def set_axes_labels(graph, xlabel, ylabel, zlabel, **kwds):
r"""
Set axes labels for a 3D graphics object ``graph``.
This is a workaround for the lack of axes labels in 3D plots.
This sets the labels as :func:`~sage.plot.plot3d.shapes2.text3d`
objects at locations determined from the bounding box of the
graphic object ``graph``.
INPUT:
- ``graph`` -- :class:`~sage.plot.plot3d.base.Graphics3d`;
a 3D graphic object
- ``xlabel`` -- string for the x-axis label
- ``ylabel`` -- string for the y-axis label
- ``zlabel`` -- string for the z-axis label
- ``**kwds`` -- options (e.g. color) for text3d
OUTPUT:
- the 3D graphic object with text3d labels added
EXAMPLES::
sage: g = sphere()
sage: g.all
[Graphics3d Object]
sage: from sage.manifolds.utilities import set_axes_labels
sage: ga = set_axes_labels(g, 'X', 'Y', 'Z', color='red')
sage: ga.all # the 3D frame has now axes labels
[Graphics3d Object, Graphics3d Object,
Graphics3d Object, Graphics3d Object]
"""
from sage.plot.plot3d.shapes2 import text3d
xmin, ymin, zmin = graph.bounding_box()[0]
xmax, ymax, zmax = graph.bounding_box()[1]
dx = xmax - xmin
dy = ymax - ymin
dz = zmax - zmin
x1 = xmin + dx / 2
y1 = ymin + dy / 2
z1 = zmin + dz / 2
xmin1 = xmin - dx / 20
xmax1 = xmax + dx / 20
ymin1 = ymin - dy / 20
zmin1 = zmin - dz / 20
graph += text3d(' ' + xlabel, (x1, ymin1, zmin1), **kwds)
graph += text3d(' ' + ylabel, (xmax1, y1, zmin1), **kwds)
graph += text3d(' ' + zlabel, (xmin1, ymin1, z1), **kwds)
return graph
def set_axes_labels(graph, xlabel, ylabel, zlabel, **kwds):
r"""
Set axes labels for a 3D graphics object ``graph``.
This is a workaround for the lack of axes labels in 3D plots.
This sets the labels as :func:`~sage.plot.plot3d.shapes2.text3d`
objects at locations determined from the bounding box of the
graphic object ``graph``.
INPUT:
- ``graph`` -- :class:`~sage.plot.plot3d.base.Graphics3d`;
a 3D graphic object
- ``xlabel`` -- string for the x-axis label
- ``ylabel`` -- string for the y-axis label
- ``zlabel`` -- string for the z-axis label
- ``**kwds`` -- options (e.g. color) for text3d
OUTPUT:
- the 3D graphic object with text3d labels added
EXAMPLES::
sage: g = sphere()
sage: g.all
[Graphics3d Object]
sage: from sage.manifolds.utilities import set_axes_labels
sage: ga = set_axes_labels(g, 'X', 'Y', 'Z', color='red')
sage: ga.all # the 3D frame has now axes labels
[Graphics3d Object, Graphics3d Object,
Graphics3d Object, Graphics3d Object]
"""
from sage.plot.plot3d.shapes2 import text3d
xmin, ymin, zmin = graph.bounding_box()[0]
xmax, ymax, zmax = graph.bounding_box()[1]
dx = xmax - xmin
dy = ymax - ymin
dz = zmax - zmin
x1 = xmin + dx / 2
y1 = ymin + dy / 2
z1 = zmin + dz / 2
xmin1 = xmin - dx / 20
xmax1 = xmax + dx / 20
ymin1 = ymin - dy / 20
zmin1 = zmin - dz / 20
graph += text3d(' ' + xlabel, (x1, ymin1, zmin1), **kwds)
graph += text3d(' ' + ylabel, (xmax1, y1, zmin1), **kwds)
graph += text3d(' ' + zlabel, (xmin1, ymin1, z1), **kwds)
return graph
def set_axes_labels(graph, xlabel, ylabel, zlabel, **kwds):
r"""
Set axes labels for a 3D graphics object.
This is a workaround for the lack of axes labels in Sage 3D plots; it
sets the labels as text3d objects at locations determined from the
bounding box of the graphic object ``graph``.
INPUT:
- ``graph`` -- a 3D graphic object, as an instance of
:class:`~sage.plot.plot3d.base.Graphics3d`
- ``xlabel`` -- string for the x-axis label
- ``ylabel`` -- string for the y-axis label
- ``zlabel`` -- string for the z-axis label
- ``**kwds`` -- options (e.g. color) for text3d
OUTPUT:
- the 3D graphic object with text3d labels added.
"""
from sage.plot.plot3d.shapes2 import text3d
xmin, ymin, zmin = graph.bounding_box()[0]
xmax, ymax, zmax = graph.bounding_box()[1]
dx = xmax - xmin
dy = ymax - ymin
dz = zmax - zmin
x1 = xmin + dx / 2
y1 = ymin + dy / 2
z1 = zmin + dz / 2
xmin1 = xmin - dx / 20
xmax1 = xmax + dx / 20
ymin1 = ymin - dy / 20
zmin1 = zmin - dz / 20
graph += text3d(' ' + xlabel, (x1, ymin1, zmin1), **kwds)
graph += text3d(' ' + ylabel, (xmax1, y1, zmin1), **kwds)
graph += text3d(' ' + zlabel, (xmin1, ymin1, z1), **kwds)
return graph
def set_axes_labels(graph, xlabel, ylabel, zlabel, **kwds):
r"""
Set axes labels for a 3D graphics object.
This is a workaround for the lack of axes labels in Sage 3D plots; it
sets the labels as text3d objects at locations determined from the
bounding box of the graphic object ``graph``.
INPUT:
- ``graph`` -- a 3D graphic object, as an instance of
:class:`~sage.plot.plot3d.base.Graphics3d`
- ``xlabel`` -- string for the x-axis label
- ``ylabel`` -- string for the y-axis label
- ``zlabel`` -- string for the z-axis label
- ``**kwds`` -- options (e.g. color) for text3d
OUTPUT:
- the 3D graphic object with text3d labels added.
"""
from sage.plot.plot3d.shapes2 import text3d
xmin, ymin, zmin = graph.bounding_box()[0]
xmax, ymax, zmax = graph.bounding_box()[1]
dx = xmax - xmin
dy = ymax - ymin
dz = zmax - zmin
x1 = xmin + dx / 2
y1 = ymin + dy / 2
z1 = zmin + dz / 2
xmin1 = xmin - dx / 20
xmax1 = xmax + dx / 20
ymin1 = ymin - dy / 20
zmin1 = zmin - dz / 20
graph += text3d(' ' + xlabel, (x1, ymin1, zmin1), **kwds)
graph += text3d(' ' + ylabel, (xmax1, y1, zmin1), **kwds)
graph += text3d(' ' + zlabel, (xmin1, ymin1, z1), **kwds)
return graph
def text(self, label, position):
r"""
Return text widget with label ``label`` at position ``position``
INPUT:
- ``label`` -- a string, or a Sage object upon which latex will
be called
- ``position`` -- a position
EXAMPLES::
sage: L = RootSystem(["A",2]).root_lattice()
sage: options = L.plot_parse_options()
sage: list(options.text("coucou", [0,1]))
[Text 'coucou' at the point (0.0,1.0)]
sage: list(options.text(L.simple_root(1), [0,1]))
[Text '$\alpha_{1}$' at the point (0.0,1.0)]
sage: options = RootSystem(["A",2]).root_lattice().plot_parse_options(labels=False)
sage: options.text("coucou", [0,1])
0
sage: options = RootSystem(["B",3]).root_lattice().plot_parse_options()
sage: print options.text("coucou", [0,1,2]).x3d_str()
<Transform translation='0 1 2'>
<Shape><Text string='coucou' solid='true'/><Appearance><Material diffuseColor='0.0 0.0 0.0' shininess='1' specularColor='0.0 0.0 0.0'/></Appearance></Shape>
<BLANKLINE>
</Transform>
"""
if self.labels:
if self.dimension <= 2:
if not isinstance(label, basestring):
label = "$" + str(latex(label)) + "$"
from sage.plot.text import text
return text(label, position, fontsize=15)
elif self.dimension == 3:
# LaTeX labels not yet supported in 3D
if isinstance(label, basestring):
label = label.replace("{", "").replace("}", "").replace("$", "").replace("_", "")
else:
label = str(label)
from sage.plot.plot3d.shapes2 import text3d
return text3d(label, position)
else:
raise NotImplementedError("Plots in dimension > 3")
else:
return self.empty()
def text(self, label, position):
r"""
Return text widget with label ``label`` at position ``position``
INPUT:
- ``label`` -- a string, or a Sage object upon which latex will
be called
- ``position`` -- a position
EXAMPLES::
sage: L = RootSystem(["A",2]).root_lattice()
sage: options = L.plot_parse_options()
sage: list(options.text("coucou", [0,1]))
[Text 'coucou' at the point (0.0,1.0)]
sage: list(options.text(L.simple_root(1), [0,1]))
[Text '$\alpha_{1}$' at the point (0.0,1.0)]
sage: options = RootSystem(["A",2]).root_lattice().plot_parse_options(labels=False)
sage: options.text("coucou", [0,1])
0
sage: options = RootSystem(["B",3]).root_lattice().plot_parse_options()
sage: print options.text("coucou", [0,1,2]).x3d_str()
<Transform translation='0 1 2'>
<Shape><Text string='coucou' solid='true'/><Appearance><Material diffuseColor='0.0 0.0 0.0' shininess='1' specularColor='0.0 0.0 0.0'/></Appearance></Shape>
<BLANKLINE>
</Transform>
"""
if self.labels:
if self.dimension <= 2:
if not isinstance(label, basestring):
label = "$" + str(latex(label)) + "$"
from sage.plot.text import text
return text(label, position, fontsize=15)
elif self.dimension == 3:
# LaTeX labels not yet supported in 3D
if isinstance(label, basestring):
label = label.replace("{", "").replace("}", "").replace(
"$", "").replace("_", "")
else:
label = str(label)
from sage.plot.plot3d.shapes2 import text3d
return text3d(label, position)
else:
raise NotImplementedError("Plots in dimension > 3")
else:
return self.empty()
def plot3d(self, **kwds):
"""
Plots 2D text in 3D.
EXAMPLES::
sage: T = text("ABC",(1,1))
sage: t = T[0]
sage: s=t.plot3d()
sage: s.jmol_repr(s.testing_render_params())[0][2]
'label "ABC"'
sage: s._trans
(1.0, 1.0, 0)
"""
from sage.plot.plot3d.shapes2 import text3d
options = self._plot3d_options()
options.update(kwds)
return text3d(self.string, (self.x, self.y, 0), **options)
def plot3d(self, **kwds):
"""
Plots 2D text in 3D.
EXAMPLES::
sage: T = text("ABC",(1,1))
sage: t = T[0]
sage: s=t.plot3d()
sage: s.jmol_repr(s.testing_render_params())[0][2]
'label "ABC"'
sage: s._trans
(1.0, 1.0, 0)
"""
from sage.plot.plot3d.shapes2 import text3d
options = self._plot3d_options()
options.update(kwds)
return text3d(self.string, (self.x, self.y, 0), **options)
def show_pentaminos(box=(5, 8, 2)):
r"""
Show the 17 3-D pentaminos included in the game and the `5 \times 8
\times 2` box where 16 of them must fit.
INPUT:
- ``box`` - tuple of size three (optional, default: ``(5,8,2)``),
size of the box
OUTPUT:
3D Graphic object
EXAMPLES::
sage: from sage.games.quantumino import show_pentaminos
sage: show_pentaminos() # not tested (1s)
To remove the frame do::
sage: show_pentaminos().show(frame=False) # not tested (1s)
"""
G = Graphics()
for i, p in enumerate(pentaminos):
x = 4 * (i % 4)
y = 4 * (i / 4)
q = p + (x, y, 0)
G += q.show3d()
G += text3d(str(i), (x, y, 2))
G += cube(color='gray', opacity=0.5).scale(box).translate((17, 6, 0))
# hack to set the aspect ratio to 1
a, b = G.bounding_box()
a, b = map(vector, (a, b))
G.frame_aspect_ratio(tuple(b - a))
return G
def show_pentaminos(box=(5,8,2)):
r"""
Show the 17 3-D pentaminos included in the game and the `5 \times 8
\times 2` box where 16 of them must fit.
INPUT:
- ``box`` -- tuple of size three (optional, default: ``(5,8,2)``),
size of the box
OUTPUT:
3D Graphic object
EXAMPLES::
sage: from sage.games.quantumino import show_pentaminos
sage: show_pentaminos() # not tested (1s)
To remove the frame do::
sage: show_pentaminos().show(frame=False) # not tested (1s)
"""
G = Graphics()
for i, p in enumerate(pentaminos):
x = 4 * (i % 4)
y = 4 * (i // 4)
q = p + (x, y, 0)
G += q.show3d()
G += text3d(str(i), (x, y, 2))
G += cube(color='gray',opacity=0.5).scale(box).translate((17, 6, 0))
# hack to set the aspect ratio to 1
a, b = G.bounding_box()
a, b = map(vector, (a, b))
G.frame_aspect_ratio(tuple(b - a))
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,
color='blue',
print_label=True,
label=None,
label_color=None,
fontsize=10,
label_offset=0.1,
parameters=None,
**extra_options):
r"""
Plot the vector in a Cartesian graph based on the coordinates of some
ambient chart.
The vector is drawn in terms of two (2D graphics) or three (3D graphics)
coordinates of a given chart, called hereafter the *ambient chart*.
The vector's base point `p` (or its image `\Phi(p)` by some
differentiable mapping `\Phi`) must lie in the ambient chart's domain.
If `\Phi` is different from the identity mapping, the vector
actually depicted is `\mathrm{d}\Phi_p(v)`, where `v` is the current
vector (``self``) (see the example of a vector tangent to the
2-sphere below, where `\Phi: S^2 \to \RR^3`).
INPUT:
- ``chart`` -- (default: ``None``) the ambient chart (see above); if
``None``, it is set to the default chart of the open set containing
the point at which the vector (or the vector image via the
differential `\mathrm{d}\Phi_p` of ``mapping``) is defined
- ``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.differentiable.diff_map.DiffMap`;
differentiable mapping `\Phi` providing the link between the
point `p` at which the vector is defined and the ambient chart
``chart``: the domain of ``chart`` must contain `\Phi(p)`;
if ``None``, the identity mapping is assumed
- ``scale`` -- (default: 1) value by which the length of the arrow
representing the vector is multiplied
- ``color`` -- (default: 'blue') color of the arrow representing the
vector
- ``print_label`` -- (boolean; default: ``True``) determines whether a
label is printed next to the arrow representing the vector
- ``label`` -- (string; default: ``None``) label printed next to the
arrow representing the vector; if ``None``, the vector's symbol is
used, if any
- ``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 vector arrow and the label
- ``parameters`` -- (default: ``None``) dictionary giving the numerical
values of the parameters that may appear in the coordinate expression
of ``self`` (see example below)
- ``**extra_options`` -- extra options for the arrow plot, like
``linestyle``, ``width`` or ``arrowsize`` (see
:func:`~sage.plot.arrow.arrow2d` and
:func:`~sage.plot.plot3d.shapes.arrow3d` for details)
OUTPUT:
- a graphic object, either an instance of
:class:`~sage.plot.graphics.Graphics` for a 2D plot (i.e. based on
2 coordinates of ``chart``) or an instance of
:class:`~sage.plot.plot3d.base.Graphics3d` for a 3D plot (i.e.
based on 3 coordinates of ``chart``)
EXAMPLES:
Vector tangent to a 2-dimensional manifold::
sage: M = Manifold(2, 'M')
sage: X.<x,y> = M.chart()
sage: p = M((2,2), name='p')
sage: Tp = M.tangent_space(p)
sage: v = Tp((2, 1), name='v') ; v
Tangent vector v at Point p on the 2-dimensional differentiable
manifold M
Plot of the vector alone (arrow + label)::
sage: v.plot()
Graphics object consisting of 2 graphics primitives
Plot atop of the chart grid::
sage: X.plot() + v.plot()
Graphics object consisting of 20 graphics primitives
.. PLOT::
M = Manifold(2, 'M')
X = M.chart('x y'); x, y = X[:]
p = M((2,2), name='p'); Tp = M.tangent_space(p)
v = Tp((2, 1), name='v')
g = X.plot() + v.plot()
sphinx_plot(g)
Plots with various options::
sage: X.plot() + v.plot(color='green', scale=2, label='V')
Graphics object consisting of 20 graphics primitives
.. PLOT::
M = Manifold(2, 'M')
X = M.chart('x y'); x, y = X[:]
p = M((2,2), name='p'); Tp = M.tangent_space(p)
v = Tp((2, 1), name='v')
g = X.plot() + v.plot(color='green', scale=2, label='V')
sphinx_plot(g)
::
sage: X.plot() + v.plot(print_label=False)
Graphics object consisting of 19 graphics primitives
.. PLOT::
M = Manifold(2, 'M')
X = M.chart('x y'); x, y = X[:]
p = M((2,2), name='p'); Tp = M.tangent_space(p)
v = Tp((2, 1), name='v')
g = X.plot() + v.plot(print_label=False)
sphinx_plot(g)
::
sage: X.plot() + v.plot(color='green', label_color='black',
....: fontsize=20, label_offset=0.2)
Graphics object consisting of 20 graphics primitives
.. PLOT::
M = Manifold(2, 'M')
X = M.chart('x y'); x, y = X[:]
p = M((2,2), name='p'); Tp = M.tangent_space(p)
v = Tp((2, 1), name='v')
g = X.plot() + v.plot(color='green', label_color='black', fontsize=20, label_offset=0.2)
sphinx_plot(g)
::
sage: X.plot() + v.plot(linestyle=':', width=4, arrowsize=8,
....: fontsize=20)
Graphics object consisting of 20 graphics primitives
.. PLOT::
M = Manifold(2, 'M')
X = M.chart('x y'); x, y = X[:]
p = M((2,2), name='p'); Tp = M.tangent_space(p)
v = Tp((2, 1), name='v')
g = X.plot() + v.plot(linestyle=':', width=4, arrowsize=8, fontsize=20)
sphinx_plot(g)
Plot with specific values of some free parameters::
sage: var('a b')
(a, b)
sage: v = Tp((1+a, -b^2), name='v') ; v.display()
v = (a + 1) d/dx - b^2 d/dy
sage: X.plot() + v.plot(parameters={a: -2, b: 3})
Graphics object consisting of 20 graphics primitives
Special case of the zero vector::
sage: v = Tp.zero() ; v
Tangent vector zero at Point p on the 2-dimensional differentiable
manifold M
sage: X.plot() + v.plot()
Graphics object consisting of 19 graphics primitives
Vector tangent to a 4-dimensional manifold::
sage: M = Manifold(4, 'M')
sage: X.<t,x,y,z> = M.chart()
sage: p = M((0,1,2,3), name='p')
sage: Tp = M.tangent_space(p)
sage: v = Tp((5,4,3,2), name='v') ; v
Tangent vector v at Point p on the 4-dimensional differentiable
manifold M
We cannot make a 4D plot directly::
sage: v.plot()
Traceback (most recent call last):
...
ValueError: the number of coordinates involved in the plot must
be either 2 or 3, not 4
Rather, we have to select some chart coordinates for the plot, via
the argument ``ambient_coords``. For instance, for a 2-dimensional
plot in terms of the coordinates `(x, y)`::
sage: v.plot(ambient_coords=(x,y))
Graphics object consisting of 2 graphics primitives
.. PLOT::
M = Manifold(4, 'M')
X = M.chart('t x y z'); t,x,y,z = X[:]
p = M((0,1,2,3), name='p'); Tp = M.tangent_space(p)
v = Tp((5,4,3,2), name='v')
g = X.plot(ambient_coords=(x,y)) + v.plot(ambient_coords=(x,y))
sphinx_plot(g)
This plot involves only the components `v^x` and `v^y` of `v`.
Similarly, for a 3-dimensional plot in terms of the coordinates
`(t, x, y)`::
sage: g = v.plot(ambient_coords=(t,x,z))
sage: print(g)
Graphics3d Object
This plot involves only the components `v^t`, `v^x` and `v^z` of `v`.
A nice 3D view atop the coordinate grid is obtained via::
sage: (X.plot(ambient_coords=(t,x,z)) # long time
....: + v.plot(ambient_coords=(t,x,z),
....: label_offset=0.5, width=6))
Graphics3d Object
.. PLOT::
M = Manifold(4, 'M')
X = M.chart('t x y z'); t,x,y,z = X[:]
p = M((0,1,2,3), name='p'); Tp = M.tangent_space(p)
v = Tp((5,4,3,2), name='v')
g = X.plot(ambient_coords=(t,x,z)) + v.plot(ambient_coords=(t,x,z),
label_offset=0.5, width=6)
sphinx_plot(g)
An example of plot via a differential mapping: plot of a vector tangent
to a 2-sphere viewed in `\RR^3`::
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: R3 = Manifold(3, 'R^3')
sage: X3.<x,y,z> = R3.chart()
sage: F = S2.diff_map(R3, {(XS, X3): [sin(th)*cos(ph),
....: sin(th)*sin(ph),
....: cos(th)]}, name='F')
sage: F.display() # the standard embedding of S^2 into R^3
F: S^2 --> R^3
on U: (th, ph) |--> (x, y, z) = (cos(ph)*sin(th), sin(ph)*sin(th), cos(th))
sage: p = U.point((pi/4, 7*pi/4), name='p')
sage: v = XS.frame()[1].at(p) ; v # the coordinate vector d/dphi at p
Tangent vector d/dph at Point p on the 2-dimensional differentiable
manifold S^2
sage: graph_v = v.plot(mapping=F)
sage: graph_S2 = XS.plot(chart=X3, mapping=F, number_values=9) # long time
sage: graph_v + graph_S2 # long time
Graphics3d Object
.. PLOT::
S2 = Manifold(2, 'S^2')
U = S2.open_subset('U')
XS = U.chart(r'th:(0,pi):\theta ph:(0,2*pi):\phi')
th, ph = XS[:]
R3 = Manifold(3, 'R^3')
X3 = R3.chart('x y z')
F = S2.diff_map(R3, {(XS, X3): [sin(th)*cos(ph), sin(th)*sin(ph),
cos(th)]}, name='F')
p = U.point((pi/4, 7*pi/4), name='p')
v = XS.frame()[1].at(p)
graph_v = v.plot(mapping=F)
graph_S2 = XS.plot(chart=X3, mapping=F, number_values=9)
sphinx_plot(graph_v + graph_S2)
"""
from sage.plot.arrow import arrow2d
from sage.plot.text import text
from sage.plot.graphics import Graphics
from sage.plot.plot3d.shapes import arrow3d
from sage.plot.plot3d.shapes2 import text3d
from sage.misc.functional import numerical_approx
from sage.manifolds.differentiable.chart import DiffChart
scale = extra_options.pop("scale")
#
# The "effective" vector to be plotted
#
if mapping is None:
eff_vector = self
base_point = self._point
else:
#!# check
# For efficiency, the method FiniteRankFreeModuleMorphism._call_()
# is called instead of FiniteRankFreeModuleMorphism.__call__()
eff_vector = mapping.differential(self._point)._call_(self)
base_point = mapping(self._point)
#
# The chart w.r.t. which the vector is plotted
#
if chart is None:
chart = base_point.parent().default_chart()
elif not isinstance(chart, DiffChart):
raise TypeError("{} is not a chart".format(chart))
#
# Coordinates of the above chart w.r.t. which the vector is plotted
#
if ambient_coords is None:
ambient_coords = chart[:] # all chart coordinates are used
n_pc = len(ambient_coords)
if n_pc != 2 and n_pc != 3:
raise ValueError("the number of coordinates involved in the " +
"plot must be either 2 or 3, not {}".format(n_pc))
# indices coordinates involved in the plot:
ind_pc = [chart[:].index(pc) for pc in ambient_coords]
#
# Components of the vector w.r.t. the chart frame
#
basis = chart.frame().at(base_point)
vcomp = eff_vector.comp(basis=basis)[:]
xp = base_point.coord(chart=chart)
#
# The arrow
#
resu = Graphics()
if parameters is None:
coord_tail = [numerical_approx(xp[i]) for i in ind_pc]
coord_head = [
numerical_approx(xp[i] + scale * vcomp[i]) for i in ind_pc
]
else:
coord_tail = [
numerical_approx(xp[i].substitute(parameters)) for i in ind_pc
]
coord_head = [
numerical_approx(
(xp[i] + scale * vcomp[i]).substitute(parameters))
for i in ind_pc
]
if coord_head != coord_tail:
if n_pc == 2:
resu += arrow2d(tailpoint=coord_tail,
headpoint=coord_head,
color=color,
**extra_options)
else:
resu += arrow3d(coord_tail,
coord_head,
color=color,
**extra_options)
#
# The label
#
if print_label:
if label is None:
if n_pc == 2 and self._latex_name is not None:
label = r'$' + self._latex_name + r'$'
if n_pc == 3 and self._name is not None:
label = self._name
if label is not None:
xlab = [xh + label_offset for xh in coord_head]
if label_color is None:
label_color = color
if n_pc == 2:
resu += text(label,
xlab,
fontsize=fontsize,
color=label_color)
else:
resu += text3d(label,
xlab,
fontsize=fontsize,
color=label_color)
return resu
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
def plot(self, chart=None, ambient_coords=None, mapping=None,
color='blue', print_label=True, label=None, label_color=None,
fontsize=10, label_offset=0.1, parameters=None, **extra_options):
r"""
Plot the vector in a Cartesian graph based on the coordinates of some
ambient chart.
The vector is drawn in terms of two (2D graphics) or three (3D graphics)
coordinates of a given chart, called hereafter the *ambient chart*.
The vector's base point `p` (or its image `\Phi(p)` by some
differentiable mapping `\Phi`) must lie in the ambient chart's domain.
If `\Phi` is different from the identity mapping, the vector
actually depicted is `\mathrm{d}\Phi_p(v)`, where `v` is the current
vector (``self``) (see the example of a vector tangent to the
2-sphere below, where `\Phi: S^2 \to \RR^3`).
INPUT:
- ``chart`` -- (default: ``None``) the ambient chart (see above); if
``None``, it is set to the default chart of the open set containing
the point at which the vector (or the vector image via the
differential `\mathrm{d}\Phi_p` of ``mapping``) is defined
- ``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.differentiable.diff_map.DiffMap`;
differentiable mapping `\Phi` providing the link between the
point `p` at which the vector is defined and the ambient chart
``chart``: the domain of ``chart`` must contain `\Phi(p)`;
if ``None``, the identity mapping is assumed
- ``scale`` -- (default: 1) value by which the length of the arrow
representing the vector is multiplied
- ``color`` -- (default: 'blue') color of the arrow representing the
vector
- ``print_label`` -- (boolean; default: ``True``) determines whether a
label is printed next to the arrow representing the vector
- ``label`` -- (string; default: ``None``) label printed next to the
arrow representing the vector; if ``None``, the vector's symbol is
used, if any
- ``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 vector arrow and the label
- ``parameters`` -- (default: ``None``) dictionary giving the numerical
values of the parameters that may appear in the coordinate expression
of ``self`` (see example below)
- ``**extra_options`` -- extra options for the arrow plot, like
``linestyle``, ``width`` or ``arrowsize`` (see
:func:`~sage.plot.arrow.arrow2d` and
:func:`~sage.plot.plot3d.shapes.arrow3d` for details)
OUTPUT:
- a graphic object, either an instance of
:class:`~sage.plot.graphics.Graphics` for a 2D plot (i.e. based on
2 coordinates of ``chart``) or an instance of
:class:`~sage.plot.plot3d.base.Graphics3d` for a 3D plot (i.e.
based on 3 coordinates of ``chart``)
EXAMPLES:
Vector tangent to a 2-dimensional manifold::
sage: M = Manifold(2, 'M')
sage: X.<x,y> = M.chart()
sage: p = M((2,2), name='p')
sage: Tp = M.tangent_space(p)
sage: v = Tp((2, 1), name='v') ; v
Tangent vector v at Point p on the 2-dimensional differentiable
manifold M
Plot of the vector alone (arrow + label)::
sage: v.plot()
Graphics object consisting of 2 graphics primitives
Plot atop of the chart grid::
sage: X.plot() + v.plot()
Graphics object consisting of 20 graphics primitives
.. PLOT::
M = Manifold(2, 'M')
X = M.chart('x y'); x, y = X[:]
p = M((2,2), name='p'); Tp = M.tangent_space(p)
v = Tp((2, 1), name='v')
g = X.plot() + v.plot()
sphinx_plot(g)
Plots with various options::
sage: X.plot() + v.plot(color='green', scale=2, label='V')
Graphics object consisting of 20 graphics primitives
.. PLOT::
M = Manifold(2, 'M')
X = M.chart('x y'); x, y = X[:]
p = M((2,2), name='p'); Tp = M.tangent_space(p)
v = Tp((2, 1), name='v')
g = X.plot() + v.plot(color='green', scale=2, label='V')
sphinx_plot(g)
::
sage: X.plot() + v.plot(print_label=False)
Graphics object consisting of 19 graphics primitives
.. PLOT::
M = Manifold(2, 'M')
X = M.chart('x y'); x, y = X[:]
p = M((2,2), name='p'); Tp = M.tangent_space(p)
v = Tp((2, 1), name='v')
g = X.plot() + v.plot(print_label=False)
sphinx_plot(g)
::
sage: X.plot() + v.plot(color='green', label_color='black',
....: fontsize=20, label_offset=0.2)
Graphics object consisting of 20 graphics primitives
.. PLOT::
M = Manifold(2, 'M')
X = M.chart('x y'); x, y = X[:]
p = M((2,2), name='p'); Tp = M.tangent_space(p)
v = Tp((2, 1), name='v')
g = X.plot() + v.plot(color='green', label_color='black', fontsize=20, label_offset=0.2)
sphinx_plot(g)
::
sage: X.plot() + v.plot(linestyle=':', width=4, arrowsize=8,
....: fontsize=20)
Graphics object consisting of 20 graphics primitives
.. PLOT::
M = Manifold(2, 'M')
X = M.chart('x y'); x, y = X[:]
p = M((2,2), name='p'); Tp = M.tangent_space(p)
v = Tp((2, 1), name='v')
g = X.plot() + v.plot(linestyle=':', width=4, arrowsize=8, fontsize=20)
sphinx_plot(g)
Plot with specific values of some free parameters::
sage: var('a b')
(a, b)
sage: v = Tp((1+a, -b^2), name='v') ; v.display()
v = (a + 1) d/dx - b^2 d/dy
sage: X.plot() + v.plot(parameters={a: -2, b: 3})
Graphics object consisting of 20 graphics primitives
Special case of the zero vector::
sage: v = Tp.zero() ; v
Tangent vector zero at Point p on the 2-dimensional differentiable
manifold M
sage: X.plot() + v.plot()
Graphics object consisting of 19 graphics primitives
Vector tangent to a 4-dimensional manifold::
sage: M = Manifold(4, 'M')
sage: X.<t,x,y,z> = M.chart()
sage: p = M((0,1,2,3), name='p')
sage: Tp = M.tangent_space(p)
sage: v = Tp((5,4,3,2), name='v') ; v
Tangent vector v at Point p on the 4-dimensional differentiable
manifold M
We cannot make a 4D plot directly::
sage: v.plot()
Traceback (most recent call last):
...
ValueError: the number of coordinates involved in the plot must
be either 2 or 3, not 4
Rather, we have to select some chart coordinates for the plot, via
the argument ``ambient_coords``. For instance, for a 2-dimensional
plot in terms of the coordinates `(x, y)`::
sage: v.plot(ambient_coords=(x,y))
Graphics object consisting of 2 graphics primitives
.. PLOT::
M = Manifold(4, 'M')
X = M.chart('t x y z'); t,x,y,z = X[:]
p = M((0,1,2,3), name='p'); Tp = M.tangent_space(p)
v = Tp((5,4,3,2), name='v')
g = X.plot(ambient_coords=(x,y)) + v.plot(ambient_coords=(x,y))
sphinx_plot(g)
This plot involves only the components `v^x` and `v^y` of `v`.
Similarly, for a 3-dimensional plot in terms of the coordinates
`(t, x, y)`::
sage: g = v.plot(ambient_coords=(t,x,z))
sage: print(g)
Graphics3d Object
This plot involves only the components `v^t`, `v^x` and `v^z` of `v`.
A nice 3D view atop the coordinate grid is obtained via::
sage: (X.plot(ambient_coords=(t,x,z)) # long time
....: + v.plot(ambient_coords=(t,x,z),
....: label_offset=0.5, width=6))
Graphics3d Object
.. PLOT::
M = Manifold(4, 'M')
X = M.chart('t x y z'); t,x,y,z = X[:]
p = M((0,1,2,3), name='p'); Tp = M.tangent_space(p)
v = Tp((5,4,3,2), name='v')
g = X.plot(ambient_coords=(t,x,z)) + v.plot(ambient_coords=(t,x,z),
label_offset=0.5, width=6)
sphinx_plot(g)
An example of plot via a differential mapping: plot of a vector tangent
to a 2-sphere viewed in `\RR^3`::
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: R3 = Manifold(3, 'R^3')
sage: X3.<x,y,z> = R3.chart()
sage: F = S2.diff_map(R3, {(XS, X3): [sin(th)*cos(ph),
....: sin(th)*sin(ph),
....: cos(th)]}, name='F')
sage: F.display() # the standard embedding of S^2 into R^3
F: S^2 --> R^3
on U: (th, ph) |--> (x, y, z) = (cos(ph)*sin(th), sin(ph)*sin(th), cos(th))
sage: p = U.point((pi/4, 7*pi/4), name='p')
sage: v = XS.frame()[1].at(p) ; v # the coordinate vector d/dphi at p
Tangent vector d/dph at Point p on the 2-dimensional differentiable
manifold S^2
sage: graph_v = v.plot(mapping=F)
sage: graph_S2 = XS.plot(chart=X3, mapping=F, number_values=9) # long time
sage: graph_v + graph_S2 # long time
Graphics3d Object
.. PLOT::
S2 = Manifold(2, 'S^2')
U = S2.open_subset('U')
XS = U.chart(r'th:(0,pi):\theta ph:(0,2*pi):\phi')
th, ph = XS[:]
R3 = Manifold(3, 'R^3')
X3 = R3.chart('x y z')
F = S2.diff_map(R3, {(XS, X3): [sin(th)*cos(ph), sin(th)*sin(ph),
cos(th)]}, name='F')
p = U.point((pi/4, 7*pi/4), name='p')
v = XS.frame()[1].at(p)
graph_v = v.plot(mapping=F)
graph_S2 = XS.plot(chart=X3, mapping=F, number_values=9)
sphinx_plot(graph_v + graph_S2)
"""
from sage.plot.arrow import arrow2d
from sage.plot.text import text
from sage.plot.graphics import Graphics
from sage.plot.plot3d.shapes import arrow3d
from sage.plot.plot3d.shapes2 import text3d
from sage.misc.functional import numerical_approx
from sage.manifolds.differentiable.chart import DiffChart
scale = extra_options.pop("scale")
#
# The "effective" vector to be plotted
#
if mapping is None:
eff_vector = self
base_point = self._point
else:
#!# check
# For efficiency, the method FiniteRankFreeModuleMorphism._call_()
# is called instead of FiniteRankFreeModuleMorphism.__call__()
eff_vector = mapping.differential(self._point)._call_(self)
base_point = mapping(self._point)
#
# The chart w.r.t. which the vector is plotted
#
if chart is None:
chart = base_point.parent().default_chart()
elif not isinstance(chart, DiffChart):
raise TypeError("{} is not a chart".format(chart))
#
# Coordinates of the above chart w.r.t. which the vector is plotted
#
if ambient_coords is None:
ambient_coords = chart[:] # all chart coordinates are used
n_pc = len(ambient_coords)
if n_pc != 2 and n_pc !=3:
raise ValueError("the number of coordinates involved in the " +
"plot must be either 2 or 3, not {}".format(n_pc))
# indices coordinates involved in the plot:
ind_pc = [chart[:].index(pc) for pc in ambient_coords]
#
# Components of the vector w.r.t. the chart frame
#
basis = chart.frame().at(base_point)
vcomp = eff_vector.comp(basis=basis)[:]
xp = base_point.coord(chart=chart)
#
# The arrow
#
resu = Graphics()
if parameters is None:
coord_tail = [numerical_approx(xp[i]) for i in ind_pc]
coord_head = [numerical_approx(xp[i] + scale*vcomp[i])
for i in ind_pc]
else:
coord_tail = [numerical_approx(xp[i].substitute(parameters))
for i in ind_pc]
coord_head = [numerical_approx(
(xp[i] + scale*vcomp[i]).substitute(parameters))
for i in ind_pc]
if coord_head != coord_tail:
if n_pc == 2:
resu += arrow2d(tailpoint=coord_tail, headpoint=coord_head,
color=color, **extra_options)
else:
resu += arrow3d(coord_tail, coord_head, color=color,
**extra_options)
#
# The label
#
if print_label:
if label is None:
if n_pc == 2 and self._latex_name is not None:
label = r'$' + self._latex_name + r'$'
if n_pc == 3 and self._name is not None:
label = self._name
if label is not None:
xlab = [xh + label_offset for xh in coord_head]
if label_color is None:
label_color = color
if n_pc == 2:
resu += text(label, xlab, fontsize=fontsize,
color=label_color)
else:
resu += 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
