def healpix_diagram(a=1, e=0, shade_polar_region=True):
r"""
Return a Sage Graphics object diagramming the HEALPix projection
boundary and polar triangles for the ellipsoid with major radius `a`
and eccentricity `e`.
Inessential graphics method.
Requires Sage graphics methods.
"""
from sage.all import Graphics, line2d, point, polygon, text, RealNumber, Integer
# Make Sage types compatible with Numpy.
RealNumber = float
Integer = int
R = auth_rad(a, e)
g = Graphics()
color = 'black' # Boundary color.
shade_color = 'blue' # Polar triangles color.
dl = array((R*pi/2,0))
lu = [(-R*pi, R*pi/4),(-R*3*pi/4, R*pi/2)]
ld = [(-R*3*pi/4, R*pi/2),(-R*pi/2, R*pi/4)]
g += line2d([(-R*pi, -R*pi/4),(-R*pi, R*pi/4)], color=color)
g += line2d([(R*pi, R*pi/4),(R*pi, -R*pi/4)], linestyle = '--',
color=color)
for k in range(4):
g += line2d([array(p) + k*dl for p in lu], color=color)
g += line2d([array(p) + k*dl for p in ld], linestyle = '--',
color=color)
g += line2d([array(p) + array((k*R*pi/2 - R*pi/4, -R*3*pi/4))
for p in ld], color=color)
g += line2d([array(p) + array((k*R*pi/2 + R*pi/4, -R*3*pi/4))
for p in lu], linestyle = '--', color=color)
pn = array((-R*3*pi/4, R*pi/2))
ps = array((-R*3*pi/4, -R*pi/2))
g += point([pn + k*dl for k in range(4)] +
[ps + k*dl for k in range(4)], size=20, color=color)
g += point([pn + k*dl for k in range(1, 4)] +
[ps + k*dl for k in range(1, 4)], color='white', size=10,
zorder=3)
npp = [(-R*pi, R*pi/4), (-R*3*pi/4, R*pi/2), (-R*pi/2, R*pi/4)]
spp = [(-R*pi, -R*pi/4), (-R*3*pi/4, -R*pi/2), (-R*pi/2, -R*pi/4)]
if shade_polar_region:
for k in range(4):
g += polygon([array(p) + k*dl for p in npp], alpha=0.1,
color=shade_color)
g += text(str(k), array((-R*3*pi/4, R*5*pi/16)) + k*dl,
color='red', fontsize=20)
g += polygon([array(p) + k*dl for p in spp], alpha=0.1,
color=shade_color)
g += text(str(k), array((-R*3*pi/4, -R*5*pi/16)) + k*dl,
color='red', fontsize=20)
return g
def healpix_diagram(a=1, e=0, shade_polar_region=True):
r"""
Return a Sage Graphics object diagramming the HEALPix projection
boundary and polar triangles for the ellipsoid with major radius `a`
and eccentricity `e`.
Inessential graphics method.
Requires Sage graphics methods.
"""
from sage.all import Graphics, line2d, point, polygon, text, RealNumber, Integer
# Make Sage types compatible with Numpy.
RealNumber = float
Integer = int
R = auth_rad(a, e)
g = Graphics()
color = 'black' # Boundary color.
shade_color = 'blue' # Polar triangles color.
dl = array((R*pi/2,0))
lu = [(-R*pi, R*pi/4),(-R*3*pi/4, R*pi/2)]
ld = [(-R*3*pi/4, R*pi/2),(-R*pi/2, R*pi/4)]
g += line2d([(-R*pi, -R*pi/4),(-R*pi, R*pi/4)], color=color)
g += line2d([(R*pi, R*pi/4),(R*pi, -R*pi/4)], linestyle = '--',
color=color)
for k in range(4):
g += line2d([array(p) + k*dl for p in lu], color=color)
g += line2d([array(p) + k*dl for p in ld], linestyle = '--',
color=color)
g += line2d([array(p) + array((k*R*pi/2 - R*pi/4, -R*3*pi/4))
for p in ld], color=color)
g += line2d([array(p) + array((k*R*pi/2 + R*pi/4, -R*3*pi/4))
for p in lu], linestyle = '--', color=color)
pn = array((-R*3*pi/4, R*pi/2))
ps = array((-R*3*pi/4, -R*pi/2))
g += point([pn + k*dl for k in range(4)] +
[ps + k*dl for k in range(4)], size=20, color=color)
g += point([pn + k*dl for k in range(1, 4)] +
[ps + k*dl for k in range(1, 4)], color='white', size=10,
zorder=3)
npp = [(-R*pi, R*pi/4), (-R*3*pi/4, R*pi/2), (-R*pi/2, R*pi/4)]
spp = [(-R*pi, -R*pi/4), (-R*3*pi/4, -R*pi/2), (-R*pi/2, -R*pi/4)]
if shade_polar_region:
for k in range(4):
g += polygon([array(p) + k*dl for p in npp], alpha=0.1,
color=shade_color)
g += text(str(k), array((-R*3*pi/4, R*5*pi/16)) + k*dl,
color='red', fontsize=20)
g += polygon([array(p) + k*dl for p in spp], alpha=0.1,
color=shade_color)
g += text(str(k), array((-R*3*pi/4, -R*5*pi/16)) + k*dl,
color='red', fontsize=20)
return g
def EC_nf_plot(K, ainvs, base_field_gen_name):
try:
n1 = K.signature()[0]
if n1 == 0:
return plot([])
R=[]
S=K.embeddings(RDF)
for s in S:
A=[s(c) for c in ainvs]
R.append(EC_R_plot_zone(Rx([A[4],A[3],A[1],1]),Rx([A[2],A[0]])))
xmin = min([r[0] for r in R])
xmax = max([r[1] for r in R])
ymin = min([r[2] for r in R])
ymax = max([r[3] for r in R])
cols = rainbow(n1) # Default choice of n colours
# However, these tend to be too pale, so we preset them for small values of n
if n1==1:
cols=["blue"]
elif n1==2:
cols=["red","blue"]
elif n1==3:
cols=["red","limegreen","blue"]
elif n1==4:
cols = ["red", "orange", "forestgreen", "blue"]
elif n1==5:
cols = ["red", "orange", "forestgreen", "blue", "darkviolet"]
elif n1==6:
cols = ["red", "darkorange", "gold", "forestgreen", "blue", "darkviolet"]
elif n1==7:
cols = ["red", "darkorange", "gold", "forestgreen", "blue", "darkviolet", "fuchsia"]
return sum([EC_R_plot([S[i](c) for c in ainvs], xmin, xmax, ymin, ymax, cols[i], "$" + base_field_gen_name + " \mapsto$ " + str(S[i].im_gens()[0].n(20))+"$\dots$") for i in range(n1)])
except:
return text("Unable to plot", (1, 1), fontsize=36)
def EC_nf_plot(K, ainvs, base_field_gen_name):
try:
n1 = K.signature()[0]
if n1 == 0:
return plot([])
R=[]
S=K.embeddings(RDF)
for s in S:
A=[s(c) for c in ainvs]
R.append(EC_R_plot_zone(Rx([A[4],A[3],A[1],1]),Rx([A[2],A[0]])))
xmin = min([r[0] for r in R])
xmax = max([r[1] for r in R])
ymin = min([r[2] for r in R])
ymax = max([r[3] for r in R])
cols = rainbow(n1) # Default choice of n colours
# However, these tend to be too pale, so we preset them for small values of n
if n1==1:
cols=["blue"]
elif n1==2:
cols=["red","blue"]
elif n1==3:
cols=["red","limegreen","blue"]
elif n1==4:
cols = ["red", "orange", "forestgreen", "blue"]
elif n1==5:
cols = ["red", "orange", "forestgreen", "blue", "darkviolet"]
elif n1==6:
cols = ["red", "darkorange", "gold", "forestgreen", "blue", "darkviolet"]
elif n1==7:
cols = ["red", "darkorange", "gold", "forestgreen", "blue", "darkviolet", "fuchsia"]
return sum([EC_R_plot([S[i](c) for c in ainvs], xmin, xmax, ymin, ymax, cols[i], "$" + base_field_gen_name + " \mapsto$ " + str(S[i].im_gens()[0].n(20))+"$\dots$") for i in range(n1)])
except:
return text("Unable to plot", (1, 1), fontsize=36)
def EC_nf_plot(E, base_field_gen_name):
K = E.base_field()
n1 = K.signature()[0]
if n1 == 0:
return plot([])
prec = 53
maxprec = 10 ** 6
while prec < maxprec: # Try to base change to R. May fail if resulting curve is almost singular, so increase precision.
try:
SR = K.embeddings(RealField(prec))
X = [E.base_extend(s) for s in SR]
break
except ArithmeticError:
prec *= 2
if prec >= maxprec:
return text("Unable to plot", (1, 1), fontsize="xx-large")
X = [e.plot() for e in X]
xmin = min([x.xmin() for x in X])
xmax = max([x.xmax() for x in X])
ymin = min([x.ymin() for x in X])
ymax = max([x.ymax() for x in X])
cols = ["blue", "red", "green", "orange", "brown"] # Preset colours, because rainbow tends to return too pale ones
if n1 > len(cols):
cols = rainbow(n1)
return sum([EC_R_plot([SR[i](a) for a in E.ainvs()], xmin, xmax, ymin, ymax, cols[i], "$" + base_field_gen_name + " \mapsto$ " + str(SR[i].im_gens()[0].n(20))) for i in range(n1)])
def EC_nf_plot(E,base_field_gen_name):
K = E.base_field()
n1 = K.signature()[0]
if n1 == 0:
return plot([])
prec = 53
maxprec = 10**6
while prec < maxprec: # Try to base change to R. May fail if resulting curve is almost singular, so increase precision.
try:
SR = K.embeddings(RealField(prec))
X = [E.base_extend(s) for s in SR]
break
except ArithmeticError:
prec *= 2
if prec >= maxprec:
return text("Unable to plot",(1,1),fontsize="xx-large")
X = [e.plot() for e in X]
xmin = min([x.xmin() for x in X])
xmax = max([x.xmax() for x in X])
ymin = min([x.ymin() for x in X])
ymax = max([x.ymax() for x in X])
cols = ["blue","red","green","orange","brown"] # Preset colours, because rainbow tends to return too pale ones
if n1 > len(cols):
cols = rainbow(n1)
return sum([EC_R_plot([SR[i](a) for a in E.ainvs()],xmin,xmax,ymin,ymax,cols[i],"$"+base_field_gen_name+" \mapsto$ "+str(SR[i].im_gens()[0].n(20))) for i in range(n1)])
def EC_nf_plot(E, base_field_gen_name):
K = E.base_field()
n1 = K.signature()[0]
if n1 == 0:
return plot([])
prec = 53
maxprec = 10 ** 6
while prec < maxprec: # Try to base change to RR. May fail if resulting curve is almost singular, so increase precision.
try:
SR = K.embeddings(RealField(prec))
X = [E.base_extend(s) for s in SR]
break
except ArithmeticError:
prec *= 2
if prec >= maxprec:
return text("Unable to plot", (1, 1), fontsize=36)
try:
X = [e.plot() for e in X]
except:
return text("Unable to plot", (1, 1), fontsize=36)
xmin = min([x.xmin() for x in X])
xmax = max([x.xmax() for x in X])
ymin = min([x.ymin() for x in X])
ymax = max([x.ymax() for x in X])
cols = rainbow(n1) # Default choice of n colours
# Howver, these tend to be too pale, so we preset them for small values of n
if n1==1:
cols=["blue"]
elif n1==2:
cols=["red","blue"]
elif n1==3:
cols=["red","limegreen","blue"]
elif n1==4:
cols = ["red", "orange", "forestgreen", "blue"]
elif n1==5:
cols = ["red", "orange", "forestgreen", "blue", "darkviolet"]
elif n1==6:
cols = ["red", "darkorange", "gold", "forestgreen", "blue", "darkviolet"]
elif n1==7:
cols = ["red", "darkorange", "gold", "forestgreen", "blue", "darkviolet", "fuchsia"]
return sum([EC_R_plot([SR[i](a) for a in E.ainvs()], xmin, xmax, ymin, ymax, cols[i], "$" + base_field_gen_name + " \mapsto$ " + str(SR[i].im_gens()[0].n(20))+"$\dots$") for i in range(n1)])
def show(self, unit=10, labels=True):
from sage.all import circle, text, line, Graphics
pos = self.basic_grid_embedding()
for v, (a, b) in pos.items():
pos[v] = (unit * a, unit * b)
if not labels:
verts = [circle(p, 1, fill=True) for p in pos.values()]
else:
verts = [
text(repr(v), p, fontsize=20, color='black')
for v, p in pos.items()
]
verts += [circle(p, 1.5, fill=False) for p in pos.values()]
edges = [
line([pos[e.tail], pos[e.head]]) for e in self.edges
if e not in self.dummy
]
G = sum(verts + edges, Graphics())
G.axes(False)
return G
def EC_nf_plot(E, base_field_gen_name):
K = E.base_field()
n1 = K.signature()[0]
if n1 == 0:
return plot([])
prec = 53
maxprec = 10 ** 6
while prec < maxprec: # Try to base change to RR. May fail if resulting curve is almost singular, so increase precision.
try:
SR = K.embeddings(RealField(prec))
X = [E.base_extend(s) for s in SR]
break
except ArithmeticError:
prec *= 2
if prec >= maxprec:
return text("Unable to plot", (1, 1), fontsize="xx-large")
X = [e.plot() for e in X]
xmin = min([x.xmin() for x in X])
xmax = max([x.xmax() for x in X])
ymin = min([x.ymin() for x in X])
ymax = max([x.ymax() for x in X])
cols = rainbow(n1) # Default choice of n colours
# Howver, these tend to be too pale, so we preset them for small values of n
if n1==1:
cols=["blue"]
elif n1==2:
cols=["red","blue"]
elif n1==3:
cols=["red","limegreen","blue"]
elif n1==4:
cols = ["red", "orange", "forestgreen", "blue"]
elif n1==5:
cols = ["red", "orange", "forestgreen", "blue", "darkviolet"]
elif n1==6:
cols = ["red", "darkorange", "gold", "forestgreen", "blue", "darkviolet"]
elif n1==7:
cols = ["red", "darkorange", "gold", "forestgreen", "blue", "darkviolet", "fuchsia"]
return sum([EC_R_plot([SR[i](a) for a in E.ainvs()], xmin, xmax, ymin, ymax, cols[i], "$" + base_field_gen_name + " \mapsto$ " + str(SR[i].im_gens()[0].n(20))+"$\dots$") for i in range(n1)])
def plot(self,
textvertices=False,
only_simple=False,
fontsize=18,
sort=False,
fact=0.5,
thickness=4,
edges_thickness=4,
arrowshorten=8,
linestyle_simple='solid',
linestyle_nonsimple='dashed',
arrowsize=2,
arrows=False,
**options):
r"""
Plots a SimpleModulesGraph.
"""
vertex_colors = self._vertex_colors
simple_color = self._simple_color
nonsimple_color = self._nonsimple_color
heights = self._heights
if only_simple:
for v in vertex_colors[nonsimple_color]:
if self.has_vertex(v):
self.delete_vertex(v)
# while vertex_colors[nonsimple_color].count(v)>0:
# vertex_colors[nonsimple_color].remove(v)
for j in range(len(heights)):
while heights[j].count(v) > 0:
heights[j].remove(v)
vertex_colors[nonsimple_color] = dict()
pos = dict()
labels = list()
vertices = list()
edges = list()
min_d = 0.5
widths = [
float(sum([len(str(v)) / float(6) + min_d for v in heights[i]]))
for i in range(len(heights))
]
print widths
max_w = max(widths)
if len(widths) > 1:
min_w = min([w for w in widths if w > 3])
else:
min_w = 0
print min_w, max_w, widths
max_vert = max([len(_) for _ in heights.values()])
for i in range(len(heights)):
if sort:
heights[i] = sorted(heights[i])
#print heights[i]
real_w = widths[i]
prev_w = min_w if i == 0 else widths[i - 1]
next_w = min_w if i == len(heights) - 1 else widths[i + 1]
cur_w = max(float(next_w) * 0.9, float(prev_w) * 0.9, real_w)
d = max(2 * float(cur_w - real_w) / len(heights[i]), min_d)
print real_w, cur_w
print "d = ", d
p = [-(cur_w), float(max_vert) * fact * i]
w = 0
for j in range(len(heights[i])):
v = heights[i][j]
p = [p[0] + w + 0.2, p[1]]
w = float(len(str(v))) / float(6)
p[0] = p[0] + w + d
pos[heights[i][j]] = p
c = simple_color if vertex_colors[simple_color].count(
v) > 0 else nonsimple_color
if textvertices:
ct = colors.black.rgb()
else:
ct = colors.white.rgb()
labels.append(
text("$\mathbf{" + str(v)[1:len(str(v)) - 1] + "}$",
(p[0] + 0.2, p[1]),
rgbcolor=ct,
zorder=8,
fontsize=fontsize))
print w
if textvertices:
P = line2d(
[[p[0] - w, p[1] - 0.9], [p[0] - w, p[1] + 1.1]],
rgbcolor=c,
thickness=thickness,
linestyle=linestyle_simple
if c == simple_color else linestyle_nonsimple)
P += line2d(
[[p[0] - w, p[1] + 1.1], [p[0] + w + 0.2, p[1] + 1.1]],
rgbcolor=c,
thickness=thickness,
linestyle=linestyle_simple
if c == simple_color else linestyle_nonsimple)
P += line2d([[p[0] + w + 0.2, p[1] + 1.1],
[p[0] + w + 0.2, p[1] - 0.9]],
rgbcolor=c,
thickness=thickness,
linestyle=linestyle_simple
if c == simple_color else linestyle_nonsimple)
P += line2d(
[[p[0] + w + 0.2, p[1] - 0.9], [p[0] - w, p[1] - 0.9]],
rgbcolor=c,
thickness=thickness,
linestyle=linestyle_simple
if c == simple_color else linestyle_nonsimple)
else:
polygon2d([[p[0] - w, p[1] - 0.9], [p[0] - w, p[1] + 1.1],
[p[0] + w + 0.2, p[1] + 1.1],
[p[0] + w + 0.2, p[1] - 0.9]],
fill=(not textvertices),
rgbcolor=c,
thickness=thickness,
linestyle=linestyle_simple
if c == simple_color else linestyle_nonsimple)
vertices.append(P)
for e in self.edges():
v = e[0]
if not self.has_vertex(e[0]) or not self.has_vertex(e[1]):
print "deleting edge ", e
self.delete_edge(e[0], e[1])
else:
c = simple_color if vertex_colors[simple_color].count(
v) > 0 else nonsimple_color
if arrows:
edges.append(
arrow([pos[e[0]][0], pos[e[0]][1] + 1.1],
[pos[e[1]][0], pos[e[1]][1] - 0.9],
rgbcolor=c,
zorder=-1,
arrowsize=arrowsize,
arrowshorten=arrowshorten,
width=edges_thickness,
linestyle=linestyle_simple
if c == simple_color else linestyle_nonsimple))
else:
edges.append(
line2d([[pos[e[0]][0], pos[e[0]][1] + 1.1],
[pos[e[1]][0], pos[e[1]][1] - 0.9]],
rgbcolor=c,
zorder=-1,
thickness=edges_thickness,
linestyle=linestyle_simple
if c == simple_color else linestyle_nonsimple))
print "calculation ended"
gp = self.graphplot(dpi=300,
pos=pos,
vertex_size=2050,
figsize=round(float(max_vert) * 1.5),
vertex_colors=vertex_colors)
gp._plot_components['vertex_labels'] = labels
gp._plot_components['vertices'] = vertices
gp._plot_components['edges'] = edges
#self._pos = pos
return gp.plot(**options)
def draw_fundamental_domain(N,group='Gamma0',model="H",axes=None,filename=None,**kwds):
r""" Draw fundamental domain
INPUT:
- ''model'' -- (default ''H'')
- ''H'' -- Upper halfplane
- ''D'' -- Disk model
- ''filename''-- filename to print to
- ''**kwds''-- additional arguments to matplotlib
- ''axes'' -- set geometry of output
=[x0,x1,y0,y1] -- restrict figure to [x0,x1]x[y0,y1]
EXAMPLES::
sage: G=MySubgroup(Gamma0(3))
sage: G.draw_fundamental_domain()
"""
G=eval(group+'('+str(N)+')')
#print G
name ="$"+latex(G)+"$"
## need a "nice" set of coset representatives to draw a connected fundamental domain. Only implemented for Gamma_0(N)
coset_reps = nice_coset_reps(G)
if(model=="D"):
g=draw_funddom_d(coset_reps,format,I)
else:
g=draw_funddom(coset_reps,format)
if(axes<>None):
[x0,x1,y0,y1]=axes
elif(model=="D"):
x0=-1 ; x1=1 ; y0=-1.1 ; y1=1
else:
# find the width of the fundamental domain
w=0 #self.cusp_width(Cusp(Infinity))
wmin=0 ; wmax=1
max_x = RR(0.55)
rho = CC( exp(2*pi*I/3))
for V in coset_reps:
## we also compare the real parts of where rho and infinity are mapped
r1 = (V.acton(rho)).real()
if(V[1,0]<>0):
inf1 = RR(V[0,0] / V[1,0])
else:
inf1 = 0
if(V[1 ,0 ]==0 and V[0 ,0 ]==1 ):
if(V[0 ,1 ]>wmax):
wmax=V[0 ,1 ]
if(V[0 ,1 ]<wmin):
wmin=V[0 ,1 ]
if( max(r1,inf1) > max_x):
max_x = max(r1,inf1)
#print "wmin,wmax=",wmin,wmax
#x0=-1; x1=1; y0=-0.2; y1=1.5
x0=RR(-max_x) ; x1=RR(max_x) ; y0=RR(-0.15) ; y1=RR(1.5)
## Draw the axes manually (since can't figure out how to do it automatically)
ax = line([[x0,0.0],[x1,0.0]],color='black')
#ax = ax + line([[0.0,y0],[0.0,y1]],color='black')
## ticks
ax = ax + line([[-0.5,-0.01],[-0.5,0.01]],color='black')
ax = ax + line([[0.5,-0.01],[0.5,0.01]],color='black')
g = g + ax
if model=="H":
t = text(name, (0, -0.1), fontsize=16, color='black')
else:
t = text(name, (0, -1.1), fontsize=16, color='black')
g = g + t
g.set_aspect_ratio(1)
g.set_axes_range(x0,x1,y0,y1)
g.axes(False)
if(filename<>None):
fig = g.matplotlib()
fig.set_canvas(FigureCanvasAgg(fig))
axes = fig.get_axes()[0]
axes.minorticks_off()
axes.set_yticks([])
fig.savefig(filename,**kwds)
else:
return g
def draw_fundamental_domain(N,
group='Gamma0',
model="H",
axes=None,
filename=None,
**kwds):
r""" Draw fundamental domain
INPUT:
- ''model'' -- (default ''H'')
- ''H'' -- Upper halfplane
- ''D'' -- Disk model
- ''filename''-- filename to print to
- ''**kwds''-- additional arguments to matplotlib
- ''axes'' -- set geometry of output
=[x0,x1,y0,y1] -- restrict figure to [x0,x1]x[y0,y1]
EXAMPLES::
sage: G=MySubgroup(Gamma0(3))
sage: G.draw_fundamental_domain()
"""
G = eval(group + '(' + str(N) + ')')
#print G
name = "$" + latex(G) + "$"
## need a "nice" set of coset representatives to draw a connected fundamental domain. Only implemented for Gamma_0(N)
coset_reps = nice_coset_reps(G)
if (model == "D"):
g = draw_funddom_d(coset_reps, format, I)
else:
g = draw_funddom(coset_reps, format)
if (axes <> None):
[x0, x1, y0, y1] = axes
elif (model == "D"):
x0 = -1
x1 = 1
y0 = -1.1
y1 = 1
else:
# find the width of the fundamental domain
w = 0 #self.cusp_width(Cusp(Infinity))
wmin = 0
wmax = 1
max_x = RR(0.55)
rho = CC(exp(2 * pi * I / 3))
for V in coset_reps:
## we also compare the real parts of where rho and infinity are mapped
r1 = (V.acton(rho)).real()
if (V[1, 0] <> 0):
inf1 = RR(V[0, 0] / V[1, 0])
else:
inf1 = 0
if (V[1, 0] == 0 and V[0, 0] == 1):
if (V[0, 1] > wmax):
wmax = V[0, 1]
if (V[0, 1] < wmin):
wmin = V[0, 1]
if (max(r1, inf1) > max_x):
max_x = max(r1, inf1)
#print "wmin,wmax=",wmin,wmax
#x0=-1; x1=1; y0=-0.2; y1=1.5
x0 = RR(-max_x)
x1 = RR(max_x)
y0 = RR(-0.15)
y1 = RR(1.5)
## Draw the axes manually (since can't figure out how to do it automatically)
ax = line([[x0, 0.0], [x1, 0.0]], color='black')
#ax = ax + line([[0.0,y0],[0.0,y1]],color='black')
## ticks
ax = ax + line([[-0.5, -0.01], [-0.5, 0.01]], color='black')
ax = ax + line([[0.5, -0.01], [0.5, 0.01]], color='black')
g = g + ax
if model == "H":
t = text(name, (0, -0.1), fontsize=16, color='black')
else:
t = text(name, (0, -1.1), fontsize=16, color='black')
g = g + t
g.set_aspect_ratio(1)
g.set_axes_range(x0, x1, y0, y1)
g.axes(False)
if (filename <> None):
fig = g.matplotlib()
fig.set_canvas(FigureCanvasAgg(fig))
axes = fig.get_axes()[0]
axes.minorticks_off()
axes.set_yticks([])
fig.savefig(filename, **kwds)
else:
return g
def draw_fundamental_domain(N,
group='Gamma0',
model="H",
axes=None,
filename=None,
**kwds):
r""" Draw fundamental domain
INPUT:
- ''model'' -- (default ''H'')
= ''H'' -- Upper halfplane
= ''D'' -- Disk model
- ''filename''-- filename to print to
- ''**kwds''-- additional arguments to matplotlib
- ''axes'' -- set geometry of output
=[x0,x1,y0,y1] -- restrict figure to [x0,x1]x[y0,y1]
EXAMPLES::
sage: G=MySubgroup(Gamma0(3))
sage: G.draw_fundamental_domain()
"""
if group.strip() == 'Gamma0':
G = Gamma0(N)
elif group.strip() == 'Gamma1':
G = Gamma1(N)
elif group.strip() == 'Gamma':
G = Gamma(N)
else:
raise ValueError('group must be one of: "Gamma0", "Gamma1", "Gamma"')
s = "$" + latex(G) + "$"
s = s.replace("mbox", "mathrm")
s = s.replace("Bold", "mathbb")
name = s
# name ="$\mbox{SL}_{2}(\mathbb{Z})$"
# need a "nice" set of coset representatives to draw a connected
# fundamental domain. Only implemented for Gamma_0(N)
coset_reps = nice_coset_reps(G)
# if(group=='Gamma0'):
# else:
# coset_reps = list(G.coset_reps())
from matplotlib.backends.backend_agg import FigureCanvasAgg
if (model == "D"):
g = _draw_funddom_d(coset_reps, format, I)
else:
g = _draw_funddom(coset_reps, format)
if (axes is not None):
[x0, x1, y0, y1] = axes
elif (model == "D"):
x0 = -1
x1 = 1
y0 = -1.1
y1 = 1
else:
# find the width of the fundamental domain
# w = 0 # self.cusp_width(Cusp(Infinity)) FIXME: w is never used
wmin = 0
wmax = 1
max_x = RR(0.55)
rho = CC(exp(2 * pi * I / 3))
for V in coset_reps:
## we also compare the real parts of where rho and infinity are mapped
r1 = (V.acton(rho)).real()
if (V[1, 0] != 0):
inf1 = RR(V[0, 0] / V[1, 0])
else:
inf1 = 0
if (V[1, 0] == 0 and V[0, 0] == 1):
if (V[0, 1] > wmax):
wmax = V[0, 1]
if (V[0, 1] < wmin):
wmin = V[0, 1]
if (max(r1, inf1) > max_x):
max_x = max(r1, inf1)
logging.debug("wmin,wmax=%s,%s" % (wmin, wmax))
# x0=-1; x1=1; y0=-0.2; y1=1.5
x0 = RR(-max_x)
x1 = RR(max_x)
y0 = RR(-0.15)
y1 = RR(1.5)
## Draw the axes manually (since can't figure out how to do it automatically)
ax = line([[x0, 0.0], [x1, 0.0]], color='black')
# ax = ax + line([[0.0,y0],[0.0,y1]],color='black')
## ticks
ax = ax + line([[-0.5, -0.01], [-0.5, 0.01]], color='black')
ax = ax + line([[0.5, -0.01], [0.5, 0.01]], color='black')
g = g + ax
if model == "H":
t = text(name, (0, -0.1), fontsize=16, color='black')
t = t + text(
"$ -\\frac{1}{2} $", (-0.5, -0.1), fontsize=12, color='black')
t = t + text(
"$ \\frac{1}{2} $", (0.5, -0.1), fontsize=12, color='black')
else:
t = text(name, (0, -1.1), fontsize=16, color='black')
g = g + t
g.set_aspect_ratio(1)
g.set_axes_range(x0, x1, y0, y1)
g.axes(False)
if (filename is not None):
fig = g.matplotlib()
fig.set_canvas(FigureCanvasAgg(fig))
axes = fig.get_axes()[0]
axes.minorticks_off()
axes.set_yticks([])
fig.savefig(filename, **kwds)
else:
return g
def plot(self, textvertices=False, only_simple=False, fontsize=18, sort=False, fact = 0.5, thickness=4, edges_thickness=4, arrowshorten=8,
linestyle_simple='solid', linestyle_nonsimple='dashed', arrowsize=2, arrows=False, **options):
r"""
Plots a SimpleModulesGraph.
"""
vertex_colors = self._vertex_colors
simple_color = self._simple_color
nonsimple_color = self._nonsimple_color
heights = self._heights
if only_simple:
for v in vertex_colors[nonsimple_color]:
if self.has_vertex(v):
self.delete_vertex(v)
# while vertex_colors[nonsimple_color].count(v)>0:
# vertex_colors[nonsimple_color].remove(v)
for j in range(len(heights)):
while heights[j].count(v) > 0:
heights[j].remove(v)
vertex_colors[nonsimple_color] = dict()
pos = dict()
labels = list()
vertices = list()
edges = list()
min_d = 0.5
widths = [float(sum([len(str(v)) / float(6) + min_d for v in heights[i]]))
for i in range(len(heights))]
print widths
max_w = max(widths)
if len(widths) > 1:
min_w = min([w for w in widths if w > 3])
else:
min_w = 0
print min_w, max_w, widths
max_vert = max([len(_) for _ in heights.values()])
for i in range(len(heights)):
if sort:
heights[i] = sorted(heights[i])
#print heights[i]
real_w = widths[i]
prev_w = min_w if i == 0 else widths[i - 1]
next_w = min_w if i == len(heights) - 1 else widths[i + 1]
cur_w = max(float(next_w) * 0.9, float(prev_w) * 0.9, real_w)
d = max(2 * float(cur_w - real_w) / len(heights[i]), min_d)
print real_w, cur_w
print "d = ", d
p = [-(cur_w), float(max_vert) * fact * i]
w = 0
for j in range(len(heights[i])):
v = heights[i][j]
p = [p[0] + w + 0.2, p[1]]
w = float(len(str(v))) / float(6)
p[0] = p[0] + w + d
pos[heights[i][j]] = p
c = simple_color if vertex_colors[
simple_color].count(v) > 0 else nonsimple_color
if textvertices:
ct = colors.black.rgb()
else:
ct = colors.white.rgb()
labels.append(
text("$\mathbf{" + str(v)[1:len(str(v))-1] + "}$", (p[0] + 0.2, p[1]), rgbcolor=ct, zorder=8, fontsize=fontsize))
print w
if textvertices:
P = line2d([[p[0] - w, p[1] - 0.9], [p[0] - w, p[1] + 1.1]], rgbcolor=c, thickness=thickness, linestyle=linestyle_simple if c == simple_color else linestyle_nonsimple)
P += line2d([[p[0] - w, p[1] + 1.1], [p[0] + w + 0.2, p[1] + 1.1]], rgbcolor=c, thickness=thickness, linestyle=linestyle_simple if c == simple_color else linestyle_nonsimple)
P += line2d([[p[0] + w + 0.2, p[1] + 1.1], [p[0] + w + 0.2, p[1] - 0.9]], rgbcolor=c, thickness=thickness, linestyle=linestyle_simple if c == simple_color else linestyle_nonsimple)
P += line2d([[p[0] + w + 0.2, p[1] - 0.9], [p[0] - w, p[1] - 0.9]], rgbcolor=c, thickness=thickness, linestyle=linestyle_simple if c == simple_color else linestyle_nonsimple)
else:
polygon2d([[p[0] - w, p[1] - 0.9], [p[0] - w, p[1] + 1.1], [
p[0] + w + 0.2, p[1] + 1.1], [p[0] + w + 0.2, p[1] - 0.9]], fill=(not textvertices), rgbcolor=c, thickness=thickness, linestyle=linestyle_simple if c == simple_color else linestyle_nonsimple)
vertices.append(P)
for e in self.edges():
v = e[0]
if not self.has_vertex(e[0]) or not self.has_vertex(e[1]):
print "deleting edge ", e
self.delete_edge(e[0], e[1])
else:
c = simple_color if vertex_colors[
simple_color].count(v) > 0 else nonsimple_color
if arrows:
edges.append(arrow([pos[e[0]][0], pos[e[0]][
1] + 1.1], [pos[e[1]][0], pos[e[1]][1] - 0.9], rgbcolor=c, zorder=-1, arrowsize=arrowsize, arrowshorten=arrowshorten, width=edges_thickness, linestyle=linestyle_simple if c == simple_color else linestyle_nonsimple))
else:
edges.append(line2d([[pos[e[0]][0], pos[e[0]][1] + 1.1], [pos[e[1]][0], pos[e[1]][1] - 0.9]], rgbcolor=c, zorder=-1, thickness=edges_thickness, linestyle=linestyle_simple if c == simple_color else linestyle_nonsimple))
print "calculation ended"
gp = self.graphplot(dpi=300, pos=pos, vertex_size=2050, figsize=round(
float(max_vert) * 1.5), vertex_colors=vertex_colors)
gp._plot_components['vertex_labels'] = labels
gp._plot_components['vertices'] = vertices
gp._plot_components['edges'] = edges
#self._pos = pos
return gp.plot(**options)
def rhealpix_diagram(a=1,
e=0,
north_square=0,
south_square=0,
shade_polar_region=True):
r"""
Return a Sage Graphics object diagramming the rHEALPix projection
boundary and polar triangles for the ellipsoid with major radius `a`
and eccentricity `e`.
Inessential graphics method.
Requires Sage graphics methods.
"""
from sage.all import Graphics, line2d, point, polygon, text, RealNumber, Integer
# Make Sage types compatible with Numpy.
RealNumber = float
Integer = int
R = auth_rad(a, e)
g = Graphics()
color = 'black' # Boundary color.
shade_color = 'blue' # Polar triangles color.
north = north_square
south = south_square
south_sq = [(-R * pi + R * south * pi / 2, -R * pi / 4),
(-R * pi + R * south * pi / 2, -R * 3 * pi / 4),
(-R * pi + R * (south + 1) * pi / 2, -R * 3 * pi / 4),
(-R * pi + R * (south + 1) * pi / 2, -R * pi / 4)]
north_sq = [(-R * pi + R * north * pi / 2, R * pi / 4),
(-R * pi + R * north * pi / 2, R * 3 * pi / 4),
(-R * pi + R * (north + 1) * pi / 2, R * 3 * pi / 4),
(-R * pi + R * (north + 1) * pi / 2, R * pi / 4)]
# Outline.
g += line2d(south_sq, linestyle='--', color=color)
g += line2d(north_sq, linestyle='--', color=color)
g += line2d([(R * pi, -R * pi / 4), (R * pi, R * pi / 4)],
linestyle='--',
color=color)
g += line2d([
north_sq[0], (-R * pi, R * pi / 4), (-R * pi, -R * pi / 4), south_sq[0]
],
color=color)
g += line2d([south_sq[3], (R * pi, -R * pi / 4)], color=color)
g += line2d([north_sq[3], (R * pi, R * pi / 4)], color=color)
g += point([south_sq[0], south_sq[3]], size=20, zorder=3, color=color)
g += point([north_sq[0], north_sq[3]], size=20, zorder=3, color=color)
g += point([(R * pi, -R * pi / 4), (R * pi, R * pi / 4)],
size=20,
zorder=3,
color=color)
g += point([(R * pi, -R * pi / 4), (R * pi, R * pi / 4)],
size=10,
color='white',
zorder=3)
if shade_polar_region:
# Shade.
g += polygon(south_sq, alpha=0.1, color=shade_color)
g += polygon(north_sq, alpha=0.1, color=shade_color)
# Slice square into polar triangles.
g += line2d([south_sq[0], south_sq[2]], color='lightgray')
g += line2d([south_sq[1], south_sq[3]], color='lightgray')
g += line2d([north_sq[0], north_sq[2]], color='lightgray')
g += line2d([north_sq[1], north_sq[3]], color='lightgray')
# Label polar triangles.
sp = south_sq[0] + R * array((pi / 4, -pi / 4))
np = north_sq[0] + R * array((pi / 4, pi / 4))
shift = R * 3 * pi / 16
g += text(str(south), sp + array((0, shift)), color='red', fontsize=20)
g += text(str((south + 1) % 4),
sp + array((shift, 0)),
color='red',
rotation=90,
fontsize=20)
g += text(str((south + 2) % 4),
sp + array((0, -shift)),
color='red',
rotation=180,
fontsize=20)
g += text(str((south + 3) % 4),
sp + array((-shift, 0)),
color='red',
rotation=270,
fontsize=20)
g += text(str(north), np + array((0, -shift)), color='red', fontsize=20)
g += text(str((north + 1) % 4),
np + array((shift, 0)),
color='red',
rotation=90,
fontsize=20)
g += text(str((north + 2) % 4),
np + array((0, shift)),
color='red',
rotation=180,
fontsize=20)
g += text(str((north + 3) % 4),
np + array((-shift, 0)),
color='red',
rotation=270,
fontsize=20)
return g
def draw_fundamental_domain(N, group="Gamma0", model="H", axes=None, filename=None, **kwds):
r""" Draw fundamental domain
INPUT:
- ''model'' -- (default ''H'')
= ''H'' -- Upper halfplane
= ''D'' -- Disk model
- ''filename''-- filename to print to
- ''**kwds''-- additional arguments to matplotlib
- ''axes'' -- set geometry of output
=[x0,x1,y0,y1] -- restrict figure to [x0,x1]x[y0,y1]
EXAMPLES::
sage: G=MySubgroup(Gamma0(3))
sage: G.draw_fundamental_domain()
"""
if group.strip() == "Gamma0":
G = Gamma0(N)
elif group.strip() == "Gamma1":
G = Gamma1(N)
elif group.strip() == "Gamma":
G = Gamma(N)
else:
raise ValueError('group must be one of: "Gamma0", "Gamma1", "Gamma"')
s = "$" + latex(G) + "$"
s = s.replace("mbox", "mathrm")
s = s.replace("Bold", "mathbb")
name = s
# name ="$\mbox{SL}_{2}(\mathbb{Z})$"
# need a "nice" set of coset representatives to draw a connected
# fundamental domain. Only implemented for Gamma_0(N)
coset_reps = nice_coset_reps(G)
# if(group=='Gamma0'):
# else:
# coset_reps = list(G.coset_reps())
from matplotlib.backends.backend_agg import FigureCanvasAgg
if model == "D":
g = _draw_funddom_d(coset_reps, format, I)
else:
g = _draw_funddom(coset_reps, format)
if axes is not None:
[x0, x1, y0, y1] = axes
elif model == "D":
x0 = -1
x1 = 1
y0 = -1.1
y1 = 1
else:
# find the width of the fundamental domain
# w = 0 # self.cusp_width(Cusp(Infinity)) FIXME: w is never used
wmin = 0
wmax = 1
max_x = RR(0.55)
rho = CC(exp(2 * pi * I / 3))
for V in coset_reps:
## we also compare the real parts of where rho and infinity are mapped
r1 = (V.acton(rho)).real()
if V[1, 0] != 0:
inf1 = RR(V[0, 0] / V[1, 0])
else:
inf1 = 0
if V[1, 0] == 0 and V[0, 0] == 1:
if V[0, 1] > wmax:
wmax = V[0, 1]
if V[0, 1] < wmin:
wmin = V[0, 1]
if max(r1, inf1) > max_x:
max_x = max(r1, inf1)
logging.debug("wmin,wmax=%s,%s" % (wmin, wmax))
# x0=-1; x1=1; y0=-0.2; y1=1.5
x0 = RR(-max_x)
x1 = RR(max_x)
y0 = RR(-0.15)
y1 = RR(1.5)
## Draw the axes manually (since can't figure out how to do it automatically)
ax = line([[x0, 0.0], [x1, 0.0]], color="black")
# ax = ax + line([[0.0,y0],[0.0,y1]],color='black')
## ticks
ax = ax + line([[-0.5, -0.01], [-0.5, 0.01]], color="black")
ax = ax + line([[0.5, -0.01], [0.5, 0.01]], color="black")
g = g + ax
if model == "H":
t = text(name, (0, -0.1), fontsize=16, color="black")
t = t + text("$ -\\frac{1}{2} $", (-0.5, -0.1), fontsize=12, color="black")
t = t + text("$ \\frac{1}{2} $", (0.5, -0.1), fontsize=12, color="black")
else:
t = text(name, (0, -1.1), fontsize=16, color="black")
g = g + t
g.set_aspect_ratio(1)
g.set_axes_range(x0, x1, y0, y1)
g.axes(False)
if filename is not None:
fig = g.matplotlib()
fig.set_canvas(FigureCanvasAgg(fig))
axes = fig.get_axes()[0]
axes.minorticks_off()
axes.set_yticks([])
fig.savefig(filename, **kwds)
else:
return g
def rhealpix_diagram(a=1, e=0, north_square=0, south_square=0,
shade_polar_region=True):
r"""
Return a Sage Graphics object diagramming the rHEALPix projection
boundary and polar triangles for the ellipsoid with major radius `a`
and eccentricity `e`.
Inessential graphics method.
Requires Sage graphics methods.
"""
from sage.all import Graphics, line2d, point, polygon, text, RealNumber, Integer
# Make Sage types compatible with Numpy.
RealNumber = float
Integer = int
R = auth_rad(a, e)
g = Graphics()
color = 'black' # Boundary color.
shade_color = 'blue' # Polar triangles color.
north = north_square
south = south_square
south_sq = [(-R*pi + R*south*pi/2, -R*pi/4),
(-R*pi + R*south*pi/2, -R*3*pi/4),
(-R*pi + R*(south + 1)*pi/2, -R*3*pi/4),
(-R*pi + R*(south + 1)*pi/2, -R*pi/4)]
north_sq = [(-R*pi + R*north*pi/2, R*pi/4),
(-R*pi + R*north*pi/2, R*3*pi/4),
(-R*pi + R*(north + 1)*pi/2, R*3*pi/4),
(-R*pi + R*(north + 1)*pi/2, R*pi/4)]
# Outline.
g += line2d(south_sq, linestyle = '--', color=color)
g += line2d(north_sq, linestyle = '--', color=color)
g += line2d([(R*pi, -R*pi/4), (R*pi, R*pi/4)], linestyle='--',
color=color)
g += line2d([north_sq[0], (-R*pi, R*pi/4), (-R*pi, -R*pi/4),
south_sq[0]], color=color)
g += line2d([south_sq[3], (R*pi, -R*pi/4)], color=color)
g += line2d([north_sq[3],(R*pi, R*pi/4)], color=color)
g += point([south_sq[0], south_sq[3]], size=20, zorder=3, color=color)
g += point([north_sq[0], north_sq[3]], size=20, zorder=3, color=color)
g += point([(R*pi, -R*pi/4), (R*pi, R*pi/4)], size=20, zorder=3,
color=color)
g += point([(R*pi, -R*pi/4), (R*pi, R*pi/4)], size=10, color='white',
zorder=3)
if shade_polar_region:
# Shade.
g += polygon(south_sq, alpha=0.1, color=shade_color)
g += polygon(north_sq, alpha=0.1, color=shade_color)
# Slice square into polar triangles.
g += line2d([south_sq[0], south_sq[2]], color='lightgray')
g += line2d([south_sq[1], south_sq[3]], color='lightgray')
g += line2d([north_sq[0], north_sq[2]], color='lightgray')
g += line2d([north_sq[1], north_sq[3]], color='lightgray')
# Label polar triangles.
sp = south_sq[0] + R*array((pi/4, -pi/4))
np = north_sq[0] + R*array((pi/4, pi/4))
shift = R*3*pi/16
g += text(str(south), sp + array((0, shift)), color='red',
fontsize=20)
g += text(str((south + 1) % 4), sp + array((shift, 0)),
color='red', rotation=90, fontsize=20)
g += text(str((south + 2) % 4), sp + array((0, -shift)),
color='red', rotation=180, fontsize=20)
g += text(str((south + 3) % 4), sp + array((-shift, 0)),
color='red', rotation=270, fontsize=20)
g += text(str(north), np + array((0, -shift)), color='red',
fontsize=20)
g += text(str((north + 1) % 4), np + array((shift, 0)),
color='red', rotation=90, fontsize=20)
g += text(str((north + 2) % 4), np + array((0, shift)),
color='red', rotation=180, fontsize=20)
g += text(str((north + 3) % 4), np + array((-shift, 0)),
color='red', rotation=270, fontsize=20)
return g