def plot(self):
"""
Plot the lattice polygon.
OUTPUT:
A graphics object.
EXAMPLES::
sage: from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
sage: P = LatticePolytope_PPL((1,0), (0,1), (0,0), (2,2))
sage: P.plot()
Graphics object consisting of 6 graphics primitives
sage: LatticePolytope_PPL([0], [1]).plot()
Graphics object consisting of 3 graphics primitives
sage: LatticePolytope_PPL([0]).plot()
Graphics object consisting of 2 graphics primitives
"""
from sage.plot.point import point2d
from sage.plot.polygon import polygon2d
vertices = self.ordered_vertices()
points = self.integral_points()
if self.space_dimension() == 1:
vertices = [vector(ZZ, (v[0], 0)) for v in vertices]
points = [vector(ZZ, (p[0], 0)) for p in points]
point_plot = sum(point2d(p, pointsize=100, color='red')
for p in points)
polygon_plot = polygon2d(vertices, alpha=0.2, color='green',
zorder=-1, thickness=2)
return polygon_plot + point_plot
def plot(self):
"""
Plot the lattice polygon.
OUTPUT:
A graphics object.
EXAMPLES::
sage: from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
sage: P = LatticePolytope_PPL((1,0), (0,1), (0,0), (2,2))
sage: P.plot()
Graphics object consisting of 6 graphics primitives
sage: LatticePolytope_PPL([0], [1]).plot()
Graphics object consisting of 3 graphics primitives
sage: LatticePolytope_PPL([0]).plot()
Graphics object consisting of 2 graphics primitives
"""
from sage.plot.point import point2d
from sage.plot.polygon import polygon2d
vertices = self.ordered_vertices()
points = self.integral_points()
if self.space_dimension() == 1:
vertices = [vector(ZZ, (v[0], 0)) for v in vertices]
points = [vector(ZZ, (p[0], 0)) for p in points]
point_plot = sum(
point2d(p, pointsize=100, color='red') for p in points)
polygon_plot = polygon2d(vertices,
alpha=0.2,
color='green',
zorder=-1,
thickness=2)
return polygon_plot + point_plot
def plot(self, **options):
r"""
Plot the point (which might involve drawing several dots).
The options are passed to point2d.
If no "zorder" option is provided then we set "zorder" to 50.
"""
if "zorder" not in options:
options["zorder"]=50
return point2d(points=self.points(), **options)
def plot(self, translation=None):
r"""
Plot the polygon with the origin at ``translation``.
"""
from sage.plot.point import point2d
from sage.plot.line import line2d
from sage.plot.polygon import polygon2d
from sage.modules.free_module import VectorSpace
V = VectorSpace(RR,2)
P = self.vertices(translation)
return point2d(P, color='red') + line2d(P + (P[0],), color='orange') + polygon2d(P, alpha=0.3)
def plot(self, **options):
r"""
Plot the point (which might involve drawing several dots).
The options are passed to point2d.
If no "zorder" option is provided then we set "zorder" to 50.
"""
if "zorder" not in options:
options["zorder"]=50
return point2d(points=self.points(), **options)
def plot_points(self, points, **options):
r"""
Plot the points in the given collection of points.
The options are passed to point2d.
If no "zorder" option is provided then we set "zorder" to 50.
By default coordinates are taken in the underlying surface. Call with coordinates="graphical"
to use graphical coordinates instead.
"""
if "zorder" not in options:
options["zorder"]=50
if "coordinates" not in options:
points2 = [self.transform(point) for point in points]
elif options["coordinates"]=="graphical":
points2=[V(point) for point in points]
del options["coordinates"]
else:
raise ValueError("Invalid value of 'coordinates' option")
return point2d(points=points2, **options)
def plot_points(self, points, **options):
r"""
Plot the points in the given collection of points.
The options are passed to point2d.
If no "zorder" option is provided then we set "zorder" to 50.
By default coordinates are taken in the underlying surface. Call with coordinates="graphical"
to use graphical coordinates instead.
"""
if "zorder" not in options:
options["zorder"] = 50
if "coordinates" not in options:
points2 = [self.transform(point) for point in points]
elif options["coordinates"] == "graphical":
points2 = [V(point) for point in points]
del options["coordinates"]
else:
raise ValueError("Invalid value of 'coordinates' option")
return point2d(points=points2, **options)
def make_chart(self, region=None):
G = Graphics()
smin, smax, nmin, nmax = [], [], [], []
for d in self.chartgens(region):
x, y = self.get_coords(d)
G += point2d((x, y), zorder=10)
smin.append(y)
smax.append(y)
nmin.append(x)
nmax.append(x)
smin = [
min(smin),
]
smax = [
max(smax),
]
nmin = [
min(nmin),
]
nmax = [
max(nmax),
]
for l in self.chartlines():
try:
pts, lineargs = self.lineinfo[l["line"]]
x1, y1 = self.get_coords(self.geninfo(int(l["src"])))
x2, y2 = self.get_coords(self.geninfo(int(l["tar"])))
G += line2d([(x1, y1), (x2, y2)], **lineargs)
except:
print("line not understood:", l)
off = 0.5
G.ymin(smin[0] - off)
G.ymax(smax[0] + off)
G.xmin(nmin[0] - off)
G.xmax(nmax[0] + off)
return G
def plot(self, chart=None, ambient_coords=None, mapping=None, label=None, parameters=None, **kwds):
r"""
For real manifolds, plot ``self`` in a Cartesian graph based
on the coordinates of some ambient chart.
The point is drawn in terms of two (2D graphics) or three (3D graphics)
coordinates of a given chart, called hereafter the *ambient chart*.
The domain of the ambient chart must contain the point, or its image
by a continuous manifold map `\Phi`.
INPUT:
- ``chart`` -- (default: ``None``) the ambient chart (see above); if
``None``, the ambient chart is set the default chart of
``self.parent()``
- ``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``)
:class:`~sage.manifolds.continuous_map.ContinuousMap`; continuous
manifold map `\Phi` providing the link between the current point
`p` and the ambient chart ``chart``: the domain of ``chart`` must
contain `\Phi(p)`; if ``None``, the identity map is assumed
- ``label`` -- (default: ``None``) label printed next to the point;
if ``None``, the point's name is used
- ``parameters`` -- (default: ``None``) dictionary giving the numerical
values of the parameters that may appear in the point coordinates
- ``size`` -- (default: 10) size of the point once drawn as a small
disk or sphere
- ``color`` -- (default: ``'black'``) color of the point
- ``label_color`` -- (default: ``None``) color to print the label;
if ``None``, the value of ``color`` is used
- ``fontsize`` -- (default: 10) size of the font used to print the
label
- ``label_offset`` -- (default: 0.1) determines the separation between
the point and its label
OUTPUT:
- a graphic object, either an instance of
:class:`~sage.plot.graphics.Graphics` for a 2D plot (i.e. based on
2 coordinates of the ambient chart) or an instance of
:class:`~sage.plot.plot3d.base.Graphics3d` for a 3D plot (i.e.
based on 3 coordinates of the ambient chart)
EXAMPLES:
Drawing a point on a 2-dimensional manifold::
sage: M = Manifold(2, 'M', structure='topological')
sage: X.<x,y> = M.chart()
sage: p = M.point((1,3), name='p')
sage: g = p.plot(X)
sage: print(g)
Graphics object consisting of 2 graphics primitives
sage: gX = X.plot(max_range=4) # plot of the coordinate grid
sage: g + gX # display of the point atop the coordinate grid
Graphics object consisting of 20 graphics primitives
.. PLOT::
M = Manifold(2, 'M', structure='topological')
X = M.chart('x y'); x,y = X[:]
p = M.point((1,3), name='p')
g = p.plot(X)
gX = X.plot(max_range=4)
sphinx_plot(g+gX)
Actually, since ``X`` is the default chart of the open set in which
``p`` has been defined, it can be skipped in the arguments of
``plot``::
sage: g = p.plot()
sage: g + gX
Graphics object consisting of 20 graphics primitives
Call with some options::
sage: g = p.plot(chart=X, size=40, color='green', label='$P$',
....: label_color='blue', fontsize=20, label_offset=0.3)
sage: g + gX
Graphics object consisting of 20 graphics primitives
.. PLOT::
M = Manifold(2, 'M', structure='topological')
X = M.chart('x y'); x,y = X[:]
p = M.point((1,3), name='p')
g = p.plot(chart=X, size=40, color='green', label='$P$', \
label_color='blue', fontsize=20, label_offset=0.3)
gX = X.plot(max_range=4)
sphinx_plot(g+gX)
Use of the ``parameters`` option to set a numerical value of some
symbolic variable::
sage: a = var('a')
sage: q = M.point((a,2*a), name='q')
sage: gq = q.plot(parameters={a:-2}, label_offset=0.2)
sage: g + gX + gq
Graphics object consisting of 22 graphics primitives
.. PLOT::
M = Manifold(2, 'M', structure='topological')
X = M.chart('x y'); x,y = X[:]
p = M.point((1,3), name='p')
g = p.plot(chart=X, size=40, color='green', label='$P$', \
label_color='blue', fontsize=20, label_offset=0.3)
var('a')
q = M.point((a,2*a), name='q')
gq = q.plot(parameters={a:-2}, label_offset=0.2)
gX = X.plot(max_range=4)
sphinx_plot(g+gX+gq)
The numerical value is used only for the plot::
sage: q.coord()
(a, 2*a)
Drawing a point on a 3-dimensional manifold::
sage: M = Manifold(3, 'M', structure='topological')
sage: X.<x,y,z> = M.chart()
sage: p = M.point((2,1,3), name='p')
sage: g = p.plot()
sage: print(g)
Graphics3d Object
sage: gX = X.plot(nb_values=5) # coordinate mesh cube
sage: g + gX # display of the point atop the coordinate mesh
Graphics3d Object
Call with some options::
sage: g = p.plot(chart=X, size=40, color='green', label='P_1',
....: label_color='blue', fontsize=20, label_offset=0.3)
sage: g + gX
Graphics3d Object
An example of plot via a mapping: plot of a point on a 2-sphere viewed
in the 3-dimensional space ``M``::
sage: S2 = Manifold(2, 'S^2', structure='topological')
sage: U = S2.open_subset('U') # the open set covered by spherical coord.
sage: XS.<th,ph> = U.chart(r'th:(0,pi):\theta ph:(0,2*pi):\phi')
sage: p = U.point((pi/4, pi/8), name='p')
sage: F = S2.continuous_map(M, {(XS, X): [sin(th)*cos(ph),
....: sin(th)*sin(ph), cos(th)]}, name='F')
sage: F.display()
F: S^2 --> M
on U: (th, ph) |--> (x, y, z) = (cos(ph)*sin(th), sin(ph)*sin(th), cos(th))
sage: g = p.plot(chart=X, mapping=F)
sage: gS2 = XS.plot(chart=X, mapping=F, nb_values=9)
sage: g + gS2
Graphics3d Object
Use of the option ``ambient_coords`` for plots on a 4-dimensional
manifold::
sage: M = Manifold(4, 'M', structure='topological')
sage: X.<t,x,y,z> = M.chart()
sage: p = M.point((1,2,3,4), name='p')
sage: g = p.plot(X, ambient_coords=(t,x,y), label_offset=0.4) # the coordinate z is skipped
sage: gX = X.plot(X, ambient_coords=(t,x,y), nb_values=5) # long time
sage: g + gX # 3D plot # long time
Graphics3d Object
sage: g = p.plot(X, ambient_coords=(t,y,z), label_offset=0.4) # the coordinate x is skipped
sage: gX = X.plot(X, ambient_coords=(t,y,z), nb_values=5) # long time
sage: g + gX # 3D plot # long time
Graphics3d Object
sage: g = p.plot(X, ambient_coords=(y,z), label_offset=0.4) # the coordinates t and x are skipped
sage: gX = X.plot(X, ambient_coords=(y,z))
sage: g + gX # 2D plot
Graphics object consisting of 20 graphics primitives
.. PLOT::
M = Manifold(4, 'M', structure='topological')
X = M.chart('t x y z'); t,x,y,z = X[:]
p = M.point((1,2,3,4), name='p')
g = p.plot(X, ambient_coords=(y,z), label_offset=0.4)
gX = X.plot(X, ambient_coords=(y,z))
sphinx_plot(g+gX)
"""
from sage.plot.point import point2d
from sage.plot.text import text
from sage.plot.graphics import Graphics
from sage.plot.plot3d.shapes2 import point3d, text3d
from sage.manifolds.chart import Chart
if self._manifold.base_field_type() != "real":
raise NotImplementedError(
"plot of points on manifolds over fields different" " from the real field is not implemented"
)
# The ambient chart:
if chart is None:
chart = self.parent().default_chart()
elif not isinstance(chart, Chart):
raise TypeError("the argument 'chart' must be a coordinate chart")
# The effective point to be plotted:
if mapping is None:
eff_point = self
else:
eff_point = mapping(self)
# The coordinates of the ambient chart used for the plot:
if ambient_coords is None:
ambient_coords = chart[:]
elif not isinstance(ambient_coords, tuple):
ambient_coords = tuple(ambient_coords)
nca = len(ambient_coords)
if nca != 2 and nca != 3:
raise TypeError("invalid number of ambient coordinates: {}".format(nca))
# Extract the kwds options
size = kwds["size"]
color = kwds["color"]
label_color = kwds["label_color"]
fontsize = kwds["fontsize"]
label_offset = kwds["label_offset"]
# The point coordinates:
coords = eff_point.coord(chart)
xx = chart[:]
xp = [coords[xx.index(c)] for c in ambient_coords]
if parameters is not None:
xps = [coord.substitute(parameters) for coord in xp]
xp = xps
xlab = [coord + label_offset for coord in xp]
if label_color is None:
label_color = color
resu = Graphics()
if nca == 2:
if label is None:
label = r"$" + self._latex_name + r"$"
resu += point2d(xp, color=color, size=size) + text(label, xlab, fontsize=fontsize, color=label_color)
else:
if label is None:
label = self._name
resu += point3d(xp, color=color, size=size) + text3d(label, xlab, fontsize=fontsize, color=label_color)
return resu
def plot(self,
chart=None,
ambient_coords=None,
mapping=None,
size=10,
color='black',
label=None,
label_color=None,
fontsize=10,
label_offset=0.1,
parameters=None):
r"""
Plot the current point (``self``) in a Cartesian graph based on the
coordinates of some ambient chart.
The point is drawn in terms of two (2D graphics) or three (3D graphics)
coordinates of a given chart, called hereafter the *ambient chart*.
The domain of the ambient chart must contain the point, or its image
by a differentiable mapping `\Phi`.
INPUT:
- ``chart`` -- (default: ``None``) the ambient chart (see above); if
``None``, the ambient chart is set the default chart of
``self.containing_set()``
- ``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 the point `p` represented by ``self``
and the ambient chart ``chart``: the domain of ``chart`` must
contain `\Phi(p)`; if ``None``, the identity mapping is assumed
- ``size`` -- (default: 10) size of the point once drawn as a small
disk or sphere
- ``color`` -- (default: 'black') color of the point
- ``label`` -- (default: ``None``) label printed next to the point;
if ``None``, the point's name is used.
- ``label_color`` -- (default: ``None``) color to print the label;
if ``None``, the value of ``color`` is used
- ``fontsize`` -- (default: 10) size of the font used to print the
label
- ``label_offset`` -- (default: 0.1) determines the separation between
the point and its label
- ``parameters`` -- (default: ``None``) dictionary giving the numerical
values of the parameters that may appear in the point coordinates
OUTPUT:
- a graphic object, either an instance of
:class:`~sage.plot.graphics.Graphics` for a 2D plot (i.e. based on
2 coordinates of the ambient chart) or an instance of
:class:`~sage.plot.plot3d.base.Graphics3d` for a 3D plot (i.e.
based on 3 coordinates of the ambient chart)
EXAMPLES:
Drawing a point on a 2-dimensional manifold::
sage: M = Manifold(2, 'M')
sage: X.<x,y> = M.chart()
sage: p = M.point((1,3), name='p')
sage: g = p.plot(X)
sage: print g
Graphics object consisting of 2 graphics primitives
sage: gX = X.plot() # plot of the coordinate grid
sage: show(g+gX) # display of the point atop the coordinate grid
Actually, since ``X`` is the default chart of the open set in which
``p`` has been defined, it can be skipped in the arguments of
``plot``::
sage: g = p.plot()
sage: show(g+gX)
Call with some options::
sage: g = p.plot(chart=X, size=40, color='green', label='$P$',
....: label_color='blue', fontsize=20, label_offset=0.3)
sage: show(g+gX)
Use of the ``parameters`` option to set a numerical value of some
symbolic variable::
sage: a = var('a')
sage: q = M.point((a,2*a), name='q')
sage: gq = q.plot(parameters={a:-2})
sage: show(g+gX+gq)
The numerical value is used only for the plot::
sage: q.coord()
(a, 2*a)
Drawing a point on a 3-dimensional manifold::
sage: M = Manifold(3, 'M')
sage: X.<x,y,z> = M.chart()
sage: p = M.point((2,1,3), name='p')
sage: g = p.plot()
sage: print g
Graphics3d Object
sage: gX = X.plot(nb_values=5) # coordinate mesh cube
sage: show(g+gX) # display of the point atop the coordinate mesh
Call with some options::
sage: g = p.plot(chart=X, size=40, color='green', label='P_1',
....: label_color='blue', fontsize=20, label_offset=0.3)
sage: show(g+gX)
An example of plot via a differential mapping: plot of a point on a
2-sphere viewed in the 3-dimensional space ``M``::
sage: S2 = Manifold(2, 'S^2')
sage: U = S2.open_subset('U') # the open set covered by spherical coord.
sage: XS.<th,ph> = U.chart(r'th:(0,pi):\theta ph:(0,2*pi):\phi')
sage: p = U.point((pi/4, pi/8), name='p')
sage: F = S2.diff_mapping(M, {(XS, X): [sin(th)*cos(ph),
....: sin(th)*sin(ph), cos(th)]}, name='F')
sage: F.display()
F: S^2 --> M
on U: (th, ph) |--> (x, y, z) = (cos(ph)*sin(th), sin(ph)*sin(th), cos(th))
sage: g = p.plot(chart=X, mapping=F)
sage: gS2 = XS.plot(chart=X, mapping=F, nb_values=9)
sage: show(g+gS2)
Use of the option ``ambient_coords`` for plots on a 4-dimensional
manifold::
sage: M = Manifold(4, 'M')
sage: X.<t,x,y,z> = M.chart()
sage: p = M.point((1,2,3,4), name='p')
sage: g = p.plot(X, ambient_coords=(t,x,y)) # the coordinate z is skipped
sage: gX = X.plot(X, ambient_coords=(t,x,y), nb_values=5)
sage: show(g+gX) # 3D plot
sage: g = p.plot(X, ambient_coords=(t,y,z)) # the coordinate x is skipped
sage: gX = X.plot(X, ambient_coords=(t,y,z), nb_values=5)
sage: show(g+gX) # 3D plot
sage: g = p.plot(X, ambient_coords=(y,z)) # the coordinates t and x are skipped
sage: gX = X.plot(X, ambient_coords=(y,z))
sage: show(g+gX) # 2D plot
"""
from sage.plot.point import point2d
from sage.plot.text import text
from sage.plot.graphics import Graphics
from sage.plot.plot3d.shapes2 import point3d, text3d
from sage.geometry.manifolds.chart import Chart
# The ambient chart:
if chart is None:
chart = self.containing_set().default_chart()
elif not isinstance(chart, Chart):
raise TypeError("the argument 'chart' must be a coordinate chart")
# The effective point to be plotted:
if mapping is None:
eff_point = self
else:
eff_point = mapping(self)
# The coordinates of the ambient chart used for the plot:
if ambient_coords is None:
ambient_coords = chart._xx
elif not isinstance(ambient_coords, tuple):
ambient_coords = tuple(ambient_coords)
nca = len(ambient_coords)
if nca != 2 and nca != 3:
raise TypeError("Bad number of ambient coordinates: " + str(nca))
# The point coordinates:
coords = eff_point.coord(chart)
xx = chart[:]
xp = [coords[xx.index(c)] for c in ambient_coords]
if parameters is not None:
xps = [coord.substitute(parameters) for coord in xp]
xp = xps
xlab = [coord + label_offset for coord in xp]
if label_color is None:
label_color = color
resu = Graphics()
if nca == 2:
if label is None:
label = r'$' + self._latex_name + r'$'
resu += point2d(xp, color=color, size=size) + \
text(label, xlab, fontsize=fontsize, color=label_color)
else:
if label is None:
label = self._name
resu += point3d(xp, color=color, size=size) + \
text3d(label, xlab, fontsize=fontsize, color=label_color)
return resu
def plot(self,
chart=None,
ambient_coords=None,
mapping=None,
label=None,
parameters=None,
**kwds):
r"""
For real manifolds, plot ``self`` in a Cartesian graph based
on the coordinates of some ambient chart.
The point is drawn in terms of two (2D graphics) or three (3D graphics)
coordinates of a given chart, called hereafter the *ambient chart*.
The domain of the ambient chart must contain the point, or its image
by a continuous manifold map `\Phi`.
INPUT:
- ``chart`` -- (default: ``None``) the ambient chart (see above); if
``None``, the ambient chart is set the default chart of
``self.parent()``
- ``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``)
:class:`~sage.manifolds.continuous_map.ContinuousMap`; continuous
manifold map `\Phi` providing the link between the current point
`p` and the ambient chart ``chart``: the domain of ``chart`` must
contain `\Phi(p)`; if ``None``, the identity map is assumed
- ``label`` -- (default: ``None``) label printed next to the point;
if ``None``, the point's name is used
- ``parameters`` -- (default: ``None``) dictionary giving the numerical
values of the parameters that may appear in the point coordinates
- ``size`` -- (default: 10) size of the point once drawn as a small
disk or sphere
- ``color`` -- (default: ``'black'``) color of the point
- ``label_color`` -- (default: ``None``) color to print the label;
if ``None``, the value of ``color`` is used
- ``fontsize`` -- (default: 10) size of the font used to print the
label
- ``label_offset`` -- (default: 0.1) determines the separation between
the point and its label
OUTPUT:
- a graphic object, either an instance of
:class:`~sage.plot.graphics.Graphics` for a 2D plot (i.e. based on
2 coordinates of the ambient chart) or an instance of
:class:`~sage.plot.plot3d.base.Graphics3d` for a 3D plot (i.e.
based on 3 coordinates of the ambient chart)
EXAMPLES:
Drawing a point on a 2-dimensional manifold::
sage: M = Manifold(2, 'M', structure='topological')
sage: X.<x,y> = M.chart()
sage: p = M.point((1,3), name='p')
sage: g = p.plot(X)
sage: print(g)
Graphics object consisting of 2 graphics primitives
sage: gX = X.plot(max_range=4) # plot of the coordinate grid
sage: g + gX # display of the point atop the coordinate grid
Graphics object consisting of 20 graphics primitives
.. PLOT::
M = Manifold(2, 'M', structure='topological')
X = M.chart('x y'); x,y = X[:]
p = M.point((1,3), name='p')
g = p.plot(X)
gX = X.plot(max_range=4)
sphinx_plot(g+gX)
Actually, since ``X`` is the default chart of the open set in which
``p`` has been defined, it can be skipped in the arguments of
``plot``::
sage: g = p.plot()
sage: g + gX
Graphics object consisting of 20 graphics primitives
Call with some options::
sage: g = p.plot(chart=X, size=40, color='green', label='$P$',
....: label_color='blue', fontsize=20, label_offset=0.3)
sage: g + gX
Graphics object consisting of 20 graphics primitives
.. PLOT::
M = Manifold(2, 'M', structure='topological')
X = M.chart('x y'); x,y = X[:]
p = M.point((1,3), name='p')
g = p.plot(chart=X, size=40, color='green', label='$P$', \
label_color='blue', fontsize=20, label_offset=0.3)
gX = X.plot(max_range=4)
sphinx_plot(g+gX)
Use of the ``parameters`` option to set a numerical value of some
symbolic variable::
sage: a = var('a')
sage: q = M.point((a,2*a), name='q')
sage: gq = q.plot(parameters={a:-2}, label_offset=0.2)
sage: g + gX + gq
Graphics object consisting of 22 graphics primitives
.. PLOT::
M = Manifold(2, 'M', structure='topological')
X = M.chart('x y'); x,y = X[:]
p = M.point((1,3), name='p')
g = p.plot(chart=X, size=40, color='green', label='$P$', \
label_color='blue', fontsize=20, label_offset=0.3)
var('a')
q = M.point((a,2*a), name='q')
gq = q.plot(parameters={a:-2}, label_offset=0.2)
gX = X.plot(max_range=4)
sphinx_plot(g+gX+gq)
The numerical value is used only for the plot::
sage: q.coord()
(a, 2*a)
Drawing a point on a 3-dimensional manifold::
sage: M = Manifold(3, 'M', structure='topological')
sage: X.<x,y,z> = M.chart()
sage: p = M.point((2,1,3), name='p')
sage: g = p.plot()
sage: print(g)
Graphics3d Object
sage: gX = X.plot(number_values=5) # coordinate mesh cube
sage: g + gX # display of the point atop the coordinate mesh
Graphics3d Object
Call with some options::
sage: g = p.plot(chart=X, size=40, color='green', label='P_1',
....: label_color='blue', fontsize=20, label_offset=0.3)
sage: g + gX
Graphics3d Object
An example of plot via a mapping: plot of a point on a 2-sphere viewed
in the 3-dimensional space ``M``::
sage: S2 = Manifold(2, 'S^2', structure='topological')
sage: U = S2.open_subset('U') # the open set covered by spherical coord.
sage: XS.<th,ph> = U.chart(r'th:(0,pi):\theta ph:(0,2*pi):\phi')
sage: p = U.point((pi/4, pi/8), name='p')
sage: F = S2.continuous_map(M, {(XS, X): [sin(th)*cos(ph),
....: sin(th)*sin(ph), cos(th)]}, name='F')
sage: F.display()
F: S^2 --> M
on U: (th, ph) |--> (x, y, z) = (cos(ph)*sin(th), sin(ph)*sin(th), cos(th))
sage: g = p.plot(chart=X, mapping=F)
sage: gS2 = XS.plot(chart=X, mapping=F, number_values=9)
sage: g + gS2
Graphics3d Object
Use of the option ``ambient_coords`` for plots on a 4-dimensional
manifold::
sage: M = Manifold(4, 'M', structure='topological')
sage: X.<t,x,y,z> = M.chart()
sage: p = M.point((1,2,3,4), name='p')
sage: g = p.plot(X, ambient_coords=(t,x,y), label_offset=0.4) # the coordinate z is skipped
sage: gX = X.plot(X, ambient_coords=(t,x,y), number_values=5) # long time
sage: g + gX # 3D plot # long time
Graphics3d Object
sage: g = p.plot(X, ambient_coords=(t,y,z), label_offset=0.4) # the coordinate x is skipped
sage: gX = X.plot(X, ambient_coords=(t,y,z), number_values=5) # long time
sage: g + gX # 3D plot # long time
Graphics3d Object
sage: g = p.plot(X, ambient_coords=(y,z), label_offset=0.4) # the coordinates t and x are skipped
sage: gX = X.plot(X, ambient_coords=(y,z))
sage: g + gX # 2D plot
Graphics object consisting of 20 graphics primitives
.. PLOT::
M = Manifold(4, 'M', structure='topological')
X = M.chart('t x y z'); t,x,y,z = X[:]
p = M.point((1,2,3,4), name='p')
g = p.plot(X, ambient_coords=(y,z), label_offset=0.4)
gX = X.plot(X, ambient_coords=(y,z))
sphinx_plot(g+gX)
"""
from sage.plot.point import point2d
from sage.plot.text import text
from sage.plot.graphics import Graphics
from sage.plot.plot3d.shapes2 import point3d, text3d
from sage.manifolds.chart import Chart
if self._manifold.base_field_type() != 'real':
raise NotImplementedError(
'plot of points on manifolds over fields different'
' from the real field is not implemented')
# The ambient chart:
if chart is None:
chart = self.parent().default_chart()
elif not isinstance(chart, Chart):
raise TypeError("the argument 'chart' must be a coordinate chart")
# The effective point to be plotted:
if mapping is None:
eff_point = self
else:
eff_point = mapping(self)
# The coordinates of the ambient chart used for the plot:
if ambient_coords is None:
ambient_coords = chart[:]
elif not isinstance(ambient_coords, tuple):
ambient_coords = tuple(ambient_coords)
nca = len(ambient_coords)
if nca != 2 and nca != 3:
raise TypeError(
"invalid number of ambient coordinates: {}".format(nca))
# Extract the kwds options
size = kwds['size']
color = kwds['color']
label_color = kwds['label_color']
fontsize = kwds['fontsize']
label_offset = kwds['label_offset']
# The point coordinates:
coords = eff_point.coord(chart)
xx = chart[:]
xp = [coords[xx.index(c)] for c in ambient_coords]
if parameters is not None:
xps = [coord.substitute(parameters) for coord in xp]
xp = xps
xlab = [coord + label_offset for coord in xp]
if label_color is None:
label_color = color
resu = Graphics()
if nca == 2:
if label is None:
label = r'$' + self._latex_name + r'$'
resu += (point2d(xp, color=color, size=size) +
text(label, xlab, fontsize=fontsize, color=label_color))
else:
if label is None:
label = self._name
resu += (point3d(xp, color=color, size=size) +
text3d(label, xlab, fontsize=fontsize, color=label_color))
return resu
def plot(self, chart=None, ambient_coords=None, mapping=None, size=10,
color='black', label=None, label_color=None, fontsize=10,
label_offset=0.1, parameters=None):
r"""
Plot the current point (``self``) in a Cartesian graph based on the
coordinates of some ambient chart.
The point is drawn in terms of two (2D graphics) or three (3D graphics)
coordinates of a given chart, called hereafter the *ambient chart*.
The domain of the ambient chart must contain the point, or its image
by a differentiable mapping `\Phi`.
INPUT:
- ``chart`` -- (default: ``None``) the ambient chart (see above); if
``None``, the ambient chart is set the default chart of
``self.containing_set()``
- ``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 the point `p` represented by ``self``
and the ambient chart ``chart``: the domain of ``chart`` must
contain `\Phi(p)`; if ``None``, the identity mapping is assumed
- ``size`` -- (default: 10) size of the point once drawn as a small
disk or sphere
- ``color`` -- (default: 'black') color of the point
- ``label`` -- (default: ``None``) label printed next to the point;
if ``None``, the point's name is used.
- ``label_color`` -- (default: ``None``) color to print the label;
if ``None``, the value of ``color`` is used
- ``fontsize`` -- (default: 10) size of the font used to print the
label
- ``label_offset`` -- (default: 0.1) determines the separation between
the point and its label
- ``parameters`` -- (default: ``None``) dictionary giving the numerical
values of the parameters that may appear in the point coordinates
OUTPUT:
- a graphic object, either an instance of
:class:`~sage.plot.graphics.Graphics` for a 2D plot (i.e. based on
2 coordinates of the ambient chart) or an instance of
:class:`~sage.plot.plot3d.base.Graphics3d` for a 3D plot (i.e.
based on 3 coordinates of the ambient chart)
EXAMPLES:
Drawing a point on a 2-dimensional manifold::
sage: M = Manifold(2, 'M')
sage: X.<x,y> = M.chart()
sage: p = M.point((1,3), name='p')
sage: g = p.plot(X)
sage: print g
Graphics object consisting of 2 graphics primitives
sage: gX = X.plot() # plot of the coordinate grid
sage: show(g+gX) # display of the point atop the coordinate grid
Actually, since ``X`` is the default chart of the open set in which
``p`` has been defined, it can be skipped in the arguments of
``plot``::
sage: g = p.plot()
sage: show(g+gX)
Call with some options::
sage: g = p.plot(chart=X, size=40, color='green', label='$P$',
....: label_color='blue', fontsize=20, label_offset=0.3)
sage: show(g+gX)
Use of the ``parameters`` option to set a numerical value of some
symbolic variable::
sage: a = var('a')
sage: q = M.point((a,2*a), name='q')
sage: gq = q.plot(parameters={a:-2})
sage: show(g+gX+gq)
The numerical value is used only for the plot::
sage: q.coord()
(a, 2*a)
Drawing a point on a 3-dimensional manifold::
sage: M = Manifold(3, 'M')
sage: X.<x,y,z> = M.chart()
sage: p = M.point((2,1,3), name='p')
sage: g = p.plot()
sage: print g
Graphics3d Object
sage: gX = X.plot(nb_values=5) # coordinate mesh cube
sage: show(g+gX) # display of the point atop the coordinate mesh
Call with some options::
sage: g = p.plot(chart=X, size=40, color='green', label='P_1',
....: label_color='blue', fontsize=20, label_offset=0.3)
sage: show(g+gX)
An example of plot via a differential mapping: plot of a point on a
2-sphere viewed in the 3-dimensional space ``M``::
sage: S2 = Manifold(2, 'S^2')
sage: U = S2.open_subset('U') # the open set covered by spherical coord.
sage: XS.<th,ph> = U.chart(r'th:(0,pi):\theta ph:(0,2*pi):\phi')
sage: p = U.point((pi/4, pi/8), name='p')
sage: F = S2.diff_mapping(M, {(XS, X): [sin(th)*cos(ph),
....: sin(th)*sin(ph), cos(th)]}, name='F')
sage: F.display()
F: S^2 --> M
on U: (th, ph) |--> (x, y, z) = (cos(ph)*sin(th), sin(ph)*sin(th), cos(th))
sage: g = p.plot(chart=X, mapping=F)
sage: gS2 = XS.plot(chart=X, mapping=F, nb_values=9)
sage: show(g+gS2)
Use of the option ``ambient_coords`` for plots on a 4-dimensional
manifold::
sage: M = Manifold(4, 'M')
sage: X.<t,x,y,z> = M.chart()
sage: p = M.point((1,2,3,4), name='p')
sage: g = p.plot(X, ambient_coords=(t,x,y)) # the coordinate z is skipped
sage: gX = X.plot(X, ambient_coords=(t,x,y), nb_values=5)
sage: show(g+gX) # 3D plot
sage: g = p.plot(X, ambient_coords=(t,y,z)) # the coordinate x is skipped
sage: gX = X.plot(X, ambient_coords=(t,y,z), nb_values=5)
sage: show(g+gX) # 3D plot
sage: g = p.plot(X, ambient_coords=(y,z)) # the coordinates t and x are skipped
sage: gX = X.plot(X, ambient_coords=(y,z))
sage: show(g+gX) # 2D plot
"""
from sage.plot.point import point2d
from sage.plot.text import text
from sage.plot.graphics import Graphics
from sage.plot.plot3d.shapes2 import point3d, text3d
from sage.geometry.manifolds.chart import Chart
# The ambient chart:
if chart is None:
chart = self.containing_set().default_chart()
elif not isinstance(chart, Chart):
raise TypeError("the argument 'chart' must be a coordinate chart")
# The effective point to be plotted:
if mapping is None:
eff_point = self
else:
eff_point = mapping(self)
# The coordinates of the ambient chart used for the plot:
if ambient_coords is None:
ambient_coords = chart._xx
elif not isinstance(ambient_coords, tuple):
ambient_coords = tuple(ambient_coords)
nca = len(ambient_coords)
if nca != 2 and nca !=3:
raise TypeError("Bad number of ambient coordinates: " + str(nca))
# The point coordinates:
coords = eff_point.coord(chart)
xx = chart[:]
xp = [coords[xx.index(c)] for c in ambient_coords]
if parameters is not None:
xps = [coord.substitute(parameters) for coord in xp]
xp = xps
xlab = [coord + label_offset for coord in xp]
if label_color is None:
label_color = color
resu = Graphics()
if nca == 2:
if label is None:
label = r'$' + self._latex_name + r'$'
resu += point2d(xp, color=color, size=size) + \
text(label, xlab, fontsize=fontsize, color=label_color)
else:
if label is None:
label = self._name
resu += point3d(xp, color=color, size=size) + \
text3d(label, xlab, fontsize=fontsize, color=label_color)
return resu