def arrow(tailpoint=None, headpoint=None, **kwds):
"""
Returns either a 2-dimensional or 3-dimensional arrow depending
on value of points. Given a 1-dimensional arrow, returns a
1-dimensional arrow plotted on a 2-dimensional plane.
For information regarding additional arguments, see either arrow2d?
or arrow3d?.
EXAMPLES::
sage: arrow((0), (1))
sage: arrow([0], [1])
sage: arrow((0,0), (1,1))
sage: arrow((0,0,1), (1,1,1))
"""
if headpoint is not None and tailpoint is not None:
if (hasattr(headpoint, "__len__") and hasattr(tailpoint, "__len__")
and len(headpoint) != len(tailpoint)) or (
hasattr(headpoint, "__len__")
and not hasattr(tailpoint, "__len__")
or not hasattr(headpoint, "__len__")
and hasattr(tailpoint, "__len__")):
raise TypeError(
'Arrow requires headpoint and tailpoint to be of the same dimension.'
)
try:
return arrow2d(tailpoint, headpoint, **kwds)
except ValueError:
from sage.plot.plot3d.shapes import arrow3d
return arrow3d(tailpoint, headpoint, **kwds)
def arrow(tailpoint=None, headpoint=None, **kwds):
"""
Returns either a 2-dimensional or 3-dimensional arrow depending
on value of points.
For information regarding additional arguments, see either arrow2d?
or arrow3d?.
EXAMPLES::
sage: arrow((0,0), (1,1))
Graphics object consisting of 1 graphics primitive
.. PLOT::
sphinx_plot(arrow((0,0), (1,1)))
::
sage: arrow((0,0,1), (1,1,1))
Graphics3d Object
.. PLOT::
sphinx_plot(arrow((0,0,1), (1,1,1)))
"""
try:
return arrow2d(tailpoint, headpoint, **kwds)
except ValueError:
from sage.plot.plot3d.shapes import arrow3d
return arrow3d(tailpoint, headpoint, **kwds)
def arrow(tailpoint=None, headpoint=None, **kwds):
"""
Returns either a 2-dimensional or 3-dimensional arrow depending
on value of points. Given a 1-dimensional arrow, returns a
1-dimensional arrow plotted on a 2-dimensional plane.
For information regarding additional arguments, see either arrow2d?
or arrow3d?.
EXAMPLES::
sage: arrow((0), (1))
sage: arrow([0], [1])
sage: arrow((0,0), (1,1))
sage: arrow((0,0,1), (1,1,1))
"""
if headpoint is not None and tailpoint is not None:
if (
hasattr(headpoint, "__len__") and hasattr(tailpoint, "__len__") and
len(headpoint) != len(tailpoint)
) or (
hasattr(headpoint, "__len__") and not hasattr(tailpoint, "__len__") or
not hasattr(headpoint, "__len__") and hasattr(tailpoint, "__len__")
):
raise TypeError('Arrow requires headpoint and tailpoint to be of the same dimension.')
try:
return arrow2d(tailpoint, headpoint, **kwds)
except ValueError:
from sage.plot.plot3d.shapes import arrow3d
return arrow3d(tailpoint, headpoint, **kwds)
def arrow(tailpoint=None, headpoint=None, **kwds):
"""
Returns either a 2-dimensional or 3-dimensional arrow depending
on value of points.
For information regarding additional arguments, see either arrow2d?
or arrow3d?.
EXAMPLES::
sage: arrow((0,0), (1,1))
Graphics object consisting of 1 graphics primitive
.. PLOT::
sphinx_plot(arrow((0,0), (1,1)))
::
sage: arrow((0,0,1), (1,1,1))
Graphics3d Object
.. PLOT::
sphinx_plot(arrow((0,0,1), (1,1,1)))
"""
try:
return arrow2d(tailpoint, headpoint, **kwds)
except ValueError:
from sage.plot.plot3d.shapes import arrow3d
return arrow3d(tailpoint, headpoint, **kwds)
def arrow(tailpoint=None, headpoint=None, **kwds):
"""
Returns either a 2-dimensional or 3-dimensional arrow depending
on value of points.
For information regarding additional arguments, see either arrow2d?
or arrow3d?.
EXAMPLES::
sage: arrow((0,0), (1,1))
sage: arrow((0,0,1), (1,1,1))
"""
try:
return arrow2d(tailpoint, headpoint, **kwds)
except ValueError:
from sage.plot.plot3d.shapes import arrow3d
return arrow3d(tailpoint, headpoint, **kwds)
def arrow(tailpoint=None, headpoint=None, **kwds):
"""
Returns either a 2-dimensional or 3-dimensional arrow depending
on value of points.
For information regarding additional arguments, see either arrow2d?
or arrow3d?.
EXAMPLES::
sage: arrow((0,0), (1,1))
sage: arrow((0,0,1), (1,1,1))
"""
try:
return arrow2d(tailpoint, headpoint, **kwds)
except ValueError:
from sage.plot.plot3d.shapes import arrow3d
return arrow3d(tailpoint, headpoint, **kwds)
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,
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
