain.lower_bound(), self._domain.upper_bound())
elif not isinstance(prange, (tuple, list)):
raise TypeError("{} is neither a tuple nor a list".format(prange))
elif len(prange) != 2:
raise ValueError("the argument prange must be a tuple/list " +
"of 2 elements")
tmin = prange[0]
tmax = prange[1]
if tmin == -Infinity:
tmin = -max_value
elif not include_end_point[0]:
tmin = tmin + end_point_offset[0]
if tmax == Infinity:
tmax = max_value
elif not include_end_point[1]:
tmax = tmax - end_point_offset[1]
tmin = numerical_approx(tmin)
tmax = numerical_approx(tmax)
#
# The coordinate expression of the effective curve
#
canon_chart = self._domain.canonical_chart()
transf = None
for chart_pair in eff_curve._coord_expression:
if chart_pair == (canon_chart, chart):
transf = eff_curve._coord_expression[chart_pair]
break
else:
# Search for a subchart
for chart_pair in eff_curve._coord_expression:
for schart in chart._subcharts:
if chart_pair == (canon_chart, schart):
transf = eff_curve._coord_expression[chart_pair]
if transf is None:
raise ValueError("No expression has been found for " +
"{} in terms of {}".format(self, format))
#
# List of points for the plot curve
#
plot_curve = []
dt = (tmax - tmin) / (plot_points - 1)
t = tmin
if parameters is None:
for i in range(plot_points):
x = transf(t, simplify=False)
plot_curve.append( [numerical_approx(x[j]) for j in ind_pc] )
t += dt
else:
for i in range(plot_points):
x = transf(t, simplify=False)
plot_curve.append(
[numerical_approx( x[j].substitute(parameters) )
for j in ind_pc] )
t += dt
#
# The plot
#
resu = Graphics()
resu += line(plot_curve, color=color, linestyle=style,
thickness=thickness)
if n_pc==2: # 2D graphic
resu.set_aspect_ratio(aspect_ratio)
if label_axes:
# We update the dictionary _extra_kwds (options to be passed
# to show()), instead of using the method
# Graphics.axes_labels() since the latter is not robust w.r.t.
# graph addition
resu._extra_kwds['axes_labels'] = [r'$'+latex(pc)+r'$'
for pc in ambient_coords]
else: # 3D graphic
if aspect_ratio == 'automatic':
aspect_ratio = 1
resu.aspect_ratio(aspect_ratio)
if label_axes:
labels = [str(pc) for pc in ambient_coords]
resu = set_axes_labels(resu, *labels)
return resu
def plot(self,
chart=None,
ambient_coords=None,
mapping=None,
prange=None,
include_end_point=(True, True),
end_point_offset=(0.001, 0.001),
parameters=None,
color='red',
style='-',
label_axes=True,
**kwds):
r"""
Plot the current curve in a Cartesian graph based on the
coordinates of some ambient chart.
The curve is drawn in terms of two (2D graphics) or three (3D graphics)
coordinates of a given chart, called hereafter the *ambient chart*.
The ambient chart's domain must overlap with the curve's codomain or
with the codomain of the composite curve `\Phi\circ c`, where `c` is
the current curve and `\Phi` some manifold differential map (argument
``mapping`` below).
INPUT:
- ``chart`` -- (default: ``None``) the ambient chart (see above);
if ``None``, the default chart of the codomain of the curve (or of
the curve composed with `\Phi`) is used
- ``ambient_coords`` -- (default: ``None``) tuple containing the 2
or 3 coordinates of the ambient chart in terms of which the plot
is performed; if ``None``, all the coordinates of the ambient chart
are considered
- ``mapping`` -- (default: ``None``) differentiable mapping `\Phi`
(instance of
:class:`~sage.manifolds.differentiable.diff_map.DiffMap`)
providing the link between the curve and the ambient chart ``chart``
(cf. above); if ``None``, the ambient chart is supposed to be defined
on the codomain of the curve.
- ``prange`` -- (default: ``None``) range of the curve parameter for
the plot; if ``None``, the entire parameter range declared during the
curve construction is considered (with -Infinity
replaced by ``-max_range`` and +Infinity by ``max_range``)
- ``include_end_point`` -- (default: ``(True, True)``) determines
whether the end points of ``prange`` are included in the plot
- ``end_point_offset`` -- (default: ``(0.001, 0.001)``) offsets from
the end points when they are not included in the plot: if
``include_end_point[0] == False``, the minimal value of the curve
parameter used for the plot is ``prange[0] + end_point_offset[0]``,
while if ``include_end_point[1] == False``, the maximal value is
``prange[1] - end_point_offset[1]``.
- ``max_range`` -- (default: 8) numerical value substituted to
+Infinity if the latter is the upper bound of the parameter range;
similarly ``-max_range`` is the numerical valued substituted for
-Infinity
- ``parameters`` -- (default: ``None``) dictionary giving the numerical
values of the parameters that may appear in the coordinate expression
of the curve
- ``color`` -- (default: 'red') color of the drawn curve
- ``style`` -- (default: '-') color of the drawn curve; NB: ``style``
is effective only for 2D plots
- ``thickness`` -- (default: 1) thickness of the drawn curve
- ``plot_points`` -- (default: 75) number of points to plot the curve
- ``label_axes`` -- (default: ``True``) boolean determining whether the
labels of the coordinate axes of ``chart`` shall be added to the
graph; can be set to ``False`` if the graph is 3D and must be
superposed with another graph.
- ``aspect_ratio`` -- (default: ``'automatic'``) aspect ratio of the
plot; the default value (``'automatic'``) applies only for 2D plots;
for 3D plots, the default value is ``1`` instead
OUTPUT:
- a graphic object, either an instance of
:class:`~sage.plot.graphics.Graphics` for a 2D plot (i.e. based on
2 coordinates of ``chart``) or an instance of
:class:`~sage.plot.plot3d.base.Graphics3d` for a 3D plot (i.e.
based on 3 coordinates of ``chart``)
EXAMPLES:
Plot of the lemniscate of Gerono::
sage: R2 = Manifold(2, 'R^2')
sage: X.<x,y> = R2.chart()
sage: R.<t> = RealLine()
sage: c = R2.curve([sin(t), sin(2*t)/2], (t, 0, 2*pi), name='c')
sage: c.plot() # 2D plot
Graphics object consisting of 1 graphics primitive
.. PLOT::
R2 = Manifold(2, 'R^2')
X = R2.chart('x y')
t = RealLine().canonical_coordinate()
c = R2.curve([sin(t), sin(2*t)/2], (t, 0, 2*pi), name='c')
g = c.plot()
sphinx_plot(g)
Plot for a subinterval of the curve's domain::
sage: c.plot(prange=(0,pi))
Graphics object consisting of 1 graphics primitive
.. PLOT::
R2 = Manifold(2, 'R^2')
X = R2.chart('x y')
t = RealLine().canonical_coordinate()
c = R2.curve([sin(t), sin(2*t)/2], (t, 0, 2*pi), name='c')
g = c.plot(prange=(0,pi))
sphinx_plot(g)
Plot with various options::
sage: c.plot(color='green', style=':', thickness=3, aspect_ratio=1)
Graphics object consisting of 1 graphics primitive
.. PLOT::
R2 = Manifold(2, 'R^2')
X = R2.chart('x y')
t = RealLine().canonical_coordinate()
c = R2.curve([sin(t), sin(2*t)/2], (t, 0, 2*pi), name='c')
g = c.plot(color='green', style=':', thickness=3, aspect_ratio=1)
sphinx_plot(g)
Plot via a mapping to another manifold: loxodrome of a sphere viewed
in `\RR^3`::
sage: S2 = Manifold(2, 'S^2')
sage: U = S2.open_subset('U')
sage: XS.<th,ph> = U.chart(r'th:(0,pi):\theta ph:(0,2*pi):\phi')
sage: R3 = Manifold(3, 'R^3')
sage: X3.<x,y,z> = R3.chart()
sage: F = S2.diff_map(R3, {(XS, X3): [sin(th)*cos(ph),
....: sin(th)*sin(ph), cos(th)]}, name='F')
sage: F.display()
F: S^2 --> R^3
on U: (th, ph) |--> (x, y, z) = (cos(ph)*sin(th), sin(ph)*sin(th), cos(th))
sage: c = S2.curve([2*atan(exp(-t/10)), t], (t, -oo, +oo), name='c')
sage: graph_c = c.plot(mapping=F, max_range=40,
....: plot_points=200, thickness=2, label_axes=False) # 3D plot
sage: graph_S2 = XS.plot(X3, mapping=F, number_values=11, color='black') # plot of the sphere
sage: show(graph_c + graph_S2) # the loxodrome + the sphere
.. PLOT::
S2 = Manifold(2, 'S^2')
U = S2.open_subset('U')
XS = U.chart(r'th:(0,pi):\theta ph:(0,2*pi):\phi')
th, ph = XS[:]
R3 = Manifold(3, 'R^3')
X3 = R3.chart('x y z')
F = S2.diff_map(R3, {(XS, X3): [sin(th)*cos(ph), sin(th)*sin(ph),
cos(th)]}, name='F')
t = RealLine().canonical_coordinate()
c = S2.curve([2*atan(exp(-t/10)), t], (t, -oo, +oo), name='c')
graph_c = c.plot(mapping=F, max_range=40, plot_points=200,
thickness=2, label_axes=False)
graph_S2 = XS.plot(X3, mapping=F, number_values=11, color='black')
sphinx_plot(graph_c + graph_S2)
Example of use of the argument ``parameters``: we define a curve with
some symbolic parameters ``a`` and ``b``::
sage: a, b = var('a b')
sage: c = R2.curve([a*cos(t) + b, a*sin(t)], (t, 0, 2*pi), name='c')
To make a plot, we set spectific values for ``a`` and ``b`` by means
of the Python dictionary ``parameters``::
sage: c.plot(parameters={a: 2, b: -3}, aspect_ratio=1)
Graphics object consisting of 1 graphics primitive
.. PLOT::
R2 = Manifold(2, 'R^2')
X = R2.chart('x y')
t = RealLine().canonical_coordinate()
a, b = var('a b')
c = R2.curve([a*cos(t) + b, a*sin(t)], (t, 0, 2*pi), name='c')
g = c.plot(parameters={a: 2, b: -3}, aspect_ratio=1)
sphinx_plot(g)
"""
from sage.rings.infinity import Infinity
from sage.misc.functional import numerical_approx
from sage.plot.graphics import Graphics
from sage.plot.line import line
from sage.manifolds.chart import RealChart
from sage.manifolds.utilities import set_axes_labels
#
# Get the @options from kwds
#
thickness = kwds.pop('thickness')
plot_points = kwds.pop('plot_points')
max_range = kwds.pop('max_range')
aspect_ratio = kwds.pop('aspect_ratio')
#
# The "effective" curve to be plotted
#
if mapping is None:
eff_curve = self
else:
eff_curve = mapping.restrict(self.codomain()) * self
#
# The chart w.r.t. which the curve is plotted
#
if chart is None:
chart = eff_curve._codomain.default_chart()
elif not isinstance(chart, RealChart):
raise TypeError("{} is not a real chart".format(chart))
#
# Coordinates of the above chart w.r.t. which the curve is plotted
#
if ambient_coords is None:
ambient_coords = chart[:] # all chart coordinates are used
n_pc = len(ambient_coords)
if n_pc != 2 and n_pc != 3:
raise ValueError("the number of coordinates involved in the " +
"plot must be either 2 or 3, not {}".format(n_pc))
# indices of plot coordinates
ind_pc = [chart[:].index(pc) for pc in ambient_coords]
#
# Parameter range for the plot
#
if prange is None:
prange = (self._domain.lower_bound(), self._domain.upper_bound())
elif not isinstance(prange, (tuple, list)):
raise TypeError("{} is neither a tuple nor a list".format(prange))
elif len(prange) != 2:
raise ValueError("the argument prange must be a tuple/list " +
"of 2 elements")
tmin = prange[0]
tmax = prange[1]
if tmin == -Infinity:
tmin = -max_range
elif not include_end_point[0]:
tmin = tmin + end_point_offset[0]
if tmax == Infinity:
tmax = max_range
elif not include_end_point[1]:
tmax = tmax - end_point_offset[1]
tmin = numerical_approx(tmin)
tmax = numerical_approx(tmax)
#
# The coordinate expression of the effective curve
#
canon_chart = self._domain.canonical_chart()
transf = None
for chart_pair in eff_curve._coord_expression:
if chart_pair == (canon_chart, chart):
transf = eff_curve._coord_expression[chart_pair]
break
else:
# Search for a subchart
for chart_pair in eff_curve._coord_expression:
for schart in chart._subcharts:
if chart_pair == (canon_chart, schart):
transf = eff_curve._coord_expression[chart_pair]
if transf is None:
raise ValueError("No expression has been found for " +
"{} in terms of {}".format(self, chart))
#
# List of points for the plot curve
#
plot_curve = []
dt = (tmax - tmin) / (plot_points - 1)
t = tmin
if parameters is None:
for i in range(plot_points):
x = transf(t, simplify=False)
plot_curve.append([numerical_approx(x[j]) for j in ind_pc])
t += dt
else:
for i in range(plot_points):
x = transf(t, simplify=False)
plot_curve.append([
numerical_approx(x[j].substitute(parameters))
for j in ind_pc
])
t += dt
#
# The plot
#
resu = Graphics()
resu += line(plot_curve,
color=color,
linestyle=style,
thickness=thickness)
if n_pc == 2: # 2D graphic
resu.set_aspect_ratio(aspect_ratio)
if label_axes:
# We update the dictionary _extra_kwds (options to be passed
# to show()), instead of using the method
# Graphics.axes_labels() since the latter is not robust w.r.t.
# graph addition
resu._extra_kwds['axes_labels'] = [
r'$' + latex(pc) + r'$' for pc in ambient_coords
]
else: # 3D graphic
if aspect_ratio == 'automatic':
aspect_ratio = 1
resu.aspect_ratio(aspect_ratio)
if label_axes:
labels = [str(pc) for pc in ambient_coords]
resu = set_axes_labels(resu, *labels)
return resu
def _graphics(self, plot_curve, ambient_coords, thickness=1,
aspect_ratio='automatic', color='red', style='-',
label_axes=True):
r"""
Plot a 2D or 3D curve in a Cartesian graph with axes labeled by
the ambient coordinates; it is invoked by the methods
:meth:`plot` of
:class:`~sage.manifolds.differentiable.curve.DifferentiableCurve`,
and its subclasses
(:class:`~sage.manifolds.differentiable.integrated_curve.IntegratedCurve`,
:class:`~sage.manifolds.differentiable.integrated_curve.IntegratedAutoparallelCurve`,
and
:class:`~sage.manifolds.differentiable.integrated_curve.IntegratedGeodesic`).
TESTS::
sage: M = Manifold(2, 'R^2')
sage: X.<x,y> = M.chart()
sage: R.<t> = RealLine()
sage: c = M.curve([cos(t), sin(t)], (t, 0, 2*pi), name='c')
sage: graph = c._graphics([[1,2], [3,4]], [x,y])
sage: graph._objects[0].xdata == [1,3]
True
sage: graph._objects[0].ydata == [2,4]
True
sage: graph._objects[0]._options['thickness'] == 1
True
sage: graph._extra_kwds['aspect_ratio'] == 'automatic'
True
sage: graph._objects[0]._options['rgbcolor'] == 'red'
True
sage: graph._objects[0]._options['linestyle'] == '-'
True
sage: l = [r'$'+latex(x)+r'$', r'$'+latex(y)+r'$']
sage: graph._extra_kwds['axes_labels'] == l
True
"""
from sage.plot.graphics import Graphics
from sage.plot.line import line
from sage.manifolds.utilities import set_axes_labels
#
# The plot
#
n_pc = len(ambient_coords)
resu = Graphics()
resu += line(plot_curve, color=color, linestyle=style,
thickness=thickness)
if n_pc == 2: # 2D graphic
resu.set_aspect_ratio(aspect_ratio)
if label_axes:
# We update the dictionary _extra_kwds (options to be passed
# to show()), instead of using the method
# Graphics.axes_labels() since the latter is not robust w.r.t.
# graph addition
resu._extra_kwds['axes_labels'] = [r'$'+latex(pc)+r'$'
for pc in ambient_coords]
else: # 3D graphic
if aspect_ratio == 'automatic':
aspect_ratio = 1
resu.aspect_ratio(aspect_ratio)
if label_axes:
labels = [str(pc) for pc in ambient_coords]
resu = set_axes_labels(resu, *labels)
return resu
def _graphics(self, plot_curve, ambient_coords, thickness=1,
aspect_ratio='automatic', color='red', style='-',
label_axes=True):
r"""
Plot a 2D or 3D curve in a Cartesian graph with axes labeled by
the ambient coordinates; it is invoked by the methods
:meth:`plot` of
:class:`~sage.manifolds.differentiable.curve.DifferentiableCurve`,
and its subclasses
(:class:`~sage.manifolds.differentiable.integrated_curve.IntegratedCurve`,
:class:`~sage.manifolds.differentiable.integrated_curve.IntegratedAutoparallelCurve`,
and
:class:`~sage.manifolds.differentiable.integrated_curve.IntegratedGeodesic`).
TESTS::
sage: M = Manifold(2, 'R^2')
sage: X.<x,y> = M.chart()
sage: R.<t> = RealLine()
sage: c = M.curve([cos(t), sin(t)], (t, 0, 2*pi), name='c')
sage: graph = c._graphics([[1,2], [3,4]], [x,y])
sage: graph._objects[0].xdata == [1,3]
True
sage: graph._objects[0].ydata == [2,4]
True
sage: graph._objects[0]._options['thickness'] == 1
True
sage: graph._extra_kwds['aspect_ratio'] == 'automatic'
True
sage: graph._objects[0]._options['rgbcolor'] == 'red'
True
sage: graph._objects[0]._options['linestyle'] == '-'
True
sage: l = [r'$'+latex(x)+r'$', r'$'+latex(y)+r'$']
sage: graph._extra_kwds['axes_labels'] == l
True
"""
from sage.plot.graphics import Graphics
from sage.plot.line import line
from sage.manifolds.utilities import set_axes_labels
#
# The plot
#
n_pc = len(ambient_coords)
resu = Graphics()
resu += line(plot_curve, color=color, linestyle=style,
thickness=thickness)
if n_pc == 2: # 2D graphic
resu.set_aspect_ratio(aspect_ratio)
if label_axes:
# We update the dictionary _extra_kwds (options to be passed
# to show()), instead of using the method
# Graphics.axes_labels() since the latter is not robust w.r.t.
# graph addition
resu._extra_kwds['axes_labels'] = [r'$'+latex(pc)+r'$'
for pc in ambient_coords]
else: # 3D graphic
if aspect_ratio == 'automatic':
aspect_ratio = 1
resu.aspect_ratio(aspect_ratio)
if label_axes:
labels = [str(pc) for pc in ambient_coords]
resu = set_axes_labels(resu, *labels)
return resu
def legend_3d(hyperplane_arrangement, hyperplane_colors, length):
r"""
Create plot of a 3d legend for an arrangement of planes in 3-space. The
``length`` parameter determines whether short or long labels are used in
the legend.
INPUT:
- ``hyperplane_arrangement`` -- a hyperplane arrangement
- ``hyperplane_colors`` -- list of colors
- ``length`` -- either ``'short'`` or ``'long'``
OUTPUT:
- A graphics object.
EXAMPLES::
sage: a = hyperplane_arrangements.semiorder(3)
sage: from sage.geometry.hyperplane_arrangement.plot import legend_3d
sage: legend_3d(a, list(colors.values())[:6],length='long')
Graphics object consisting of 6 graphics primitives
sage: b = hyperplane_arrangements.semiorder(4)
sage: c = b.essentialization()
sage: legend_3d(c, list(colors.values())[:12], length='long')
Graphics object consisting of 12 graphics primitives
sage: legend_3d(c, list(colors.values())[:12], length='short')
Graphics object consisting of 12 graphics primitives
sage: p = legend_3d(c, list(colors.values())[:12], length='short')
sage: p.set_legend_options(ncol=4)
sage: type(p)
<class 'sage.plot.graphics.Graphics'>
"""
if hyperplane_arrangement.dimension() != 3:
raise ValueError('arrangements must be in 3-space')
hyps = hyperplane_arrangement.hyperplanes()
N = len(hyperplane_arrangement)
if length == 'short':
labels = [' ' + str(i) for i in range(N)]
else:
labels = [
' ' + hyps[i]._repr_linear(include_zero=False) for i in range(N)
]
p = Graphics()
for i in range(N):
p += line([(0, 0), (0, 0)],
color=hyperplane_colors[i],
thickness=8,
legend_label=labels[i],
axes=False)
p.set_legend_options(title='Hyperplanes',
loc='center',
labelspacing=0.4,
fancybox=True,
font_size='x-large',
ncol=2)
p.legend(True)
return p
def plot_phase_trajectories(func,
inits,
xbound,
ybound,
tbound=(0, 10, 100),
use_sage=False,
axis=None,
f_kwargs={},
odeint_kwargs={},
plt_kwargs={'color': 'blue'}):
"""function for plotting the phase space trajectories of ordinary
differential equations (ODEs) of the form dy/dt = f(y, t), where y can be
a n-dim vector (see scipy.integrate.odeint for reference).
Inspired by plotdf; reference: github.com/jmoy
Parameters:
func - RHS of ODE; callable(t, y)
inits - initial values for plotting trajectories
xbound, ybound - sequences with len = 2 gives min and max values for x
and y respectivly
tbound - sequence with len = 3 gives min, max and step for t values to
calculate the trajectories for
use_sage - whether to use sage for plotting instead of matplotlib
axis - matplotlib axis to draw plot in; if None, current will be used
odeint_kwargs - dict containing kwargs for scipy.integrate.odeint
plt_kwargs - dict containing kwargs for matplotlib.pyplot; can also be
used for sage-Graphics objects
Returns:
list containing matplotlib-artist objects (Line2D)
or one sage-Graphics object
"""
f = lambda x, t: func(t, x, **f_kwargs)
if use_sage:
from sage.plot.graphics import Graphics
from sage.plot.line import line
elif axis is None:
axis = plt.gca()
def f_neg(x, t):
return -f(x, t)
artists = [] if not use_sage else Graphics()
t = np.linspace(*tbound)
for i in inits:
sol_fwd = odeint(f, i, t, **odeint_kwargs) # forward solution
sol_bwd = odeint(f_neg, i, t, **odeint_kwargs) # backward solution
sol = np.vstack((np.flipud(sol_bwd),
sol_fwd)) # flip sol_bwd and put both together
sol_x = sol[:, 0] # left column of sol
sol_y = sol[:, 1] # right column of sol
sol_x_masked = np.ma.masked_outside(sol_x,
*xbound) # mask data to prevent
sol_y_masked = np.ma.masked_outside(sol_y,
*ybound) # blow-up of solution
if not use_sage:
artists.append(axis.plot(sol_x_masked, sol_y_masked, **plt_kwargs))
else:
artists += line(zip(sol_x_masked, sol_y_masked),
plt_kwargs,
xmin=xbound[0],
xmax=xbound[1])
if not use_sage:
plt.xlim(xbound)
plt.ylim(ybound)
return artists
def _plot(self, geosub, color=None):
r"""
EXAMPLES::
sage: from EkEkstar import kFace, GeoSub
sage: sub = {1:[1,2], 2:[1,3], 3:[1]}
sage: geosub = GeoSub(sub,2, dual=True)
sage: _ = kFace((10,21,33), (1,))._plot(geosub) # case A
sage: _ = kFace((10,21,33), (1,2), dual=True)._plot(geosub) # case C
sage: _ = kFace((10,21,33), (1,), dual=True)._plot(geosub) # case E
::
sage: sub = {1: [1, 3, 2, 3], 2: [2, 3], 3: [3, 2, 3, 1, 3, 2, 3]}
sage: geosub = GeoSub(sub, 2, dual=True)
sage: _ = kFace((10,21,33), (1,2), dual=True)._plot(geosub) # case B
::
sage: sub = {1:[1,2,3,3,3,3], 2:[1,3], 3:[1]}
sage: geosub = GeoSub(sub,2, dual=True)
sage: _ = kFace((0,0,0),(1,2), dual=True)._plot(geosub) # case A
"""
v = self.vector()
t = self.type()
if color != None:
col = color
else:
col = self._color
G = Graphics()
K = geosub.field()
b = K.gen()
num = geosub._sigma_dict.keys()
if self.is_dual():
h = list(set(num) - set(t))
B = b
vec = geosub.dominant_left_eigenvector()
emb = geosub.contracting_eigenvalues_indices()
else:
h = list(t)
B = b**(-1) # TODO: this seems useless (why?)
vec = -geosub.dominant_left_eigenvector()
emb = geosub.dilating_eigenvalues_indices()
el = v * vec
iter = 0
conjugates = geosub.complex_embeddings()
if len(h) == 1:
if conjugates[emb[0]].is_real() == True:
bp = zero_vector(CC, len(emb))
for i in range(len(emb)):
bp[i] = K(el).complex_embeddings()[emb[i]]
bp1 = zero_vector(CC, len(emb))
for i in range(len(emb)):
bp1[i] = K(
(el + vec[h[0] - 1])).complex_embeddings()[emb[i]]
if len(emb) == 1:
return line([bp[0], bp1[0]], color=col, thickness=3)
else:
return line([bp, bp1], color=col, thickness=3)
else:
bp = K(el).complex_embeddings()[emb[0]]
bp1 = K((el + vec[h[0] - 1])).complex_embeddings()[emb[0]]
return line([bp, bp1], color=col, thickness=3)
elif len(h) == 2:
if conjugates[emb[0]].is_real() == True:
bp = (K(el).complex_embeddings()[emb[0]],
K(el).complex_embeddings()[emb[1]])
bp1 = (K(el + vec[h[0] - 1]).complex_embeddings()[emb[0]],
K(el + vec[h[0] - 1]).complex_embeddings()[emb[1]])
bp2 = (K(el + vec[h[0] - 1] +
vec[h[1] - 1]).complex_embeddings()[emb[0]],
K(el + vec[h[0] - 1] +
vec[h[1] - 1]).complex_embeddings()[emb[1]])
bp3 = (K(el + vec[h[1] - 1]).complex_embeddings()[emb[0]],
K(el + vec[h[1] - 1]).complex_embeddings()[emb[1]])
return polygon2d([bp, bp1, bp2, bp3],
color=col,
thickness=.1,
alpha=0.8)
else:
bp = K(el).complex_embeddings()[emb[0]]
bp1 = K(el + vec[h[0] - 1]).complex_embeddings()[emb[0]]
bp2 = K(el + vec[h[0] - 1] +
vec[h[1] - 1]).complex_embeddings()[emb[0]]
bp3 = K(el + vec[h[1] - 1]).complex_embeddings()[emb[0]]
return polygon2d([[bp[0], bp[1]], [bp1[0], bp1[1]],
[bp2[0], bp2[1]], [bp3[0], bp3[1]]],
color=col,
thickness=.1,
alpha=0.8)
else:
raise NotImplementedError(
"Plotting is implemented only for patches in two or three dimensions."
)
return G
def plot_cluster_fan_stereographically(self,
northsign=1,
north=None,
right=None,
colors=None,
d_vectors=False):
from sage.plot.graphics import Graphics
from sage.plot.point import point
from sage.misc.flatten import flatten
from sage.plot.line import line
from sage.misc.functional import norm
if self.rk != 3:
raise ValueError("Can only stereographically project fans in 3d.")
if not self.is_finite() and self._depth == infinity:
raise ValueError(
"For infinite algebras you must specify the depth.")
if north == None:
if self.is_affine():
north = vector(self.delta())
else:
north = vector((-1, -1, -1))
if right == None:
if self.is_affine():
right = vector(self.gamma())
else:
right = vector((1, 0, 0))
if colors == None:
colors = dict([(0, 'red'), (1, 'green'), (2, 'blue'), (3, 'cyan'),
(4, 'yellow')])
G = Graphics()
roots = list(self.g_vectors())
compatible = []
while roots:
x = roots.pop()
if x in self.initial_cluster() and d_vectors:
x1 = -self.simple_roots()[list(
self.initial_cluster()).index(x)]
else:
x1 = x
for y in roots:
if self.compatibility_degree(x, y) == 0:
if y in self.initial_cluster() and d_vectors:
y1 = -self.simple_roots()[list(
self.initial_cluster()).index(y)]
else:
y1 = y
compatible.append((x1, y1))
for (u, v) in compatible:
G += _stereo_arc(vector(u),
vector(v),
vector(u + v),
north=northsign * north,
right=right,
thickness=0.5,
color='black')
for i in range(3):
orbit = self.ith_orbit(i)
if d_vectors:
orbit[0] = -self.simple_roots()[list(
self.initial_cluster()).index(orbit[0])]
for j in orbit:
G += point(_stereo_coordinates(vector(orbit[j]),
north=northsign * north,
right=right),
color=colors[i],
zorder=len(G))
if self.is_affine():
tube_vectors = map(vector, flatten(self.affine_tubes()))
for v in tube_vectors:
G += point(_stereo_coordinates(v,
north=northsign * north,
right=right),
color=colors[3],
zorder=len(G))
if north != vector(self.delta()):
G += _stereo_arc(tube_vectors[0],
tube_vectors[1],
vector(self.delta()),
north=northsign * north,
right=right,
thickness=2,
color=colors[4],
zorder=0)
else:
# FIXME: refactor this before publishing
tube_projections = [
_stereo_coordinates(v,
north=northsign * north,
right=right) for v in tube_vectors
]
t = min(
(G.get_minmax_data()['xmax'], G.get_minmax_data()['ymax']))
G += line([
tube_projections[0], tube_projections[0] + t *
(_normalize(tube_projections[0] - tube_projections[1]))
],
thickness=2,
color=colors[4],
zorder=0)
G += line([
tube_projections[1], tube_projections[1] + t *
(_normalize(tube_projections[1] - tube_projections[0]))
],
thickness=2,
color=colors[4],
zorder=0)
G.set_aspect_ratio(1)
G._show_axes = False
return G
def plot(self):
r"""
Return a graphical object of the Fully Packed Loop
EXAMPLES:
Here is the fully packed loop for
.. MATH::
\begin{pmatrix} 0&1&1 \\ 1&-1&1 \\ 0&1&0 \end{pmatrix}:
.. PLOT::
:width: 200 px
A = AlternatingSignMatrix([[0, 1, 0], [1, -1, 1], [0, 1, 0]])
fpl = FullyPackedLoop(A)
p = fpl.plot()
sphinx_plot(p)
Here is how Sage represents this::
sage: A = AlternatingSignMatrix([[0, 1, 0], [1, -1, 1], [0, 1, 0]])
sage: fpl = FullyPackedLoop(A)
sage: print(fpl.plot().description())
Line defined by 2 points: [(-1.0, 1.0), (0.0, 1.0)]
Line defined by 2 points: [(0.0, 0.0), (0.0, -1.0)]
Line defined by 2 points: [(0.0, 0.0), (1.0, 0.0)]
Line defined by 2 points: [(0.0, 2.0), (0.0, 3.0)]
Line defined by 2 points: [(0.0, 2.0), (0.0, 3.0)]
Line defined by 2 points: [(0.0, 2.0), (1.0, 2.0)]
Line defined by 2 points: [(1.0, 1.0), (0.0, 1.0)]
Line defined by 2 points: [(1.0, 1.0), (2.0, 1.0)]
Line defined by 2 points: [(2.0, 0.0), (1.0, 0.0)]
Line defined by 2 points: [(2.0, 0.0), (2.0, -1.0)]
Line defined by 2 points: [(2.0, 2.0), (1.0, 2.0)]
Line defined by 2 points: [(2.0, 2.0), (2.0, 3.0)]
Line defined by 2 points: [(2.0, 2.0), (2.0, 3.0)]
Line defined by 2 points: [(3.0, 1.0), (2.0, 1.0)]
Line defined by 2 points: [(3.0, 1.0), (2.0, 1.0)]
Here are the other 3 by 3 Alternating Sign Matrices and their corresponding
fully packed loops:
.. MATH::
A = \begin{pmatrix} 1&0&0 \\ 0&1&0 \\ 0&0&1 \\ \end{pmatrix}
gives:
.. PLOT::
:width: 200 px
A = AlternatingSignMatrix([[1, 0, 0], [0, 1, 0], [0, 0, 1]])
fpl = FullyPackedLoop(A)
p = fpl.plot()
sphinx_plot(p)
.. MATH::
A = \begin{pmatrix} 1&0&0 \\ 0&0&1 \\ 0&1&0 \\ \end{pmatrix}
gives:
.. PLOT::
:width: 200 px
A = AlternatingSignMatrix([[1, 0, 0], [0, 0, 1], [0, 1, 0]])
fpl = FullyPackedLoop(A)
p = fpl.plot()
sphinx_plot(p)
.. MATH::
A = \begin{pmatrix} 0&1&0\\ 1&0&0\\ 0&0&1\\ \end{pmatrix}
gives:
.. PLOT::
:width: 200 px
A = AlternatingSignMatrix([[0, 1, 0], [1, 0, 0], [0, 0, 1]])
fpl = FullyPackedLoop(A)
p = fpl.plot()
sphinx_plot(p)
.. MATH::
A = \begin{pmatrix} 0&1&0\\ 0&0&1\\ 1&0&0\\ \end{pmatrix}
gives:
.. PLOT::
:width: 200 px
A = AlternatingSignMatrix([[0, 1, 0], [0, 0, 1], [1, 0, 0]])
fpl = FullyPackedLoop(A)
p = fpl.plot()
sphinx_plot(p)
.. MATH::
A = \begin{pmatrix} 0&0&1\\ 1&0&0\\ 0&1&0\\ \end{pmatrix}
gives:
.. PLOT::
:width: 200 px
A = AlternatingSignMatrix([[0, 0, 1], [1, 0, 0], [0, 1, 0]])
fpl = FullyPackedLoop(A)
p = fpl.plot()
sphinx_plot(p)
.. MATH::
A = \begin{pmatrix} 0&0&1\\ 0&1&0\\ 1&0&0\\ \end{pmatrix}
gives:
.. PLOT::
:width: 200 px
A = AlternatingSignMatrix([[0, 0, 1], [0, 1, 0], [1, 0, 0]])
fpl = FullyPackedLoop(A)
p = fpl.plot()
sphinx_plot(p)
EXAMPLES::
sage: A = AlternatingSignMatrix([[0, 1, 0, 0], [1, -1, 0, 1], \
[0, 1, 0, 0],[0, 0, 1, 0]])
sage: fpl = FullyPackedLoop(A)
sage: print(fpl.plot().description())
Line defined by 2 points: [(-1.0, 0.0), (0.0, 0.0)]
Line defined by 2 points: [(-1.0, 2.0), (0.0, 2.0)]
Line defined by 2 points: [(0.0, 1.0), (0.0, 0.0)]
Line defined by 2 points: [(0.0, 1.0), (1.0, 1.0)]
Line defined by 2 points: [(0.0, 3.0), (0.0, 4.0)]
Line defined by 2 points: [(0.0, 3.0), (0.0, 4.0)]
Line defined by 2 points: [(0.0, 3.0), (1.0, 3.0)]
Line defined by 2 points: [(1.0, 0.0), (1.0, -1.0)]
Line defined by 2 points: [(1.0, 0.0), (2.0, 0.0)]
Line defined by 2 points: [(1.0, 2.0), (0.0, 2.0)]
Line defined by 2 points: [(1.0, 2.0), (2.0, 2.0)]
Line defined by 2 points: [(2.0, 1.0), (1.0, 1.0)]
Line defined by 2 points: [(2.0, 1.0), (2.0, 2.0)]
Line defined by 2 points: [(2.0, 3.0), (1.0, 3.0)]
Line defined by 2 points: [(2.0, 3.0), (2.0, 4.0)]
Line defined by 2 points: [(2.0, 3.0), (2.0, 4.0)]
Line defined by 2 points: [(3.0, 0.0), (2.0, 0.0)]
Line defined by 2 points: [(3.0, 0.0), (3.0, -1.0)]
Line defined by 2 points: [(3.0, 2.0), (3.0, 1.0)]
Line defined by 2 points: [(3.0, 2.0), (3.0, 3.0)]
Line defined by 2 points: [(4.0, 1.0), (3.0, 1.0)]
Line defined by 2 points: [(4.0, 1.0), (3.0, 1.0)]
Line defined by 2 points: [(4.0, 3.0), (3.0, 3.0)]
Line defined by 2 points: [(4.0, 3.0), (3.0, 3.0)]
Here is the plot:
.. PLOT::
:width: 300 px
A = AlternatingSignMatrix([[0, 1, 0, 0], [1, -1, 0, 1], [0, 1, 0, 0],[0, 0, 1, 0]])
fpl = FullyPackedLoop(A)
p = fpl.plot()
sphinx_plot(p)
"""
G = Graphics()
n=len(self._six_vertex_model)-1
for j,row in enumerate(reversed(self._six_vertex_model)):
for i,entry in enumerate(row):
if i == 0 and (i+j+n+1) % 2 ==0:
G+= line([(i-1,j),(i,j)])
if i == n and (i+j+n+1) % 2 ==0:
G+= line([(i+1,j),(i,j)])
if j == 0 and (i+j+n) % 2 ==0:
G+= line([(i,j),(i,j-1)])
if j == n and (i+j+n) % 2 ==0:
G+= line([(i,j),(i,j+1)])
if entry == 0: # LR
if (i+j+n) % 2==0:
G += line([(i,j), (i+1,j)])
else:
G += line([(i,j),(i,j+1)])
elif entry == 1: # LU
if (i+j+n) % 2 ==0:
G += line([(i,j), (i,j+1)])
else:
G += line([(i+1,j), (i,j)])
elif entry == 2: # LD
if (i+j+n) % 2 == 0:
pass
else:
G += line([(i,j+1), (i,j)])
G += line([(i+1,j), (i,j)])
elif entry == 3: # UD
if (i+j+n) % 2 == 0:
G += line([(i,j), (i,j+1)])
else:
G += line([(i+1,j), (i,j)])
elif entry == 4: # UR
if (i+j+n) % 2 ==0:
G += line([(i,j), (i,j+1)])
G += line([(i,j), (i+1,j)])
else:
pass
elif entry == 5: # RD
if (i+j+n) % 2 ==0:
G += line([(i,j), (i+1,j)])
else:
G += line([(i,j+1), (i,j)])
G.axes(False)
return G
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 show(self, **args):
r"""
Displays the pseudoline arrangement as a wiring diagram.
INPUT:
- ``**args`` -- any arguments to be forwarded to the ``show`` method. In
particular, to tune the dimensions, use the ``figsize`` argument
(example below).
EXAMPLES::
sage: from sage.geometry.pseudolines import PseudolineArrangement
sage: permutations = [[3, 2, 1], [3, 2, 0], [3, 1, 0], [2, 1, 0]]
sage: p = PseudolineArrangement(permutations)
sage: p.show(figsize=[7,5])
TESTS::
sage: from sage.geometry.pseudolines import PseudolineArrangement
sage: permutations = [[3, 2, 1], [3, 2, 0], [3, 0, 1], [2, 0, 1]]
sage: p = PseudolineArrangement(permutations)
sage: p.show()
Traceback (most recent call last):
...
ValueError: There has been a problem while plotting the figure...
"""
x = 1
from sage.plot.line import line
from sage.plot.text import text
lines = [[(0, self._n - 1 - i)] for i in range(self._n)]
for i, j in self.transpositions():
iy = lines[i][-1][1]
jy = lines[j][-1][1]
lines[i].append((x, iy))
lines[j].append((x, jy))
if abs(iy - jy) != 1:
raise ValueError(
"There has been a problem while plotting the figure. It " +
"seems that the lines are not correctly ordered. Please " +
"check the pseudolines modules documentation, there is a "
+ "warning about that. ")
lines[i].append((x + 2, jy))
lines[j].append((x + 2, iy))
x += 2
L = line([(1, 1)])
for i, l in enumerate(lines):
l.append((x + 2, l[-1][1]))
L += line(l)
L += text(str(i), (0, l[0][1] + .3), horizontal_alignment="right")
L += text(str(i), (x + 2, l[-1][1] + .3),
horizontal_alignment="left")
return L.show(axes=False, **args)
def plot_paths_2d(self,
start_sys,
end_sys,
input_ring,
c_skew=.001,
endpoints=True,
saved_start=None,
rand_colors=False):
"""
This returns a graphics object of solution paths in the complex plane.
INPUT:
- start_sys -- a square polynomial system, given as a list of polynomials
- end_sys -- same type as start_sys
- input_ring -- for coercion of the variables into the desired ring.
- c_skew -- optional. the imaginary part of homotopy multiplier; nonzero values
are often necessary to avoid intermediate path collisions
- endpoints -- optional. Whether to draw in the ends of paths as points.
- saved_start -- optional. A phc output file. If not given, start system solutions
are computed via the phc.blackbox function.
OUTPUT:
- lines and points of solution paths
EXAMPLES::
sage: from sage.interfaces.phc import *
sage: from sage.structure.sage_object import SageObject
sage: R2.<x,y> = PolynomialRing(QQ,2)
sage: start_sys = [x^5-y^2,y^5-1]
sage: sol = phc.blackbox(start_sys, R2) # optional -- phc
sage: start_save = sol.save_as_start() # optional -- phc
sage: end_sys = [x^5-25,y^5-x^2] # optional -- phc
sage: testing = phc.plot_paths_2d(start_sys, end_sys, R2) # optional -- phc
sage: type(testing) # optional -- phc (normally use plot here)
<class 'sage.plot.graphics.Graphics'>
"""
paths = phc.path_track(start_sys,
end_sys,
input_ring,
c_skew=c_skew,
saved_start=saved_start)
path_lines = []
sol_pts = []
if rand_colors:
r_color = {}
for a_var in input_ring.gens():
var_name = str(a_var)
r_color[var_name] = (random(), random(), random())
for a_sol in paths:
for a_var in input_ring.gens():
var_name = str(a_var)
temp_line = []
for data in a_sol:
temp_line.append(
[data[var_name].real(), data[var_name].imag()])
if rand_colors:
path_lines.append(
line(temp_line, rgbcolor=r_color[var_name]))
else:
path_lines.append(line(temp_line))
if endpoints:
sol_pts = []
for a_sol in paths:
for a_var in input_ring.gens():
var_name = str(a_var)
sol_pts.append(
point([
a_sol[0][var_name].real(),
a_sol[0][var_name].imag()
]))
sol_pts.append(
point([
a_sol[-1][var_name].real(),
a_sol[-1][var_name].imag()
]))
return sum(sol_pts) + sum(path_lines)
else:
return sum(path_lines)
def plot(self, size=[[0],[0]], projection='usual', simple_roots=True, fundamental_weights=True, alcovewalks=[]):
r"""
Return a graphics object built from a space of weight(space/lattice).
There is a different technic to plot if the Cartan type is affine or not.
The graphics returned is a Graphics object.
This function is experimental, and is subject to short term evolutions.
EXAMPLES::
By default, the plot returned has no axes and the ratio between axes is 1.
sage: G = RootSystem(['C',2]).weight_lattice().plot()
sage: G.axes(True)
sage: G.set_aspect_ratio(2)
For a non affine Cartan type, the plot method work for type with 2 generators,
it will draw the hyperlane(line for this dimension) accrow the fundamentals weights.
sage: G = RootSystem(['A',2]).weight_lattice().plot()
sage: G = RootSystem(['B',2]).weight_lattice().plot()
sage: G = RootSystem(['G',2]).weight_lattice().plot()
The plot returned has a size of one fundamental polygon by default. We can
ask plot to give a bigger plot by using the argument size
sage: G = RootSystem(['G',2,1]).weight_space().plot(size = [[0..1],[-1..1]])
sage: G = RootSystem(['A',2,1]).weight_space().plot(size = [[-1..1],[-1..1]])
A very important argument is the projection which will draw the plot. There are
some usual projections is this method. If you want to draw in the plane a very
special Cartan type, Sage will ask you to specify the projection. The projection
is a matrix over a ring. In practice, calcul over float is a good way to draw.
sage: L = RootSystem(['A',2,1]).weight_space()
sage: G = L.plot(projection=matrix(RR, [[0,0.5,-0.5],[0,0.866,0.866]]))
sage: G = RootSystem(['C',2,1]).weight_space().plot()
By default, the plot method draw the simple roots, this can be disabled by setting
the argument simple_roots=False
sage: G = RootSystem(['A',2]).weight_space().plot(simple_roots=False)
By default, the plot method draw the fundamental weights,this can be disabled by
setting the argument fundamental_weights=False
sage: G = RootSystem(['A',2]).weight_space().plot(fundamental_weights=False, simple_roots=False)
There is in a plot an argument to draw alcoves walks. The good way to do this is
to use the crystals theory. the plot method contains only the drawing part...
sage: L = RootSystem(['A',2,1]).weight_space()
sage: G = L.plot(size=[[-1..1],[-1..1]],alcovewalks=[[0,2,0,1,2,1,2,0,2,1]])
"""
from sage.plot.all import Graphics
from sage.plot.line import line
from cartan_type import CartanType
from sage.matrix.constructor import matrix
from sage.rings.all import QQ, RR
from sage.plot.arrow import arrow
from sage.plot.point import point
# We begin with an empty plot G
G = Graphics()
ct = self.cartan_type()
n = ct.n
# Define a set of colors
# TODO : Colors in option ?
colors=[(0,1,0),(1,0,0),(0,0,1),(1,1,0),(0,1,1),(1,0,1)]
# plot the affine types:
if ct.is_affine():
# Check the projection
# TODO : try to have usual_projection for main plotable types
if projection == 'usual':
if ct == CartanType(['A',2,1]):
projection = matrix(RR, [[0,0.5,-0.5],[0,0.866,0.866]])
elif ct == CartanType(['C',2,1]):
projection = matrix(QQ, [[0,1,1],[0,0,1]])
elif ct == CartanType(['G',2,1]):
projection = matrix(RR, [[0,0.5,0],[0,0.866,1.732]])
else:
raise 'There is no usual projection for this Cartan type, you have to give one in argument'
assert(n + 1 == projection.ncols())
assert(2 == projection.nrows())
# Check the size is correct with the lattice
assert(len(size) == n)
# Select the center of the translated fundamental polygon to plot
translation_factors = ct.translation_factors()
simple_roots = self.simple_roots()
translation_vectors = [translation_factors[i]*simple_roots[i] for i in ct.classical().index_set()]
initial = [[]]
for i in range(n):
prod_list = []
for elem in size[i]:
for partial_list in initial:
prod_list.append( [elem]+partial_list );
initial = prod_list;
part_lattice = []
for combinaison in prod_list:
elem_lattice = self.zero()
for i in range(n):
elem_lattice = elem_lattice + combinaison[i]*translation_vectors[i]
part_lattice.append(elem_lattice)
# Get the vertices of the fundamental alcove
fundamental_weights = self.fundamental_weights()
vertices = map(lambda x: (1/x.level())*x, fundamental_weights.list())
# Recup the group which act on the fundamental polygon
classical = self.weyl_group().classical()
for center in part_lattice:
for w in classical:
# for each center of polygon and each element of classical
# parabolic subgroup, we have to draw an alcove.
#first, iterate over pairs of fundamental weights, drawing lines border of polygons:
for i in range(1,n+1):
for j in range(i+1,n+1):
p1=projection*((w.action(vertices[i])).to_vector() + center.to_vector())
p2=projection*((w.action(vertices[j])).to_vector() + center.to_vector())
G+=line([p1,p2],rgbcolor=(0,0,0),thickness=2)
#next, get all lines from point to a fundamental weight, that separe different
#chanber in a same polygon (important: associate a color with a fundamental weight)
pcenter = projection*(center.to_vector())
for i in range(1,n+1):
p3=projection*((w.action(vertices[i])).to_vector() + center.to_vector())
G+=line([p3,pcenter], rgbcolor=colors[n-i+1])
#Draw alcovewalks
#FIXME : The good way to draw this is to use the alcoves walks works made in Cristals
#The code here just draw like example and import the good things.
rho = (1/self.rho().level())*self.rho()
W = self.weyl_group()
for walk in alcovewalks:
target = W.from_reduced_word(walk).action(rho)
for i in range(len(walk)):
walk.pop()
origin = W.from_reduced_word(walk).action(rho)
G+=arrow(projection*(origin.to_vector()),projection*(target.to_vector()), rgbcolor=(0.6,0,0.6), width=1, arrowsize=5)
target = origin
else:
# non affine plot
# Check the projection
# TODO : try to have usual_projection for main plotable types
if projection == 'usual':
if ct == CartanType(['A',2]):
projection = matrix(RR, [[0.5,-0.5],[0.866,0.866]])
elif ct == CartanType(['B',2]):
projection = matrix(QQ, [[1,0],[1,1]])
elif ct == CartanType(['C',2]):
projection = matrix(QQ, [[1,1],[0,1]])
elif ct == CartanType(['G',2]):
projection = matrix(RR, [[0.5,0],[0.866,1.732]])
else:
raise 'There is no usual projection for this Cartan type, you have to give one in argument'
# Get the fundamental weights
fundamental_weights = self.fundamental_weights()
WeylGroup = self.weyl_group()
#Draw not the alcove but the cones delimited by the hyperplanes
#The size of the line depend of the fundamental weights.
pcenter = projection*(self.zero().to_vector())
for w in WeylGroup:
for i in range(1,n+1):
p3=3*projection*((w.action(fundamental_weights[i])).to_vector())
G+=line([p3,pcenter], rgbcolor=colors[n-i+1])
#Draw the simple roots
if simple_roots:
SimpleRoots = self.simple_roots()
if ct.is_affine():
G+=arrow((0,0), projection*(SimpleRoots[0].to_vector()), rgbcolor=(0,0,0))
for j in range(1,n+1):
G+=arrow((0,0),projection*(SimpleRoots[j].to_vector()), rgbcolor=colors[j])
#Draw the fundamental weights
if fundamental_weights:
FundWeight = self.fundamental_weights()
for j in range(1,n+1):
G+=point(projection*(FundWeight[j].to_vector()), rgbcolor=colors[j], pointsize=60)
G.set_aspect_ratio(1)
G.axes(False)
return G
def plot(self,
size=[[0], [0]],
projection='usual',
simple_roots=True,
fundamental_weights=True,
alcovewalks=[]):
r"""
Return a graphics object built from a space of weight(space/lattice).
There is a different technic to plot if the Cartan type is affine or not.
The graphics returned is a Graphics object.
This function is experimental, and is subject to short term evolutions.
EXAMPLES::
By default, the plot returned has no axes and the ratio between axes is 1.
sage: G = RootSystem(['C',2]).weight_lattice().plot()
sage: G.axes(True)
sage: G.set_aspect_ratio(2)
For a non affine Cartan type, the plot method work for type with 2 generators,
it will draw the hyperlane(line for this dimension) accrow the fundamentals weights.
sage: G = RootSystem(['A',2]).weight_lattice().plot()
sage: G = RootSystem(['B',2]).weight_lattice().plot()
sage: G = RootSystem(['G',2]).weight_lattice().plot()
The plot returned has a size of one fundamental polygon by default. We can
ask plot to give a bigger plot by using the argument size
sage: G = RootSystem(['G',2,1]).weight_space().plot(size = [[0..1],[-1..1]])
sage: G = RootSystem(['A',2,1]).weight_space().plot(size = [[-1..1],[-1..1]])
A very important argument is the projection which will draw the plot. There are
some usual projections is this method. If you want to draw in the plane a very
special Cartan type, Sage will ask you to specify the projection. The projection
is a matrix over a ring. In practice, calcul over float is a good way to draw.
sage: L = RootSystem(['A',2,1]).weight_space()
sage: G = L.plot(projection=matrix(RR, [[0,0.5,-0.5],[0,0.866,0.866]]))
sage: G = RootSystem(['C',2,1]).weight_space().plot()
By default, the plot method draw the simple roots, this can be disabled by setting
the argument simple_roots=False
sage: G = RootSystem(['A',2]).weight_space().plot(simple_roots=False)
By default, the plot method draw the fundamental weights,this can be disabled by
setting the argument fundamental_weights=False
sage: G = RootSystem(['A',2]).weight_space().plot(fundamental_weights=False, simple_roots=False)
There is in a plot an argument to draw alcoves walks. The good way to do this is
to use the crystals theory. the plot method contains only the drawing part...
sage: L = RootSystem(['A',2,1]).weight_space()
sage: G = L.plot(size=[[-1..1],[-1..1]],alcovewalks=[[0,2,0,1,2,1,2,0,2,1]])
"""
from sage.plot.all import Graphics
from sage.plot.line import line
from cartan_type import CartanType
from sage.matrix.constructor import matrix
from sage.rings.all import QQ, RR
from sage.plot.arrow import arrow
from sage.plot.point import point
# We begin with an empty plot G
G = Graphics()
ct = self.cartan_type()
n = ct.n
# Define a set of colors
# TODO : Colors in option ?
colors = [(0, 1, 0), (1, 0, 0), (0, 0, 1), (1, 1, 0), (0, 1, 1),
(1, 0, 1)]
# plot the affine types:
if ct.is_affine():
# Check the projection
# TODO : try to have usual_projection for main plotable types
if projection == 'usual':
if ct == CartanType(['A', 2, 1]):
projection = matrix(
RR, [[0, 0.5, -0.5], [0, 0.866, 0.866]])
elif ct == CartanType(['C', 2, 1]):
projection = matrix(QQ, [[0, 1, 1], [0, 0, 1]])
elif ct == CartanType(['G', 2, 1]):
projection = matrix(RR,
[[0, 0.5, 0], [0, 0.866, 1.732]])
else:
raise 'There is no usual projection for this Cartan type, you have to give one in argument'
assert (n + 1 == projection.ncols())
assert (2 == projection.nrows())
# Check the size is correct with the lattice
assert (len(size) == n)
# Select the center of the translated fundamental polygon to plot
translation_factors = ct.translation_factors()
simple_roots = self.simple_roots()
translation_vectors = [
translation_factors[i] * simple_roots[i]
for i in ct.classical().index_set()
]
initial = [[]]
for i in range(n):
prod_list = []
for elem in size[i]:
for partial_list in initial:
prod_list.append([elem] + partial_list)
initial = prod_list
part_lattice = []
for combinaison in prod_list:
elem_lattice = self.zero()
for i in range(n):
elem_lattice = elem_lattice + combinaison[
i] * translation_vectors[i]
part_lattice.append(elem_lattice)
# Get the vertices of the fundamental alcove
fundamental_weights = self.fundamental_weights()
vertices = map(lambda x: (1 / x.level()) * x,
fundamental_weights.list())
# Recup the group which act on the fundamental polygon
classical = self.weyl_group().classical()
for center in part_lattice:
for w in classical:
# for each center of polygon and each element of classical
# parabolic subgroup, we have to draw an alcove.
#first, iterate over pairs of fundamental weights, drawing lines border of polygons:
for i in range(1, n + 1):
for j in range(i + 1, n + 1):
p1 = projection * (
(w.action(vertices[i])).to_vector() +
center.to_vector())
p2 = projection * (
(w.action(vertices[j])).to_vector() +
center.to_vector())
G += line([p1, p2],
rgbcolor=(0, 0, 0),
thickness=2)
#next, get all lines from point to a fundamental weight, that separe different
#chanber in a same polygon (important: associate a color with a fundamental weight)
pcenter = projection * (center.to_vector())
for i in range(1, n + 1):
p3 = projection * (
(w.action(vertices[i])).to_vector() +
center.to_vector())
G += line([p3, pcenter],
rgbcolor=colors[n - i + 1])
#Draw alcovewalks
#FIXME : The good way to draw this is to use the alcoves walks works made in Cristals
#The code here just draw like example and import the good things.
rho = (1 / self.rho().level()) * self.rho()
W = self.weyl_group()
for walk in alcovewalks:
target = W.from_reduced_word(walk).action(rho)
for i in range(len(walk)):
walk.pop()
origin = W.from_reduced_word(walk).action(rho)
G += arrow(projection * (origin.to_vector()),
projection * (target.to_vector()),
rgbcolor=(0.6, 0, 0.6),
width=1,
arrowsize=5)
target = origin
else:
# non affine plot
# Check the projection
# TODO : try to have usual_projection for main plotable types
if projection == 'usual':
if ct == CartanType(['A', 2]):
projection = matrix(RR, [[0.5, -0.5], [0.866, 0.866]])
elif ct == CartanType(['B', 2]):
projection = matrix(QQ, [[1, 0], [1, 1]])
elif ct == CartanType(['C', 2]):
projection = matrix(QQ, [[1, 1], [0, 1]])
elif ct == CartanType(['G', 2]):
projection = matrix(RR, [[0.5, 0], [0.866, 1.732]])
else:
raise 'There is no usual projection for this Cartan type, you have to give one in argument'
# Get the fundamental weights
fundamental_weights = self.fundamental_weights()
WeylGroup = self.weyl_group()
#Draw not the alcove but the cones delimited by the hyperplanes
#The size of the line depend of the fundamental weights.
pcenter = projection * (self.zero().to_vector())
for w in WeylGroup:
for i in range(1, n + 1):
p3 = 3 * projection * (
(w.action(fundamental_weights[i])).to_vector())
G += line([p3, pcenter], rgbcolor=colors[n - i + 1])
#Draw the simple roots
if simple_roots:
SimpleRoots = self.simple_roots()
if ct.is_affine():
G += arrow((0, 0),
projection * (SimpleRoots[0].to_vector()),
rgbcolor=(0, 0, 0))
for j in range(1, n + 1):
G += arrow((0, 0),
projection * (SimpleRoots[j].to_vector()),
rgbcolor=colors[j])
#Draw the fundamental weights
if fundamental_weights:
FundWeight = self.fundamental_weights()
for j in range(1, n + 1):
G += point(projection * (FundWeight[j].to_vector()),
rgbcolor=colors[j],
pointsize=60)
G.set_aspect_ratio(1)
G.axes(False)
return G
