def point(points, **kwds):
"""
Return either a 2-dimensional or 3-dimensional point or sum of points.
INPUT:
- ``points`` - either a single point (as a tuple), a list of
points, a single complex number, or a list of complex numbers.
For information regarding additional arguments, see either point2d?
or point3d?.
.. SEEALSO::
:func:`sage.plot.point.point2d`, :func:`sage.plot.plot3d.shapes2.point3d`
EXAMPLES::
sage: point((1,2))
Graphics object consisting of 1 graphics primitive
::
sage: point((1,2,3))
Graphics3d Object
::
sage: point([(0,0), (1,1)])
Graphics object consisting of 1 graphics primitive
::
sage: point([(0,0,1), (1,1,1)])
Graphics3d Object
Extra options will get passed on to show(), as long as they are valid::
sage: point([(cos(theta), sin(theta)) for theta in srange(0, 2*pi, pi/8)], frame=True)
Graphics object consisting of 1 graphics primitive
sage: point([(cos(theta), sin(theta)) for theta in srange(0, 2*pi, pi/8)]).show(frame=True) # These are equivalent
TESTS:
One can now use iterators (:trac:`13890`)::
sage: point(iter([(1,1,1)]))
Graphics3d Object
sage: point(iter([(1,2),(3,5)]))
Graphics object consisting of 1 graphics primitive
"""
if isinstance(points, Iterator):
points = list(points)
try:
return point2d(points, **kwds)
except (ValueError, TypeError):
from sage.plot.plot3d.shapes2 import point3d
return point3d(points, **kwds)
def point(points, **kwds):
"""
Returns either a 2-dimensional or 3-dimensional point or sum of points.
INPUT:
- ``points`` - either a single point (as a tuple), a list of
points, a single complex number, or a list of complex numbers.
For information regarding additional arguments, see either point2d?
or point3d?.
.. SEEALSO::
:func:`sage.plot.point.point2d`, :func:`sage.plot.plot3d.shapes2.point3d`
EXAMPLES::
sage: point((1,2))
Graphics object consisting of 1 graphics primitive
::
sage: point((1,2,3))
Graphics3d Object
::
sage: point([(0,0), (1,1)])
Graphics object consisting of 1 graphics primitive
::
sage: point([(0,0,1), (1,1,1)])
Graphics3d Object
Extra options will get passed on to show(), as long as they are valid::
sage: point([(cos(theta), sin(theta)) for theta in srange(0, 2*pi, pi/8)], frame=True)
Graphics object consisting of 1 graphics primitive
sage: point([(cos(theta), sin(theta)) for theta in srange(0, 2*pi, pi/8)]).show(frame=True) # These are equivalent
TESTS:
One can now use iterators (:trac:`13890`)::
sage: point(iter([(1,1,1)]))
Graphics3d Object
sage: point(iter([(1,2),(3,5)]))
Graphics object consisting of 1 graphics primitive
"""
if isinstance(points, collections.Iterator):
points = list(points)
try:
return point2d(points, **kwds)
except (ValueError, TypeError):
from sage.plot.plot3d.shapes2 import point3d
return point3d(points, **kwds)
def point(points, **kwds):
"""
Returns either a 2-dimensional or 3-dimensional point or sum of points.
INPUT:
- ``points`` - either a single point (as a tuple), a list of
points, a single complex number, or a list of complex numbers.
For information regarding additional arguments, see either point2d?
or point3d?.
EXAMPLES::
sage: point((1,2))
Graphics object consisting of 1 graphics primitive
::
sage: point((1,2,3))
Graphics3d Object
::
sage: point([(0,0), (1,1)])
Graphics object consisting of 1 graphics primitive
::
sage: point([(0,0,1), (1,1,1)])
Graphics3d Object
Extra options will get passed on to show(), as long as they are valid::
sage: point([(cos(theta), sin(theta)) for theta in srange(0, 2*pi, pi/8)], frame=True)
Graphics object consisting of 1 graphics primitive
sage: point([(cos(theta), sin(theta)) for theta in srange(0, 2*pi, pi/8)]).show(frame=True) # These are equivalent
"""
try:
return point2d(points, **kwds)
except (ValueError, TypeError):
from sage.plot.plot3d.shapes2 import point3d
return point3d(points, **kwds)
def point(points, **kwds):
"""
Returns either a 2-dimensional or 3-dimensional point or sum of points.
INPUT:
- ``points`` - either a single point (as a tuple), a list of
points, a single complex number, or a list of complex numbers.
For information regarding additional arguments, see either point2d?
or point3d?.
EXAMPLES::
sage: point((1,2))
Graphics object consisting of 1 graphics primitive
::
sage: point((1,2,3))
Graphics3d Object
::
sage: point([(0,0), (1,1)])
Graphics object consisting of 1 graphics primitive
::
sage: point([(0,0,1), (1,1,1)])
Graphics3d Object
Extra options will get passed on to show(), as long as they are valid::
sage: point([(cos(theta), sin(theta)) for theta in srange(0, 2*pi, pi/8)], frame=True)
Graphics object consisting of 1 graphics primitive
sage: point([(cos(theta), sin(theta)) for theta in srange(0, 2*pi, pi/8)]).show(frame=True) # These are equivalent
"""
try:
return point2d(points, **kwds)
except (ValueError, TypeError):
from sage.plot.plot3d.shapes2 import point3d
return point3d(points, **kwds)
def plot3d(self, z=0, **kwds):
"""
Plots a two-dimensional point in 3-D, with default height zero.
INPUT:
- ``z`` - optional 3D height above `xy`-plane. May be a list
if self is a list of points.
EXAMPLES:
One point::
sage: A=point((1,1))
sage: a=A[0];a
Point set defined by 1 point(s)
sage: b=a.plot3d()
One point with a height::
sage: A=point((1,1))
sage: a=A[0];a
Point set defined by 1 point(s)
sage: b=a.plot3d(z=3)
sage: b.loc[2]
3.0
Multiple points::
sage: P=point([(0,0), (1,1)])
sage: p=P[0]; p
Point set defined by 2 point(s)
sage: q=p.plot3d(size=22)
Multiple points with different heights::
sage: P=point([(0,0), (1,1)])
sage: p=P[0]
sage: q=p.plot3d(z=[2,3])
sage: q.all[0].loc[2]
2.0
sage: q.all[1].loc[2]
3.0
Note that keywords passed must be valid point3d options::
sage: A=point((1,1),size=22)
sage: a=A[0];a
Point set defined by 1 point(s)
sage: b=a.plot3d()
sage: b.size
22
sage: b=a.plot3d(pointsize=23) # only 2D valid option
sage: b.size
22
sage: b=a.plot3d(size=23) # correct keyword
sage: b.size
23
TESTS:
Heights passed as a list should have same length as
number of points::
sage: P=point([(0,0), (1,1), (2,3)])
sage: p=P[0]
sage: q=p.plot3d(z=2)
sage: q.all[1].loc[2]
2.0
sage: q=p.plot3d(z=[2,-2])
Traceback (most recent call last):
...
ValueError: Incorrect number of heights given
"""
from sage.plot.plot3d.base import Graphics3dGroup
from sage.plot.plot3d.shapes2 import point3d
options = self._plot3d_options()
options.update(kwds)
zdata = []
if isinstance(z, list):
zdata = z
else:
zdata = [z] * len(self.xdata)
if len(zdata) == len(self.xdata):
all = [
point3d(list(zip(self.xdata, self.ydata, zdata)), **options)
]
if len(all) == 1:
return all[0]
else:
return Graphics3dGroup(all)
else:
raise ValueError('Incorrect number of heights given')
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 plot3d(self, z=0, **kwds):
"""
Plots a two-dimensional point in 3-D, with default height zero.
INPUT:
- ``z`` - optional 3D height above `xy`-plane. May be a list
if self is a list of points.
EXAMPLES:
One point::
sage: A=point((1,1))
sage: a=A[0];a
Point set defined by 1 point(s)
sage: b=a.plot3d()
One point with a height::
sage: A=point((1,1))
sage: a=A[0];a
Point set defined by 1 point(s)
sage: b=a.plot3d(z=3)
sage: b.loc[2]
3.0
Multiple points::
sage: P=point([(0,0), (1,1)])
sage: p=P[0]; p
Point set defined by 2 point(s)
sage: q=p.plot3d(size=22)
Multiple points with different heights::
sage: P=point([(0,0), (1,1)])
sage: p=P[0]
sage: q=p.plot3d(z=[2,3])
sage: q.all[0].loc[2]
2.0
sage: q.all[1].loc[2]
3.0
Note that keywords passed must be valid point3d options::
sage: A=point((1,1),size=22)
sage: a=A[0];a
Point set defined by 1 point(s)
sage: b=a.plot3d()
sage: b.size
22
sage: b=a.plot3d(pointsize=23) # only 2D valid option
sage: b.size
22
sage: b=a.plot3d(size=23) # correct keyword
sage: b.size
23
TESTS:
Heights passed as a list should have same length as
number of points::
sage: P=point([(0,0), (1,1), (2,3)])
sage: p=P[0]
sage: q=p.plot3d(z=2)
sage: q.all[1].loc[2]
2.0
sage: q=p.plot3d(z=[2,-2])
Traceback (most recent call last):
...
ValueError: Incorrect number of heights given
"""
from sage.plot.plot3d.base import Graphics3dGroup
from sage.plot.plot3d.shapes2 import point3d
options = self._plot3d_options()
options.update(kwds)
zdata=[]
if isinstance(z, list):
zdata=z
else:
zdata=[z]*len(self.xdata)
if len(zdata)==len(self.xdata):
all = [point3d([(x, y, z) for x, y, z in zip(self.xdata, self.ydata, zdata)], **options)]
if len(all) == 1:
return all[0]
else:
return Graphics3dGroup(all)
else:
raise ValueError('Incorrect number of heights given')
def plot(self,
chart=None,
ambient_coords=None,
mapping=None,
label=None,
parameters=None,
**kwds):
r"""
For real manifolds, plot ``self`` in a Cartesian graph based
on the coordinates of some ambient chart.
The point is drawn in terms of two (2D graphics) or three (3D graphics)
coordinates of a given chart, called hereafter the *ambient chart*.
The domain of the ambient chart must contain the point, or its image
by a continuous manifold map `\Phi`.
INPUT:
- ``chart`` -- (default: ``None``) the ambient chart (see above); if
``None``, the ambient chart is set the default chart of
``self.parent()``
- ``ambient_coords`` -- (default: ``None``) tuple containing the 2
or 3 coordinates of the ambient chart in terms of which the plot
is performed; if ``None``, all the coordinates of the ambient
chart are considered
- ``mapping`` -- (default: ``None``)
:class:`~sage.manifolds.continuous_map.ContinuousMap`; continuous
manifold map `\Phi` providing the link between the current point
`p` and the ambient chart ``chart``: the domain of ``chart`` must
contain `\Phi(p)`; if ``None``, the identity map is assumed
- ``label`` -- (default: ``None``) label printed next to the point;
if ``None``, the point's name is used
- ``parameters`` -- (default: ``None``) dictionary giving the numerical
values of the parameters that may appear in the point coordinates
- ``size`` -- (default: 10) size of the point once drawn as a small
disk or sphere
- ``color`` -- (default: ``'black'``) color of the point
- ``label_color`` -- (default: ``None``) color to print the label;
if ``None``, the value of ``color`` is used
- ``fontsize`` -- (default: 10) size of the font used to print the
label
- ``label_offset`` -- (default: 0.1) determines the separation between
the point and its label
OUTPUT:
- a graphic object, either an instance of
:class:`~sage.plot.graphics.Graphics` for a 2D plot (i.e. based on
2 coordinates of the ambient chart) or an instance of
:class:`~sage.plot.plot3d.base.Graphics3d` for a 3D plot (i.e.
based on 3 coordinates of the ambient chart)
EXAMPLES:
Drawing a point on a 2-dimensional manifold::
sage: M = Manifold(2, 'M', structure='topological')
sage: X.<x,y> = M.chart()
sage: p = M.point((1,3), name='p')
sage: g = p.plot(X)
sage: print(g)
Graphics object consisting of 2 graphics primitives
sage: gX = X.plot(max_range=4) # plot of the coordinate grid
sage: g + gX # display of the point atop the coordinate grid
Graphics object consisting of 20 graphics primitives
.. PLOT::
M = Manifold(2, 'M', structure='topological')
X = M.chart('x y'); x,y = X[:]
p = M.point((1,3), name='p')
g = p.plot(X)
gX = X.plot(max_range=4)
sphinx_plot(g+gX)
Actually, since ``X`` is the default chart of the open set in which
``p`` has been defined, it can be skipped in the arguments of
``plot``::
sage: g = p.plot()
sage: g + gX
Graphics object consisting of 20 graphics primitives
Call with some options::
sage: g = p.plot(chart=X, size=40, color='green', label='$P$',
....: label_color='blue', fontsize=20, label_offset=0.3)
sage: g + gX
Graphics object consisting of 20 graphics primitives
.. PLOT::
M = Manifold(2, 'M', structure='topological')
X = M.chart('x y'); x,y = X[:]
p = M.point((1,3), name='p')
g = p.plot(chart=X, size=40, color='green', label='$P$', \
label_color='blue', fontsize=20, label_offset=0.3)
gX = X.plot(max_range=4)
sphinx_plot(g+gX)
Use of the ``parameters`` option to set a numerical value of some
symbolic variable::
sage: a = var('a')
sage: q = M.point((a,2*a), name='q')
sage: gq = q.plot(parameters={a:-2}, label_offset=0.2)
sage: g + gX + gq
Graphics object consisting of 22 graphics primitives
.. PLOT::
M = Manifold(2, 'M', structure='topological')
X = M.chart('x y'); x,y = X[:]
p = M.point((1,3), name='p')
g = p.plot(chart=X, size=40, color='green', label='$P$', \
label_color='blue', fontsize=20, label_offset=0.3)
var('a')
q = M.point((a,2*a), name='q')
gq = q.plot(parameters={a:-2}, label_offset=0.2)
gX = X.plot(max_range=4)
sphinx_plot(g+gX+gq)
The numerical value is used only for the plot::
sage: q.coord()
(a, 2*a)
Drawing a point on a 3-dimensional manifold::
sage: M = Manifold(3, 'M', structure='topological')
sage: X.<x,y,z> = M.chart()
sage: p = M.point((2,1,3), name='p')
sage: g = p.plot()
sage: print(g)
Graphics3d Object
sage: gX = X.plot(number_values=5) # coordinate mesh cube
sage: g + gX # display of the point atop the coordinate mesh
Graphics3d Object
Call with some options::
sage: g = p.plot(chart=X, size=40, color='green', label='P_1',
....: label_color='blue', fontsize=20, label_offset=0.3)
sage: g + gX
Graphics3d Object
An example of plot via a mapping: plot of a point on a 2-sphere viewed
in the 3-dimensional space ``M``::
sage: S2 = Manifold(2, 'S^2', structure='topological')
sage: U = S2.open_subset('U') # the open set covered by spherical coord.
sage: XS.<th,ph> = U.chart(r'th:(0,pi):\theta ph:(0,2*pi):\phi')
sage: p = U.point((pi/4, pi/8), name='p')
sage: F = S2.continuous_map(M, {(XS, X): [sin(th)*cos(ph),
....: sin(th)*sin(ph), cos(th)]}, name='F')
sage: F.display()
F: S^2 --> M
on U: (th, ph) |--> (x, y, z) = (cos(ph)*sin(th), sin(ph)*sin(th), cos(th))
sage: g = p.plot(chart=X, mapping=F)
sage: gS2 = XS.plot(chart=X, mapping=F, number_values=9)
sage: g + gS2
Graphics3d Object
Use of the option ``ambient_coords`` for plots on a 4-dimensional
manifold::
sage: M = Manifold(4, 'M', structure='topological')
sage: X.<t,x,y,z> = M.chart()
sage: p = M.point((1,2,3,4), name='p')
sage: g = p.plot(X, ambient_coords=(t,x,y), label_offset=0.4) # the coordinate z is skipped
sage: gX = X.plot(X, ambient_coords=(t,x,y), number_values=5) # long time
sage: g + gX # 3D plot # long time
Graphics3d Object
sage: g = p.plot(X, ambient_coords=(t,y,z), label_offset=0.4) # the coordinate x is skipped
sage: gX = X.plot(X, ambient_coords=(t,y,z), number_values=5) # long time
sage: g + gX # 3D plot # long time
Graphics3d Object
sage: g = p.plot(X, ambient_coords=(y,z), label_offset=0.4) # the coordinates t and x are skipped
sage: gX = X.plot(X, ambient_coords=(y,z))
sage: g + gX # 2D plot
Graphics object consisting of 20 graphics primitives
.. PLOT::
M = Manifold(4, 'M', structure='topological')
X = M.chart('t x y z'); t,x,y,z = X[:]
p = M.point((1,2,3,4), name='p')
g = p.plot(X, ambient_coords=(y,z), label_offset=0.4)
gX = X.plot(X, ambient_coords=(y,z))
sphinx_plot(g+gX)
"""
from sage.plot.point import point2d
from sage.plot.text import text
from sage.plot.graphics import Graphics
from sage.plot.plot3d.shapes2 import point3d, text3d
from sage.manifolds.chart import Chart
if self._manifold.base_field_type() != 'real':
raise NotImplementedError(
'plot of points on manifolds over fields different'
' from the real field is not implemented')
# The ambient chart:
if chart is None:
chart = self.parent().default_chart()
elif not isinstance(chart, Chart):
raise TypeError("the argument 'chart' must be a coordinate chart")
# The effective point to be plotted:
if mapping is None:
eff_point = self
else:
eff_point = mapping(self)
# The coordinates of the ambient chart used for the plot:
if ambient_coords is None:
ambient_coords = chart[:]
elif not isinstance(ambient_coords, tuple):
ambient_coords = tuple(ambient_coords)
nca = len(ambient_coords)
if nca != 2 and nca != 3:
raise TypeError(
"invalid number of ambient coordinates: {}".format(nca))
# Extract the kwds options
size = kwds['size']
color = kwds['color']
label_color = kwds['label_color']
fontsize = kwds['fontsize']
label_offset = kwds['label_offset']
# The point coordinates:
coords = eff_point.coord(chart)
xx = chart[:]
xp = [coords[xx.index(c)] for c in ambient_coords]
if parameters is not None:
xps = [coord.substitute(parameters) for coord in xp]
xp = xps
xlab = [coord + label_offset for coord in xp]
if label_color is None:
label_color = color
resu = Graphics()
if nca == 2:
if label is None:
label = r'$' + self._latex_name + r'$'
resu += (point2d(xp, color=color, size=size) +
text(label, xlab, fontsize=fontsize, color=label_color))
else:
if label is None:
label = self._name
resu += (point3d(xp, color=color, size=size) +
text3d(label, xlab, fontsize=fontsize, color=label_color))
return resu
def plot(self, chart=None, ambient_coords=None, mapping=None, size=10,
color='black', label=None, label_color=None, fontsize=10,
label_offset=0.1, parameters=None):
r"""
Plot the current point (``self``) in a Cartesian graph based on the
coordinates of some ambient chart.
The point is drawn in terms of two (2D graphics) or three (3D graphics)
coordinates of a given chart, called hereafter the *ambient chart*.
The domain of the ambient chart must contain the point, or its image
by a differentiable mapping `\Phi`.
INPUT:
- ``chart`` -- (default: ``None``) the ambient chart (see above); if
``None``, the ambient chart is set the default chart of
``self.containing_set()``
- ``ambient_coords`` -- (default: ``None``) tuple containing the 2 or 3
coordinates of the ambient chart in terms of which the plot is
performed; if ``None``, all the coordinates of the ambient chart are
considered
- ``mapping`` -- (default: ``None``) differentiable mapping `\Phi`
(instance of
:class:`~sage.geometry.manifolds.diffmapping.DiffMapping`)
providing the link between the point `p` represented by ``self``
and the ambient chart ``chart``: the domain of ``chart`` must
contain `\Phi(p)`; if ``None``, the identity mapping is assumed
- ``size`` -- (default: 10) size of the point once drawn as a small
disk or sphere
- ``color`` -- (default: 'black') color of the point
- ``label`` -- (default: ``None``) label printed next to the point;
if ``None``, the point's name is used.
- ``label_color`` -- (default: ``None``) color to print the label;
if ``None``, the value of ``color`` is used
- ``fontsize`` -- (default: 10) size of the font used to print the
label
- ``label_offset`` -- (default: 0.1) determines the separation between
the point and its label
- ``parameters`` -- (default: ``None``) dictionary giving the numerical
values of the parameters that may appear in the point coordinates
OUTPUT:
- a graphic object, either an instance of
:class:`~sage.plot.graphics.Graphics` for a 2D plot (i.e. based on
2 coordinates of the ambient chart) or an instance of
:class:`~sage.plot.plot3d.base.Graphics3d` for a 3D plot (i.e.
based on 3 coordinates of the ambient chart)
EXAMPLES:
Drawing a point on a 2-dimensional manifold::
sage: M = Manifold(2, 'M')
sage: X.<x,y> = M.chart()
sage: p = M.point((1,3), name='p')
sage: g = p.plot(X)
sage: print g
Graphics object consisting of 2 graphics primitives
sage: gX = X.plot() # plot of the coordinate grid
sage: show(g+gX) # display of the point atop the coordinate grid
Actually, since ``X`` is the default chart of the open set in which
``p`` has been defined, it can be skipped in the arguments of
``plot``::
sage: g = p.plot()
sage: show(g+gX)
Call with some options::
sage: g = p.plot(chart=X, size=40, color='green', label='$P$',
....: label_color='blue', fontsize=20, label_offset=0.3)
sage: show(g+gX)
Use of the ``parameters`` option to set a numerical value of some
symbolic variable::
sage: a = var('a')
sage: q = M.point((a,2*a), name='q')
sage: gq = q.plot(parameters={a:-2})
sage: show(g+gX+gq)
The numerical value is used only for the plot::
sage: q.coord()
(a, 2*a)
Drawing a point on a 3-dimensional manifold::
sage: M = Manifold(3, 'M')
sage: X.<x,y,z> = M.chart()
sage: p = M.point((2,1,3), name='p')
sage: g = p.plot()
sage: print g
Graphics3d Object
sage: gX = X.plot(nb_values=5) # coordinate mesh cube
sage: show(g+gX) # display of the point atop the coordinate mesh
Call with some options::
sage: g = p.plot(chart=X, size=40, color='green', label='P_1',
....: label_color='blue', fontsize=20, label_offset=0.3)
sage: show(g+gX)
An example of plot via a differential mapping: plot of a point on a
2-sphere viewed in the 3-dimensional space ``M``::
sage: S2 = Manifold(2, 'S^2')
sage: U = S2.open_subset('U') # the open set covered by spherical coord.
sage: XS.<th,ph> = U.chart(r'th:(0,pi):\theta ph:(0,2*pi):\phi')
sage: p = U.point((pi/4, pi/8), name='p')
sage: F = S2.diff_mapping(M, {(XS, X): [sin(th)*cos(ph),
....: sin(th)*sin(ph), cos(th)]}, name='F')
sage: F.display()
F: S^2 --> M
on U: (th, ph) |--> (x, y, z) = (cos(ph)*sin(th), sin(ph)*sin(th), cos(th))
sage: g = p.plot(chart=X, mapping=F)
sage: gS2 = XS.plot(chart=X, mapping=F, nb_values=9)
sage: show(g+gS2)
Use of the option ``ambient_coords`` for plots on a 4-dimensional
manifold::
sage: M = Manifold(4, 'M')
sage: X.<t,x,y,z> = M.chart()
sage: p = M.point((1,2,3,4), name='p')
sage: g = p.plot(X, ambient_coords=(t,x,y)) # the coordinate z is skipped
sage: gX = X.plot(X, ambient_coords=(t,x,y), nb_values=5)
sage: show(g+gX) # 3D plot
sage: g = p.plot(X, ambient_coords=(t,y,z)) # the coordinate x is skipped
sage: gX = X.plot(X, ambient_coords=(t,y,z), nb_values=5)
sage: show(g+gX) # 3D plot
sage: g = p.plot(X, ambient_coords=(y,z)) # the coordinates t and x are skipped
sage: gX = X.plot(X, ambient_coords=(y,z))
sage: show(g+gX) # 2D plot
"""
from sage.plot.point import point2d
from sage.plot.text import text
from sage.plot.graphics import Graphics
from sage.plot.plot3d.shapes2 import point3d, text3d
from sage.geometry.manifolds.chart import Chart
# The ambient chart:
if chart is None:
chart = self.containing_set().default_chart()
elif not isinstance(chart, Chart):
raise TypeError("the argument 'chart' must be a coordinate chart")
# The effective point to be plotted:
if mapping is None:
eff_point = self
else:
eff_point = mapping(self)
# The coordinates of the ambient chart used for the plot:
if ambient_coords is None:
ambient_coords = chart._xx
elif not isinstance(ambient_coords, tuple):
ambient_coords = tuple(ambient_coords)
nca = len(ambient_coords)
if nca != 2 and nca !=3:
raise TypeError("Bad number of ambient coordinates: " + str(nca))
# The point coordinates:
coords = eff_point.coord(chart)
xx = chart[:]
xp = [coords[xx.index(c)] for c in ambient_coords]
if parameters is not None:
xps = [coord.substitute(parameters) for coord in xp]
xp = xps
xlab = [coord + label_offset for coord in xp]
if label_color is None:
label_color = color
resu = Graphics()
if nca == 2:
if label is None:
label = r'$' + self._latex_name + r'$'
resu += point2d(xp, color=color, size=size) + \
text(label, xlab, fontsize=fontsize, color=label_color)
else:
if label is None:
label = self._name
resu += point3d(xp, color=color, size=size) + \
text3d(label, xlab, fontsize=fontsize, color=label_color)
return resu
