def main4():
print("MAIN 4:")
zeta = CDF(exp(Integer(2) * pi * I / Integer(5)))
print(zeta)
v = [legendre_symbol(n, Integer(5)) * zeta**(Integer(2) * n)
for n in range(Integer(1), Integer(5))]
S = sum([point(tuple(z), pointsize=Integer(100)) for z in v])
G = point(tuple(sum(v)), pointsize=Integer(100), rgbcolor='red')
(S + G).save(filename="gauss_sum.png")
print()
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 eqn_list_to_curve_plot(L, rat_pts):
xpoly_rng = PolynomialRing(QQ, 'x')
poly_tup = [xpoly_rng(tup) for tup in L]
f = poly_tup[0]
h = poly_tup[1]
g = f + h**2 / 4
if len(g.real_roots()) == 0 and g(0) < 0:
return text(r"$X(\mathbb{R})=\emptyset$", (1, 1), fontsize=50)
X0 = [real(z[0]) for z in g.base_extend(CC).roots()
] + [real(z[0]) for z in g.derivative().base_extend(CC).roots()]
a, b = inflate_interval(min(X0), max(X0), 1.5)
groots = [a] + g.real_roots() + [b]
if b - a < 1e-7:
a = -3
b = 3
groots = [a, b]
ngints = len(groots) - 1
plotzones = []
npts = 100
for j in range(ngints):
c = groots[j]
d = groots[j + 1]
if g((c + d) / 2) < 0:
continue
(c, d) = inflate_interval(c, d, 1.1)
s = (d - c) / npts
u = c
yvals = []
for i in range(npts + 1):
v = g(u)
if v > 0:
v = sqrt(v)
w = -h(u) / 2
yvals.append(w + v)
yvals.append(w - v)
u += s
(m, M) = inflate_interval(min(yvals), max(yvals), 1.2)
plotzones.append((c, d, m, M))
x = var('x')
y = var('y')
plot = sum(
implicit_plot(y**2 + y * h(x) - f(x), (x, R[0], R[1]), (y, R[2], R[3]),
aspect_ratio='automatic',
plot_points=500,
zorder=1) for R in plotzones)
xmin = min([R[0] for R in plotzones])
xmax = max([R[1] for R in plotzones])
ymin = min([R[2] for R in plotzones])
ymax = max([R[3] for R in plotzones])
for P in rat_pts:
(x, y, z) = P
z = ZZ(z)
if z: # Do not attempt to plot points at infinity
x = ZZ(x) / z
y = ZZ(y) / z**3
if x >= xmin and x <= xmax and y >= ymin and y <= ymax:
plot += point((x, y), color='red', size=40, zorder=2)
return plot
def main6():
c = continued_fraction([Integer(1)] * (8))
v = [(i, c.p(i) / c.q(i)) for i in range(c.length())]
P = point(v, rgbcolor=(0, 0, 1), pointsize=40)
L = line(v, rgbcolor=(0.5, 0.5, 0.5))
L2 = line([(Integer(0), c.value()), (c.length() - Integer(1), c.value())],
thickness=0.5, rgbcolor=(0.7, 0, 0))
(L + L2 + P).save(filename="continued_fraction.png")
print("Graph is saved as continued_fraction.png")
def eqn_list_to_curve_plot(L,rat_pts):
xpoly_rng = PolynomialRing(QQ,'x')
poly_tup = [xpoly_rng(tup) for tup in L]
f = poly_tup[0]
h = poly_tup[1]
g = f+h**2/4
if len(g.real_roots())==0 and g(0)<0:
return text("$X(\mathbb{R})=\emptyset$",(1,1),fontsize=50)
X0 = [real(z[0]) for z in g.base_extend(CC).roots()]+[real(z[0]) for z in g.derivative().base_extend(CC).roots()]
a,b = inflate_interval(min(X0),max(X0),1.5)
groots = [a]+g.real_roots()+[b]
if b-a<1e-7:
a=-3
b=3
groots=[a,b]
ngints = len(groots)-1
plotzones = []
npts = 100
for j in range(ngints):
c = groots[j]
d = groots[j+1]
if g((c+d)/2)<0:
continue
(c,d) = inflate_interval(c,d,1.1)
s = (d-c)/npts
u = c
yvals = []
for i in range(npts+1):
v = g(u)
if v>0:
v = sqrt(v)
w = -h(u)/2
yvals.append(w+v)
yvals.append(w-v)
u += s
(m,M) = inflate_interval(min(yvals),max(yvals),1.2)
plotzones.append((c,d,m,M))
x = var('x')
y = var('y')
plot=sum(implicit_plot(y**2 + y*h(x) - f(x), (x,R[0],R[1]),(y,R[2],R[3]), aspect_ratio='automatic', plot_points=500, zorder=1) for R in plotzones)
xmin=min([R[0] for R in plotzones])
xmax=max([R[1] for R in plotzones])
ymin=min([R[2] for R in plotzones])
ymax=max([R[3] for R in plotzones])
for P in rat_pts:
(x,y,z)=eval(P.replace(':',','))
z=ZZ(z)
if z: # Do not attempt to plot points at infinity
x=ZZ(x)/z
y=ZZ(y)/z**3
if x >= xmin and x <= xmax and y >= ymin and y <= ymax:
plot += point((x,y),color='red',size=40,zorder=2)
return plot
def displayFlightTrajectoryInfo(self, flight, render=True, threed=False):
"""Display info and render trajectory."""
T0 = pd.to_datetime('2020-07-01T00')
dt = pd.to_timedelta(int(flight.cycleTime), 'S')
times = pd.date_range(start=T0,
end=T0 + dt,
freq='30S',
tz='America/Detroit')
print(' Flight Info')
print('=' * 80)
print('%d Second cycles, for %d cycles with %d left over' %
(flight.cycleTime, (24 * 3600) / flight.cycleTime,
(24 * 3600) % flight.cycleTime))
poses = flight.toPoses(times.to_series())
print('altitude min=%d mean=%d max=%d' %
(poses.z.min(), poses.z.mean(), poses.z.max()))
if render:
if threed:
show(flight._trajectory.render())
else:
points = [
point(
(
poses['x'][poses.index[i]],
poses['y'][poses.index[i]],
#poses['z'][poses.index[i]]
),
color=hue(i / len(poses.index)),
size=100) for i in range(len(poses.index))
]
points.extend([point((x, y)) for x, y, z in self._scene.users])
show(sum(points), aspect_ratio=1)
print()
print()
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 plot_polygon(Delta, remove_points=[]):
P = Delta.integral_points()
AUT = polygon_symmetries(Delta)
F = fundamental_domain(Delta, AUT)
fundam = fundamental_points(F, AUT)
plt = Delta.plot(fill="yellow", point=False, zorder=-10)
plt += F.plot(fill=(1,0.9,0), point=False, zorder=-9)
for pt in P:
style = {'pointsize': 80, 'color': (0.4,0.5,1)}
if pt in remove_points:
style['color'] = (0.7,0,0)
style['marker'] = "x"
elif Delta.interior_contains(pt):
if pt in fundam:
style['color'] = (0.2,0.8,0)
else:
style['color'] = (0,1,0)
plt += point(pt, **style)
return plt
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 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 render(self, **kwargs):
"""render our ground users."""
return sum([point(p, **kwargs) for p in self.users])