def graphical(ao):
r"""
Return a sage graphics representation of the given acyclic orientation.
"""
from sage.all import Graphics, point, line
assert is_orientation(ao)
n, ascents, descents = ao
# compute heights
height = [0]*n
for _ in range(n):
for (i, j) in ascents:
height[j] = max(height[j], height[i]+1)
for (i, j) in descents:
height[i] = max(height[i], height[j]+1)
# compute the extent of each interval
left = range(n)
right = range(n)
for (i, j) in ascents:
right[i] += .75
left[j] -= .75
for (i, j) in descents:
right[i] += .75
left[j] -= .75
result = Graphics()
# put in the ascents and descents
for (i, j) in ascents:
result += line([(i, height[i]), (j, height[j])],
color='red', thickness=2)
for (i, j) in descents:
result += line([(i, height[i]), (j, height[j])],
color='blue', thickness=2)
# then put in the intervals
for i in range(n):
result += line([(left[i], height[i]), (right[i], height[i])],
color='black', thickness=2)
result += point([(i, height[i])],
color='black', size=100)
# reset some annoying options
result.axes(False)
result.set_aspect_ratio(1)
return result
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 graph_receivers(cls, pop):
if not _nographics:
s = Graphics()
for i, (popi, popt) in enumerate(pop):
if popt == 0: color = (.5, 0, 0)
elif popt == 1: color = (0, 0, .5)
elif popt == 2: color = (0, .5, 0)
s += polygon([(i, 0), ((i + 1), 0), ((i + 1), popi),
(i, popi)],
rgbcolor=color)
return s
else:
return None
def graphical(path):
r"""
Return a sage graphics representation of `path`.
"""
from sage.all import Graphics, polygon
assert is_dyck_path(path)
result = Graphics()
# first a row of triangles
for i in range(len(path)):
result += polygon([(2*i, 0),
(2*i+1, 1),
(2*i+2, 0)],
color='black', fill=False, thickness=2)
# then all the boxes
for i, j in boxes_under_path(path):
result += polygon([(i+j, j-i),
(i+j+1, j-i+1),
(i+j+2, j-i),
(i+j+1, j-i-1)],
color='black', fill=False, thickness=2)
# reset some annoying options
result.axes(False)
result.set_aspect_ratio(1)
return result
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 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