def plot_arc(radius, p, q, **opts):
# TODO: THIS SHOULD USE THE EXISTING PLOT OF ARCS!
# plot the arc from p to q differently depending on the type of self
p = ZZ(p)
q = ZZ(q)
t = var('t')
if p - q in [1, -1]:
def f(t):
return (radius * cos(t), radius * sin(t))
(p, q) = sorted([p, q])
angle_p = vertex_to_angle(p)
angle_q = vertex_to_angle(q)
return parametric_plot(f(t), (t, angle_q, angle_p), **opts)
if self.type() == 'A':
angle_p = vertex_to_angle(p)
angle_q = vertex_to_angle(q)
if angle_p < angle_q:
angle_p += 2 * pi
internal_angle = angle_p - angle_q
if internal_angle > pi:
(angle_p, angle_q) = (angle_q + 2 * pi, angle_p)
internal_angle = angle_p - angle_q
angle_center = (angle_p + angle_q) / 2
hypotenuse = radius / cos(internal_angle / 2)
radius_arc = hypotenuse * sin(internal_angle / 2)
center = (hypotenuse * cos(angle_center),
hypotenuse * sin(angle_center))
center_angle_p = angle_p + pi / 2
center_angle_q = angle_q + 3 * pi / 2
def f(t):
return (radius_arc * cos(t) + center[0],
radius_arc * sin(t) + center[1])
return parametric_plot(f(t),
(t, center_angle_p, center_angle_q),
**opts)
elif self.type() == 'D':
if p >= q:
q += self.r()
px = -2 * pi * p / self.r() + pi / 2
qx = -2 * pi * q / self.r() + pi / 2
arc_radius = (px - qx) / 2
arc_center = qx + arc_radius
def f(t):
return exp(I * ((cos(t) + I * sin(t)) * arc_radius +
arc_center)) * radius
return parametric_plot((real_part(f(t)), imag_part(f(t))),
(t, 0, pi), **opts)
def show(self, boundary=True, **options):
r"""
Plot ``self``.
EXAMPLES::
sage: HyperbolicPlane().PD().get_geodesic(0, 1).show()
Graphics object consisting of 2 graphics primitives
"""
opts = dict([('axes', False), ('aspect_ratio', 1)])
opts.update(self.graphics_options())
opts.update(options)
end_1, end_2 = [CC(k.coordinates()) for k in self.endpoints()]
bd_1, bd_2 = [CC(k.coordinates()) for k in self.ideal_endpoints()]
# Check to see if it's a line
if bool(real(bd_1)*imag(bd_2) - real(bd_2)*imag(bd_1))**2 < EPSILON:
from sage.plot.line import line
pic = line([(real(bd_1),imag(bd_1)),(real(bd_2),imag(bd_2))],
**opts)
else:
# If we are here, we know it's not a line
# So we compute the center and radius of the circle
center = (1/(real(bd_1)*imag(bd_2) - real(bd_2)*imag(bd_1)) *
((imag(bd_2)-imag(bd_1)) + (real(bd_1)-real(bd_2))*I))
radius = RR(abs(bd_1 - center)) # abs is Euclidean distance
# Now we calculate the angles for the parametric plot
theta1 = CC(end_1 - center).arg()
theta2 = CC(end_2 - center).arg()
if theta2 < theta1:
theta1, theta2 = theta2, theta1
from sage.calculus.var import var
from sage.plot.plot import parametric_plot
x = var('x')
mid = (theta1 + theta2)/2.0
if (radius*cos(mid) + real(center))**2 + \
(radius*sin(mid) + imag(center))**2 > 1.0:
# Swap theta1 and theta2
tmp = theta1 + 2*pi
theta1 = theta2
theta2 = tmp
pic = parametric_plot((radius*cos(x) + real(center),
radius*sin(x) + imag(center)),
(x, theta1, theta2), **opts)
else:
pic = parametric_plot((radius*cos(x) + real(center),
radius*sin(x) + imag(center)),
(x, theta1, theta2), **opts)
if boundary:
bd_pic = self._model.get_background_graphic()
pic = bd_pic + pic
return pic
def 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 _circ_arc(t0, t1, c, r, num_pts=500):
r""" Circular arc
INPUTS:
- ''t0'' -- starting parameter
- ''t1'' -- ending parameter
- ''c'' -- center point of the circle
- ''r'' -- radius of circle
- ''num_pts'' -- (default 100) number of points on polygon
OUTPUT:
- ''ca'' -- a polygonal approximation of a circular arc centered
at c and radius r, starting at t0 and ending at t1
EXAMPLES::
sage: ca=_circ_arc(0.1,0.2,0.0,1.0,100)
"""
from sage.plot.plot import parametric_plot
from sage.functions.trig import cos, sin
from sage.all import var
t00 = t0
t11 = t1
## To make sure the line is correct we reduce all arguments to the same branch,
## e.g. [0,2pi]
pi = RR.pi()
while t00 < 0.0:
t00 = t00 + RR(2.0 * pi)
while t11 < 0:
t11 = t11 + RR(2.0 * pi)
while t00 > 2 * pi:
t00 = t00 - RR(2.0 * pi)
while t11 > 2 * pi:
t11 = t11 - RR(2.0 * pi)
xc = CC(c).real()
yc = CC(c).imag()
# L0 =
# [[RR(r*cos(t00+i*(t11-t00)/num_pts))+xc,RR(r*sin(t00+i*(t11-t00)/num_pts))+yc]
# for i in range(0 ,num_pts)]
t = var("t")
if t11 > t00:
ca = parametric_plot((r * cos(t) + xc, r * sin(t) + yc), (t, t00, t11))
else:
ca = parametric_plot((r * cos(t) + xc, r * sin(t) + yc), (t, t11, t00))
return ca
def show(self, boundary=True, **options):
r"""
Plot ``self``.
EXAMPLES::
sage: HyperbolicPlane().UHP().get_geodesic(0, 1).show()
Graphics object consisting of 2 graphics primitives
"""
opts = {'axes': False, 'aspect_ratio': 1}
opts.update(self.graphics_options())
opts.update(options)
end_1, end_2 = [CC(k.coordinates()) for k in self.endpoints()]
bd_1, bd_2 = [CC(k.coordinates()) for k in self.ideal_endpoints()]
if (abs(real(end_1) - real(end_2)) < EPSILON) \
or CC(infinity) in [end_1, end_2]: #on same vertical line
# If one of the endpoints is infinity, we replace it with a
# large finite point
if end_1 == CC(infinity):
end_1 = (real(end_2), (imag(end_2) + 10))
end_2 = (real(end_2), imag(end_2))
elif end_2 == CC(infinity):
end_2 = (real(end_1), (imag(end_1) + 10))
end_1 = (real(end_1), imag(end_1))
from sage.plot.line import line
pic = line((end_1, end_2), **opts)
if boundary:
cent = min(bd_1, bd_2)
bd_dict = {'bd_min': cent - 3, 'bd_max': cent + 3}
bd_pic = self._model.get_background_graphic(**bd_dict)
pic = bd_pic + pic
return pic
else:
center = (bd_1 + bd_2)/2 # Circle center
radius = abs(bd_1 - bd_2)/2
theta1 = CC(end_1 - center).arg()
theta2 = CC(end_2 - center).arg()
if abs(theta1 - theta2) < EPSILON:
theta2 += pi
[theta1, theta2] = sorted([theta1, theta2])
from sage.calculus.var import var
from sage.plot.plot import parametric_plot
x = var('x')
pic = parametric_plot((radius*cos(x) + real(center),
radius*sin(x) + imag(center)),
(x, theta1, theta2), **opts)
if boundary:
# We want to draw a segment of the real line. The
# computations below compute the projection of the
# geodesic to the real line, and then draw a little
# to the left and right of the projection.
shadow_1, shadow_2 = [real(k) for k in [end_1, end_2]]
midpoint = (shadow_1 + shadow_2)/2
length = abs(shadow_1 - shadow_2)
bd_dict = {'bd_min': midpoint - length, 'bd_max': midpoint +
length}
bd_pic = self._model.get_background_graphic(**bd_dict)
pic = bd_pic + pic
return pic
def _circ_arc(t0, t1, c, r, num_pts=5000):
r""" Circular arc
INPUTS:
- ''t0'' -- starting parameter
- ''t1'' -- ending parameter
- ''c'' -- center point of the circle
- ''r'' -- radius of circle
- ''num_pts'' -- (default 100) number of points on polygon
OUTPUT:
- ''ca'' -- a polygonal approximation of a circular arc centered
at c and radius r, starting at t0 and ending at t1
EXAMPLES::
sage: ca=_circ_arc(0.1,0.2,0.0,1.0,100)
"""
from sage.plot.plot import line, parametric_plot
from sage.functions.trig import (cos, sin)
from sage.all import var
t00 = t0
t11 = t1
## To make sure the line is correct we reduce all arguments to the same branch,
## e.g. [0,2pi]
pi = RR.pi()
while (t00 < 0.0):
t00 = t00 + RR(2.0 * pi)
while (t11 < 0):
t11 = t11 + RR(2.0 * pi)
while (t00 > 2 * pi):
t00 = t00 - RR(2.0 * pi)
while (t11 > 2 * pi):
t11 = t11 - RR(2.0 * pi)
xc = CC(c).real()
yc = CC(c).imag()
num_pts = 3
t = var('t')
if t11 > t00:
ca = parametric_plot((r * cos(t) + xc, r * sin(t) + yc), (t, t00, t11))
else:
ca = parametric_plot((r * cos(t) + xc, r * sin(t) + yc), (t, t11, t00))
#L0 = [[RR(r*cos(t00+i*(t11-t00)/num_pts))+xc,RR(r*sin(t00+i*(t11-t00)/num_pts))+yc] for i in range(0 ,num_pts)]
#ca=line(L0)
return ca
def plot(self, coord_ranges, local_chart=None, ambient_chart=None, **kwds):
r"""
Plot of a submanifold embedded in `\RR^2` or `\RR^3`
INPUT:
- ``coord_ranges`` -- list of pairs (u_min, u_max) for each coordinate
u on the submanifold
- ``local_chart`` -- (default: None) chart on the submanifold in which
the above coordinates are defined; if none is provided, the
submanifold default chart is assumed.
- ``ambient_chart`` -- (default: None) chart on the ambient manifold
in terms of which the embedding is defined; if none is provided, the
ambient manifold default chart is assumed.
- ``**kwds`` -- (default: None) keywords passed to Sage graphic
routines
OUTPUT:
- Graphics3d object (ambient manifold = `\RR^3`) or Graphics object
(ambient manifold = `\RR^2`)
EXAMPLES:
Plot of a torus embedded in `\RR^3`::
sage: M = Manifold(3, 'R^3', r'\RR^3')
sage: c_cart.<x,y,z> = M.chart('x y z') # Cartesian coordinates on R^3
sage: T = Submanifold(M, 2, 'T')
sage: W = T.open_domain('W') # Domain of the torus covered by the cyclic coordinates (u,v)
sage: c_uv.<u,v> = W.chart(r'u:(0,2*pi) v:(0,2*pi)') # cyclic coordinates on T
sage: T.def_embedding(DiffMapping(T, M, [(2+cos(u))*cos(v),(2+cos(u))*sin(v),sin(u)]))
sage: T.plot([(0,2*pi), (0,2*pi)], aspect_ratio=1)
Plot of a helix embedded in `\RR^3`::
sage: H = Submanifold(M, 1, 'H')
sage: c_param.<t> = H.chart('t')
sage: H.def_embedding(DiffMapping(H, M, [cos(t), sin(t), t]))
sage: H.plot([0,20])
Plot of an Archimedean spiral embedded in `\RR^2`::
sage: M = Manifold(2, 'R^2', r'\RR^2')
sage: c_cart.<x,y> = M.chart('x y')
sage: S = Submanifold(M, 1, 'S')
sage: c_param.<t> = S.chart('t')
sage: S.def_embedding(DiffMapping(S, M, [t*cos(t), t*sin(t)]))
sage: S.plot([0,40])
"""
from sage.plot.plot import parametric_plot
if local_chart is None:
local_chart = self.def_chart
if ambient_chart is None:
ambient_chart = self.ambient_manifold.def_chart
if self.dim > 2:
raise ValueError("The dimension must be at most 2 " +
"for plotting.")
coord_functions = \
self.embedding.coord_expression[(local_chart, ambient_chart)].functions
coord_express = [
coord_functions[i].express
for i in range(self.ambient_manifold.dim)
]
if self.dim == 1:
urange = (local_chart.xx[0], coord_ranges[0], coord_ranges[1])
graph = parametric_plot(coord_express, urange, **kwds)
else: # self.dim = 2
urange = (local_chart.xx[0], coord_ranges[0][0],
coord_ranges[0][1])
vrange = (local_chart.xx[1], coord_ranges[1][0],
coord_ranges[1][1])
graph = parametric_plot(coord_express, urange, vrange, **kwds)
return graph
def plot(self, coord_ranges, chartname = None, **kwds):
r"""
Plot of a submanifold embedded in `\RR^2` or `\RR^3`
INPUT:
- ``coord_ranges`` -- list of pairs (u_min, u_max) for each coordinate
u on the submanifold
- ``chartname`` -- (default: None) name of the chart in which the
above coordinates are defined; if none is provided, the submanifold
default chart is assumed.
- ``**kwds`` -- (default: None) keywords passed to Sage graphic
routines
OUTPUT:
- Graphics3d object (ambient manifold = `\RR^3`) or Graphics object
(ambient manifold = `\RR^2`)
EXAMPLES:
Plot of a torus embedded in `\RR^3`::
sage: forget() # required for the doctest (to forget previous assumptions)
sage: m = Manifold(3, 'R3')
sage: c_cart = Chart(m, 'x y z', 'cart')
sage: (u,v) = var('u v')
sage: t = Submanifold(m, 2, 'u:[0,2*pi) v:[0,2*pi)', "canonical", [(2+cos(u))*cos(v),(2+cos(u))*sin(v),sin(u)], "torus")
sage: t.plot([[0,2*pi], [0,2*pi]], aspect_ratio=1)
Plot of a helix embedded in `\RR^3`::
sage: h = Submanifold(m, 1, 'u', 'u', [cos(u), sin(u), u], "helix")
sage: h.plot([0,20])
Plot of an Archimedean spiral embedded in `\RR^2`::
sage: R2 = Manifold(2, 'R2')
sage: c_cart = Chart(R2, 'x y', 'cart')
sage: t = var('t')
sage: s = Submanifold(R2, 1, 't', 't', [t*cos(t), t*sin(t)], "spiral")
sage: s.plot([0,40])
"""
from sage.plot.plot import parametric_plot
if chartname is None:
chartname = self.def_chart.name
amb = self.ambient_manifold.name
if amb == 'R3' or amb == 'R2':
if 'cart' not in self.ambient_manifold.atlas:
raise ValueError("For drawing, " + amb +
" Cartesian coordinates must be defined.")
coord_functions = \
self.embedding.coord_expression[(chartname, 'cart')].functions
chart = self.atlas[chartname]
if self.dim == 1:
urange = (chart.xx[0], coord_ranges[0], coord_ranges[1])
graph = parametric_plot(coord_functions, urange, **kwds)
elif self.dim == 2:
urange = (chart.xx[0], coord_ranges[0][0], coord_ranges[0][1])
vrange = (chart.xx[1], coord_ranges[1][0], coord_ranges[1][1])
graph = parametric_plot(coord_functions, urange, vrange, **kwds)
else:
raise ValueError("The dimension must be at most 2 " +
"for plotting.")
else:
raise NotImplementedError("Plotting is implemented only for " +
"submanifolds of R^2 or R^3.")
return graph
def plot(self, coord_ranges, local_chart=None, ambient_chart=None, **kwds):
r"""
Plot of a submanifold embedded in `\RR^2` or `\RR^3`
INPUT:
- ``coord_ranges`` -- list of pairs (u_min, u_max) for each coordinate
u on the submanifold
- ``local_chart`` -- (default: None) chart on the submanifold in which
the above coordinates are defined; if none is provided, the
submanifold default chart is assumed.
- ``ambient_chart`` -- (default: None) chart on the ambient manifold
in terms of which the embedding is defined; if none is provided, the
ambient manifold default chart is assumed.
- ``**kwds`` -- (default: None) keywords passed to Sage graphic
routines
OUTPUT:
- Graphics3d object (ambient manifold = `\RR^3`) or Graphics object
(ambient manifold = `\RR^2`)
EXAMPLES:
Plot of a torus embedded in `\RR^3`::
sage: M = Manifold(3, 'R^3', r'\RR^3')
sage: c_cart.<x,y,z> = M.chart('x y z') # Cartesian coordinates on R^3
sage: T = Submanifold(M, 2, 'T')
sage: W = T.open_domain('W') # Domain of the torus covered by the cyclic coordinates (u,v)
sage: c_uv.<u,v> = W.chart(r'u:(0,2*pi) v:(0,2*pi)') # cyclic coordinates on T
sage: T.def_embedding(DiffMapping(T, M, [(2+cos(u))*cos(v),(2+cos(u))*sin(v),sin(u)]))
sage: T.plot([(0,2*pi), (0,2*pi)], aspect_ratio=1)
Plot of a helix embedded in `\RR^3`::
sage: H = Submanifold(M, 1, 'H')
sage: c_param.<t> = H.chart('t')
sage: H.def_embedding(DiffMapping(H, M, [cos(t), sin(t), t]))
sage: H.plot([0,20])
Plot of an Archimedean spiral embedded in `\RR^2`::
sage: M = Manifold(2, 'R^2', r'\RR^2')
sage: c_cart.<x,y> = M.chart('x y')
sage: S = Submanifold(M, 1, 'S')
sage: c_param.<t> = S.chart('t')
sage: S.def_embedding(DiffMapping(S, M, [t*cos(t), t*sin(t)]))
sage: S.plot([0,40])
"""
from sage.plot.plot import parametric_plot
if local_chart is None:
local_chart = self.def_chart
if ambient_chart is None:
ambient_chart = self.ambient_manifold.def_chart
if self.dim > 2:
raise ValueError("The dimension must be at most 2 " +
"for plotting.")
coord_functions = \
self.embedding.coord_expression[(local_chart, ambient_chart)].functions
coord_express = [coord_functions[i].express for i in
range(self.ambient_manifold.dim)]
if self.dim == 1:
urange = (local_chart.xx[0], coord_ranges[0], coord_ranges[1])
graph = parametric_plot(coord_express, urange, **kwds)
else: # self.dim = 2
urange = (local_chart.xx[0], coord_ranges[0][0], coord_ranges[0][1])
vrange = (local_chart.xx[1], coord_ranges[1][0], coord_ranges[1][1])
graph = parametric_plot(coord_express, urange, vrange, **kwds)
return graph
def plot(self, coord_ranges, chartname=None, **kwds):
r"""
Plot of a submanifold embedded in `\RR^2` or `\RR^3`
INPUT:
- ``coord_ranges`` -- list of pairs (u_min, u_max) for each coordinate
u on the submanifold
- ``chartname`` -- (default: None) name of the chart in which the
above coordinates are defined; if none is provided, the submanifold
default chart is assumed.
- ``**kwds`` -- (default: None) keywords passed to Sage graphic
routines
OUTPUT:
- Graphics3d object (ambient manifold = `\RR^3`) or Graphics object
(ambient manifold = `\RR^2`)
EXAMPLES:
Plot of a torus embedded in `\RR^3`::
sage: forget() # required for the doctest (to forget previous assumptions)
sage: m = Manifold(3, 'R3')
sage: c_cart = Chart(m, 'x y z', 'cart')
sage: (u,v) = var('u v')
sage: t = Submanifold(m, 2, 'u:[0,2*pi) v:[0,2*pi)', "canonical", [(2+cos(u))*cos(v),(2+cos(u))*sin(v),sin(u)], "torus")
sage: t.plot([[0,2*pi], [0,2*pi]], aspect_ratio=1)
Plot of a helix embedded in `\RR^3`::
sage: h = Submanifold(m, 1, 'u', 'u', [cos(u), sin(u), u], "helix")
sage: h.plot([0,20])
Plot of an Archimedean spiral embedded in `\RR^2`::
sage: R2 = Manifold(2, 'R2')
sage: c_cart = Chart(R2, 'x y', 'cart')
sage: t = var('t')
sage: s = Submanifold(R2, 1, 't', 't', [t*cos(t), t*sin(t)], "spiral")
sage: s.plot([0,40])
"""
from sage.plot.plot import parametric_plot
if chartname is None:
chartname = self.def_chart.name
amb = self.ambient_manifold.name
if amb == 'R3' or amb == 'R2':
if 'cart' not in self.ambient_manifold.atlas:
raise ValueError("For drawing, " + amb +
" Cartesian coordinates must be defined.")
coord_functions = \
self.embedding.coord_expression[(chartname, 'cart')].functions
chart = self.atlas[chartname]
if self.dim == 1:
urange = (chart.xx[0], coord_ranges[0], coord_ranges[1])
graph = parametric_plot(coord_functions, urange, **kwds)
elif self.dim == 2:
urange = (chart.xx[0], coord_ranges[0][0], coord_ranges[0][1])
vrange = (chart.xx[1], coord_ranges[1][0], coord_ranges[1][1])
graph = parametric_plot(coord_functions, urange, vrange,
**kwds)
else:
raise ValueError("The dimension must be at most 2 " +
"for plotting.")
else:
raise NotImplementedError("Plotting is implemented only for " +
"submanifolds of R^2 or R^3.")
return graph
def plot_hyperplane(hyperplane, **kwds):
r"""
Return the plot of a single hyperplane.
INPUT:
- ``**kwds`` -- plot options: see below
OUTPUT:
A graphics object of the plot.
.. RUBRIC:: Plot Options
Beside the usual plot options (enter ``plot?``), the plot command for
hyperplanes includes the following:
- ``hyperplane_label`` -- Boolean value or string (default: ``True``).
If ``True``, the hyperplane is labeled with its equation, if a
string, it is labeled by that string, otherwise it is not
labeled.
- ``label_color`` -- (Default: ``'black'``) Color for hyperplane_label.
- ``label_fontsize`` -- Size for ``hyperplane_label`` font (default: 14)
(does not work in 3d, yet).
- ``label_offset`` -- (Default: 0-dim: 0.1, 1-dim: (0,1),
2-dim: (0,0,0.2)) Amount by which label is offset from
``hyperplane.point()``.
- ``point_size`` -- (Default: 50) Size of points in a zero-dimensional
arrangement or of an arrangement over a finite field.
- ``ranges`` -- Range for the parameters for the parametric plot of the
hyperplane. If a single positive number ``r`` is given for the
value of ``ranges``, then the ranges for all parameters are set to
`[-r, r]`. Otherwise, for a line in the plane, ``ranges`` has the
form ``[a, b]`` (default: [-3,3]), and for a plane in 3-space, the
``ranges`` has the form ``[[a, b], [c, d]]`` (default: [[-3,3],[-3,3]]).
(The ranges are centered around ``hyperplane.point()``.)
EXAMPLES::
sage: H1.<x> = HyperplaneArrangements(QQ)
sage: a = 3*x + 4
sage: a.plot() # indirect doctest
Graphics object consisting of 3 graphics primitives
sage: a.plot(point_size=100,hyperplane_label='hello')
Graphics object consisting of 3 graphics primitives
sage: H2.<x,y> = HyperplaneArrangements(QQ)
sage: b = 3*x + 4*y + 5
sage: b.plot()
Graphics object consisting of 2 graphics primitives
sage: b.plot(ranges=(1,5),label_offset=(2,-1))
Graphics object consisting of 2 graphics primitives
sage: opts = {'hyperplane_label':True, 'label_color':'green',
....: 'label_fontsize':24, 'label_offset':(0,1.5)}
sage: b.plot(**opts)
Graphics object consisting of 2 graphics primitives
sage: H3.<x,y,z> = HyperplaneArrangements(QQ)
sage: c = 2*x + 3*y + 4*z + 5
sage: c.plot()
Graphics3d Object
sage: c.plot(label_offset=(1,0,1), color='green', label_color='red', frame=False)
Graphics3d Object
sage: d = -3*x + 2*y + 2*z + 3
sage: d.plot(opacity=0.8)
Graphics3d Object
sage: e = 4*x + 2*z + 3
sage: e.plot(ranges=[[-1,1],[0,8]], label_offset=(2,2,1), aspect_ratio=1)
Graphics3d Object
"""
if hyperplane.base_ring().characteristic():
raise NotImplementedError('base field must have characteristic zero')
elif hyperplane.dimension() not in [
0, 1, 2
]: # dimension of hyperplane, not ambient space
raise ValueError('can only plot hyperplanes in dimensions 1, 2, 3')
# handle extra keywords
if 'hyperplane_label' in kwds:
hyp_label = kwds.pop('hyperplane_label')
if not hyp_label:
has_hyp_label = False
else:
has_hyp_label = True
else: # default
hyp_label = True
has_hyp_label = True
if has_hyp_label:
if hyp_label: # then label hyperplane with its equation
if hyperplane.dimension() == 2: # jmol does not like latex
label = hyperplane._repr_linear(include_zero=False)
else:
label = hyperplane._latex_()
else:
label = hyp_label # a string
if 'label_color' in kwds:
label_color = kwds.pop('label_color')
else:
label_color = 'black'
if 'label_fontsize' in kwds:
label_fontsize = kwds.pop('label_fontsize')
else:
label_fontsize = 14
if 'label_offset' in kwds:
has_offset = True
label_offset = kwds.pop('label_offset')
else:
has_offset = False # give default values below
if 'point_size' in kwds:
pt_size = kwds.pop('point_size')
else:
pt_size = 50
if 'ranges' in kwds:
ranges_set = True
ranges = kwds.pop('ranges')
else:
ranges_set = False # give default values below
# the extra keywords have now been handled
# now create the plot
if hyperplane.dimension() == 0: # a point on a line
x, = hyperplane.A()
d = hyperplane.b()
p = point((d / x, 0), size=pt_size, **kwds)
if has_hyp_label:
if not has_offset:
label_offset = 0.1
p += text(label, (d / x, label_offset),
color=label_color,
fontsize=label_fontsize)
p += text('', (d / x, label_offset + 0.4)) # add space at top
if 'ymax' not in kwds:
kwds['ymax'] = 0.5
elif hyperplane.dimension() == 1: # a line in the plane
pnt = hyperplane.point()
w = hyperplane.linear_part().matrix()
t = SR.var('t')
if ranges_set:
if isinstance(ranges, (list, tuple)):
t0, t1 = ranges
else: # ranges should be a single positive number
t0, t1 = -ranges, ranges
else: # default
t0, t1 = -3, 3
p = parametric_plot(pnt + t * w[0], (t, t0, t1), **kwds)
if has_hyp_label:
if has_offset:
b0, b1 = label_offset
else:
b0, b1 = 0, 0.2
label = text(label, (pnt[0] + b0, pnt[1] + b1),
color=label_color,
fontsize=label_fontsize)
p += label
elif hyperplane.dimension() == 2: # a plane in 3-space
pnt = hyperplane.point()
w = hyperplane.linear_part().matrix()
s, t = SR.var('s t')
if ranges_set:
if isinstance(ranges, (list, tuple)):
s0, s1 = ranges[0]
t0, t1 = ranges[1]
else: # ranges should be a single positive integers
s0, s1 = -ranges, ranges
t0, t1 = -ranges, ranges
else: # default
s0, s1 = -3, 3
t0, t1 = -3, 3
p = parametric_plot3d(pnt + s * w[0] + t * w[1], (s, s0, s1),
(t, t0, t1), **kwds)
if has_hyp_label:
if has_offset:
b0, b1, b2 = label_offset
else:
b0, b1, b2 = 0, 0, 0
label = text3d(label, (pnt[0] + b0, pnt[1] + b1, pnt[2] + b2),
color=label_color,
fontsize=label_fontsize)
p += label
return p
def plot_hyperplane(hyperplane, **kwds):
r"""
Return the plot of a single hyperplane.
INPUT:
- ``**kwds`` -- plot options: see below
OUTPUT:
A graphics object of the plot.
.. RUBRIC:: Plot Options
Beside the usual plot options (enter ``plot?``), the plot command for
hyperplanes includes the following:
- ``hyperplane_label`` -- Boolean value or string (default: ``True``).
If ``True``, the hyperplane is labeled with its equation, if a
string, it is labeled by that string, otherwise it is not
labeled.
- ``label_color`` -- (Default: ``'black'``) Color for hyperplane_label.
- ``label_fontsize`` -- Size for ``hyperplane_label`` font (default: 14)
(does not work in 3d, yet).
- ``label_offset`` -- (Default: 0-dim: 0.1, 1-dim: (0,1),
2-dim: (0,0,0.2)) Amount by which label is offset from
``hyperplane.point()``.
- ``point_size`` -- (Default: 50) Size of points in a zero-dimensional
arrangement or of an arrangement over a finite field.
- ``ranges`` -- Range for the parameters for the parametric plot of the
hyperplane. If a single positive number ``r`` is given for the
value of ``ranges``, then the ranges for all parameters are set to
`[-r, r]`. Otherwise, for a line in the plane, ``ranges`` has the
form ``[a, b]`` (default: [-3,3]), and for a plane in 3-space, the
``ranges`` has the form ``[[a, b], [c, d]]`` (default: [[-3,3],[-3,3]]).
(The ranges are centered around ``hyperplane.point()``.)
EXAMPLES::
sage: H1.<x> = HyperplaneArrangements(QQ)
sage: a = 3*x + 4
sage: a.plot() # indirect doctest
Graphics object consisting of 3 graphics primitives
sage: a.plot(point_size=100,hyperplane_label='hello')
Graphics object consisting of 3 graphics primitives
sage: H2.<x,y> = HyperplaneArrangements(QQ)
sage: b = 3*x + 4*y + 5
sage: b.plot()
Graphics object consisting of 2 graphics primitives
sage: b.plot(ranges=(1,5),label_offset=(2,-1))
Graphics object consisting of 2 graphics primitives
sage: opts = {'hyperplane_label':True, 'label_color':'green',
....: 'label_fontsize':24, 'label_offset':(0,1.5)}
sage: b.plot(**opts)
Graphics object consisting of 2 graphics primitives
sage: H3.<x,y,z> = HyperplaneArrangements(QQ)
sage: c = 2*x + 3*y + 4*z + 5
sage: c.plot()
Graphics3d Object
sage: c.plot(label_offset=(1,0,1), color='green', label_color='red', frame=False)
Graphics3d Object
sage: d = -3*x + 2*y + 2*z + 3
sage: d.plot(opacity=0.8)
Graphics3d Object
sage: e = 4*x + 2*z + 3
sage: e.plot(ranges=[[-1,1],[0,8]], label_offset=(2,2,1), aspect_ratio=1)
Graphics3d Object
"""
if hyperplane.base_ring().characteristic() != 0:
raise NotImplementedError('base field must have characteristic zero')
elif hyperplane.dimension() not in [0, 1, 2]: # dimension of hyperplane, not ambient space
raise ValueError('can only plot hyperplanes in dimensions 1, 2, 3')
# handle extra keywords
if 'hyperplane_label' in kwds:
hyp_label = kwds.pop('hyperplane_label')
if not hyp_label:
has_hyp_label = False
else:
has_hyp_label = True
else: # default
hyp_label = True
has_hyp_label = True
if has_hyp_label:
if hyp_label: # then label hyperplane with its equation
if hyperplane.dimension() == 2: # jmol does not like latex
label = hyperplane._repr_linear(include_zero=False)
else:
label = hyperplane._latex_()
else:
label = hyp_label # a string
if 'label_color' in kwds:
label_color = kwds.pop('label_color')
else:
label_color = 'black'
if 'label_fontsize' in kwds:
label_fontsize = kwds.pop('label_fontsize')
else:
label_fontsize = 14
if 'label_offset' in kwds:
has_offset = True
label_offset = kwds.pop('label_offset')
else:
has_offset = False # give default values below
if 'point_size' in kwds:
pt_size = kwds.pop('point_size')
else:
pt_size = 50
if 'ranges' in kwds:
ranges_set = True
ranges = kwds.pop('ranges')
else:
ranges_set = False # give default values below
# the extra keywords have now been handled
# now create the plot
if hyperplane.dimension() == 0: # a point on a line
x, = hyperplane.A()
d = hyperplane.b()
p = point((d/x,0), size = pt_size, **kwds)
if has_hyp_label:
if not has_offset:
label_offset = 0.1
p += text(label, (d/x,label_offset),
color=label_color,fontsize=label_fontsize)
p += text('',(d/x,label_offset+0.4)) # add space at top
if 'ymax' not in kwds:
kwds['ymax'] = 0.5
elif hyperplane.dimension() == 1: # a line in the plane
pnt = hyperplane.point()
w = hyperplane.linear_part().matrix()
x, y = hyperplane.A()
d = hyperplane.b()
t = SR.var('t')
if ranges_set:
if type(ranges) in [list,tuple]:
t0, t1 = ranges
else: # ranges should be a single positive number
t0, t1 = -ranges, ranges
else: # default
t0, t1 = -3, 3
p = parametric_plot(pnt+t*w[0], (t,t0,t1), **kwds)
if has_hyp_label:
if has_offset:
b0, b1 = label_offset
else:
b0, b1 = 0, 0.2
label = text(label,(pnt[0]+b0,pnt[1]+b1),
color=label_color,fontsize=label_fontsize)
p += label
elif hyperplane.dimension() == 2: # a plane in 3-space
pnt = hyperplane.point()
w = hyperplane.linear_part().matrix()
a, b, c = hyperplane.A()
d = hyperplane.b()
s,t = SR.var('s t')
if ranges_set:
if type(ranges) in [list,tuple]:
s0, s1 = ranges[0]
t0, t1 = ranges[1]
else: # ranges should be a single positive integers
s0, s1 = -ranges, ranges
t0, t1 = -ranges, ranges
else: # default
s0, s1 = -3, 3
t0, t1 = -3, 3
p = parametric_plot3d(pnt+s*w[0]+t*w[1],(s,s0,s1),(t,t0,t1),**kwds)
if has_hyp_label:
if has_offset:
b0, b1, b2 = label_offset
else:
b0, b1, b2 = 0, 0, 0
label = text3d(label,(pnt[0]+b0,pnt[1]+b1,pnt[2]+b2),
color=label_color,fontsize=label_fontsize)
p += label
return p
