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"""Plotting module for SymPy.
A plot is represented by the ``Plot`` class that contains a reference to the
backend and a list of the data series to be plotted. The data series are
instances of classes meant to simplify getting points and meshes from SymPy
expressions. ``plot_backends`` is a dictionary with all the backends.
This module gives only the essential. For all the fancy stuff use directly
the backend. You can get the backend wrapper for every plot from the
``_backend`` attribute. Moreover the data series classes have various useful
methods like ``get_points``, ``get_meshes``, etc, that may
be useful if you wish to use another plotting library.
Especially if you need publication ready graphs and this module is not enough
for you - just get the ``_backend`` attribute and add whatever you want
directly to it. In the case of matplotlib (the common way to graph data in
python) just copy ``_backend.fig`` which is the figure and ``_backend.ax``
which is the axis and work on them as you would on any other matplotlib object.
Simplicity of code takes much greater importance than performance. Do not use it
if you care at all about performance. A new backend instance is initialized
every time you call ``show()`` and the old one is left to the garbage collector.
"""
from collections.abc import Callable
from sympy.core.basic import Basic
from sympy.core.containers import Tuple
from sympy.core.expr import Expr
from sympy.core.function import arity, Function
from sympy.core.symbol import (Dummy, Symbol)
from sympy.core.sympify import sympify
from sympy.external import import_module
from sympy.printing.latex import latex
from sympy.utilities.exceptions import sympy_deprecation_warning
from sympy.utilities.iterables import is_sequence
from .experimental_lambdify import (vectorized_lambdify, lambdify)
# N.B.
# When changing the minimum module version for matplotlib, please change
# the same in the `SymPyDocTestFinder`` in `sympy/testing/runtests.py`
# Backend specific imports - textplot
from sympy.plotting.textplot import textplot
# Global variable
# Set to False when running tests / doctests so that the plots don't show.
_show = True
def unset_show():
"""
Disable show(). For use in the tests.
"""
global _show
_show = False
def _str_or_latex(label):
if isinstance(label, Basic):
return latex(label, mode='inline')
return str(label)
##############################################################################
# The public interface
##############################################################################
class Plot:
"""The central class of the plotting module.
Explanation
===========
For interactive work the function :func:`plot()` is better suited.
This class permits the plotting of SymPy expressions using numerous
backends (:external:mod:`matplotlib`, textplot, the old pyglet module for SymPy, Google
charts api, etc).
The figure can contain an arbitrary number of plots of SymPy expressions,
lists of coordinates of points, etc. Plot has a private attribute _series that
contains all data series to be plotted (expressions for lines or surfaces,
lists of points, etc (all subclasses of BaseSeries)). Those data series are
instances of classes not imported by ``from sympy import *``.
The customization of the figure is on two levels. Global options that
concern the figure as a whole (e.g. title, xlabel, scale, etc) and
per-data series options (e.g. name) and aesthetics (e.g. color, point shape,
line type, etc.).
The difference between options and aesthetics is that an aesthetic can be
a function of the coordinates (or parameters in a parametric plot). The
supported values for an aesthetic are:
- None (the backend uses default values)
- a constant
- a function of one variable (the first coordinate or parameter)
- a function of two variables (the first and second coordinate or parameters)
- a function of three variables (only in nonparametric 3D plots)
Their implementation depends on the backend so they may not work in some
backends.
If the plot is parametric and the arity of the aesthetic function permits
it the aesthetic is calculated over parameters and not over coordinates.
If the arity does not permit calculation over parameters the calculation is
done over coordinates.
Only cartesian coordinates are supported for the moment, but you can use
the parametric plots to plot in polar, spherical and cylindrical
coordinates.
The arguments for the constructor Plot must be subclasses of BaseSeries.
Any global option can be specified as a keyword argument.
The global options for a figure are:
- title : str
- xlabel : str or Symbol
- ylabel : str or Symbol
- zlabel : str or Symbol
- legend : bool
- xscale : {'linear', 'log'}
- yscale : {'linear', 'log'}
- axis : bool
- axis_center : tuple of two floats or {'center', 'auto'}
- xlim : tuple of two floats
- ylim : tuple of two floats
- aspect_ratio : tuple of two floats or {'auto'}
- autoscale : bool
- margin : float in [0, 1]
- backend : {'default', 'matplotlib', 'text'} or a subclass of BaseBackend
- size : optional tuple of two floats, (width, height); default: None
The per data series options and aesthetics are:
There are none in the base series. See below for options for subclasses.
Some data series support additional aesthetics or options:
:class:`~.LineOver1DRangeSeries`, :class:`~.Parametric2DLineSeries`, and
:class:`~.Parametric3DLineSeries` support the following:
Aesthetics:
- line_color : string, or float, or function, optional
Specifies the color for the plot, which depends on the backend being
used.
For example, if ``MatplotlibBackend`` is being used, then
Matplotlib string colors are acceptable (``"red"``, ``"r"``,
``"cyan"``, ``"c"``, ...).
Alternatively, we can use a float number, 0 < color < 1, wrapped in a
string (for example, ``line_color="0.5"``) to specify grayscale colors.
Alternatively, We can specify a function returning a single
float value: this will be used to apply a color-loop (for example,
``line_color=lambda x: math.cos(x)``).
Note that by setting line_color, it would be applied simultaneously
to all the series.
Options:
- label : str
- steps : bool
- integers_only : bool
:class:`~.SurfaceOver2DRangeSeries` and :class:`~.ParametricSurfaceSeries`
support the following:
Aesthetics:
- surface_color : function which returns a float.
"""
def __init__(self, *args,
title=None, xlabel=None, ylabel=None, zlabel=None, aspect_ratio='auto',
xlim=None, ylim=None, axis_center='auto', axis=True,
xscale='linear', yscale='linear', legend=False, autoscale=True,
margin=0, annotations=None, markers=None, rectangles=None,
fill=None, backend='default', size=None, **kwargs):
super().__init__()
# Options for the graph as a whole.
# The possible values for each option are described in the docstring of
# Plot. They are based purely on convention, no checking is done.
self.title = title
self.xlabel = xlabel
self.ylabel = ylabel
self.zlabel = zlabel
self.aspect_ratio = aspect_ratio
self.axis_center = axis_center
self.axis = axis
self.xscale = xscale
self.yscale = yscale
self.legend = legend
self.autoscale = autoscale
self.margin = margin
self.annotations = annotations
self.markers = markers
self.rectangles = rectangles
self.fill = fill
# Contains the data objects to be plotted. The backend should be smart
# enough to iterate over this list.
self._series = []
self._series.extend(args)
# The backend type. On every show() a new backend instance is created
# in self._backend which is tightly coupled to the Plot instance
# (thanks to the parent attribute of the backend).
if isinstance(backend, str):
self.backend = plot_backends[backend]
elif (type(backend) == type) and issubclass(backend, BaseBackend):
self.backend = backend
else:
raise TypeError(
"backend must be either a string or a subclass of BaseBackend")
is_real = \
lambda lim: all(getattr(i, 'is_real', True) for i in lim)
is_finite = \
lambda lim: all(getattr(i, 'is_finite', True) for i in lim)
# reduce code repetition
def check_and_set(t_name, t):
if t:
if not is_real(t):
raise ValueError(
"All numbers from {}={} must be real".format(t_name, t))
if not is_finite(t):
raise ValueError(
"All numbers from {}={} must be finite".format(t_name, t))
setattr(self, t_name, (float(t[0]), float(t[1])))
self.xlim = None
check_and_set("xlim", xlim)
self.ylim = None
check_and_set("ylim", ylim)
self.size = None
check_and_set("size", size)
def show(self):
# TODO move this to the backend (also for save)
if hasattr(self, '_backend'):
self._backend.close()
self._backend = self.backend(self)
self._backend.show()
def save(self, path):
if hasattr(self, '_backend'):
self._backend.close()
self._backend = self.backend(self)
self._backend.save(path)
def __str__(self):
series_strs = [('[%d]: ' % i) + str(s)
for i, s in enumerate(self._series)]
return 'Plot object containing:\n' + '\n'.join(series_strs)
def __getitem__(self, index):
return self._series[index]
def __setitem__(self, index, *args):
if len(args) == 1 and isinstance(args[0], BaseSeries):
self._series[index] = args
def __delitem__(self, index):
del self._series[index]
def append(self, arg):
"""Adds an element from a plot's series to an existing plot.
Examples
========
Consider two ``Plot`` objects, ``p1`` and ``p2``. To add the
second plot's first series object to the first, use the
``append`` method, like so:
.. plot::
:format: doctest
:include-source: True
>>> from sympy import symbols
>>> from sympy.plotting import plot
>>> x = symbols('x')
>>> p1 = plot(x*x, show=False)
>>> p2 = plot(x, show=False)
>>> p1.append(p2[0])
>>> p1
Plot object containing:
[0]: cartesian line: x**2 for x over (-10.0, 10.0)
[1]: cartesian line: x for x over (-10.0, 10.0)
>>> p1.show()
See Also
========
extend
"""
if isinstance(arg, BaseSeries):
self._series.append(arg)
else:
raise TypeError('Must specify element of plot to append.')
def extend(self, arg):
"""Adds all series from another plot.
Examples
========
Consider two ``Plot`` objects, ``p1`` and ``p2``. To add the
second plot to the first, use the ``extend`` method, like so:
.. plot::
:format: doctest
:include-source: True
>>> from sympy import symbols
>>> from sympy.plotting import plot
>>> x = symbols('x')
>>> p1 = plot(x**2, show=False)
>>> p2 = plot(x, -x, show=False)
>>> p1.extend(p2)
>>> p1
Plot object containing:
[0]: cartesian line: x**2 for x over (-10.0, 10.0)
[1]: cartesian line: x for x over (-10.0, 10.0)
[2]: cartesian line: -x for x over (-10.0, 10.0)
>>> p1.show()
"""
if isinstance(arg, Plot):
self._series.extend(arg._series)
elif is_sequence(arg):
self._series.extend(arg)
else:
raise TypeError('Expecting Plot or sequence of BaseSeries')
class PlotGrid:
"""This class helps to plot subplots from already created SymPy plots
in a single figure.
Examples
========
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> from sympy import symbols
>>> from sympy.plotting import plot, plot3d, PlotGrid
>>> x, y = symbols('x, y')
>>> p1 = plot(x, x**2, x**3, (x, -5, 5))
>>> p2 = plot((x**2, (x, -6, 6)), (x, (x, -5, 5)))
>>> p3 = plot(x**3, (x, -5, 5))
>>> p4 = plot3d(x*y, (x, -5, 5), (y, -5, 5))
Plotting vertically in a single line:
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> PlotGrid(2, 1, p1, p2)
PlotGrid object containing:
Plot[0]:Plot object containing:
[0]: cartesian line: x for x over (-5.0, 5.0)
[1]: cartesian line: x**2 for x over (-5.0, 5.0)
[2]: cartesian line: x**3 for x over (-5.0, 5.0)
Plot[1]:Plot object containing:
[0]: cartesian line: x**2 for x over (-6.0, 6.0)
[1]: cartesian line: x for x over (-5.0, 5.0)
Plotting horizontally in a single line:
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> PlotGrid(1, 3, p2, p3, p4)
PlotGrid object containing:
Plot[0]:Plot object containing:
[0]: cartesian line: x**2 for x over (-6.0, 6.0)
[1]: cartesian line: x for x over (-5.0, 5.0)
Plot[1]:Plot object containing:
[0]: cartesian line: x**3 for x over (-5.0, 5.0)
Plot[2]:Plot object containing:
[0]: cartesian surface: x*y for x over (-5.0, 5.0) and y over (-5.0, 5.0)
Plotting in a grid form:
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> PlotGrid(2, 2, p1, p2, p3, p4)
PlotGrid object containing:
Plot[0]:Plot object containing:
[0]: cartesian line: x for x over (-5.0, 5.0)
[1]: cartesian line: x**2 for x over (-5.0, 5.0)
[2]: cartesian line: x**3 for x over (-5.0, 5.0)
Plot[1]:Plot object containing:
[0]: cartesian line: x**2 for x over (-6.0, 6.0)
[1]: cartesian line: x for x over (-5.0, 5.0)
Plot[2]:Plot object containing:
[0]: cartesian line: x**3 for x over (-5.0, 5.0)
Plot[3]:Plot object containing:
[0]: cartesian surface: x*y for x over (-5.0, 5.0) and y over (-5.0, 5.0)
"""
def __init__(self, nrows, ncolumns, *args, show=True, size=None, **kwargs):
"""
Parameters
==========
nrows :
The number of rows that should be in the grid of the
required subplot.
ncolumns :
The number of columns that should be in the grid
of the required subplot.
nrows and ncolumns together define the required grid.
Arguments
=========
A list of predefined plot objects entered in a row-wise sequence
i.e. plot objects which are to be in the top row of the required
grid are written first, then the second row objects and so on
Keyword arguments
=================
show : Boolean
The default value is set to ``True``. Set show to ``False`` and
the function will not display the subplot. The returned instance
of the ``PlotGrid`` class can then be used to save or display the
plot by calling the ``save()`` and ``show()`` methods
respectively.
size : (float, float), optional
A tuple in the form (width, height) in inches to specify the size of
the overall figure. The default value is set to ``None``, meaning
the size will be set by the default backend.
"""
self.nrows = nrows
self.ncolumns = ncolumns
self._series = []
self.args = args
for arg in args:
self._series.append(arg._series)
self.backend = DefaultBackend
self.size = size
if show:
self.show()
def show(self):
if hasattr(self, '_backend'):
self._backend.close()
self._backend = self.backend(self)
self._backend.show()
def save(self, path):
if hasattr(self, '_backend'):
self._backend.close()
self._backend = self.backend(self)
self._backend.save(path)
def __str__(self):
plot_strs = [('Plot[%d]:' % i) + str(plot)
for i, plot in enumerate(self.args)]
return 'PlotGrid object containing:\n' + '\n'.join(plot_strs)
##############################################################################
# Data Series
##############################################################################
#TODO more general way to calculate aesthetics (see get_color_array)
### The base class for all series
class BaseSeries:
"""Base class for the data objects containing stuff to be plotted.
Explanation
===========
The backend should check if it supports the data series that is given.
(e.g. TextBackend supports only LineOver1DRangeSeries).
It is the backend responsibility to know how to use the class of
data series that is given.
Some data series classes are grouped (using a class attribute like is_2Dline)
according to the api they present (based only on convention). The backend is
not obliged to use that api (e.g. LineOver1DRangeSeries belongs to the
is_2Dline group and presents the get_points method, but the
TextBackend does not use the get_points method).
"""
# Some flags follow. The rationale for using flags instead of checking base
# classes is that setting multiple flags is simpler than multiple
# inheritance.
is_2Dline = False
# Some of the backends expect:
# - get_points returning 1D np.arrays list_x, list_y
# - get_color_array returning 1D np.array (done in Line2DBaseSeries)
# with the colors calculated at the points from get_points
is_3Dline = False
# Some of the backends expect:
# - get_points returning 1D np.arrays list_x, list_y, list_y
# - get_color_array returning 1D np.array (done in Line2DBaseSeries)
# with the colors calculated at the points from get_points
is_3Dsurface = False
# Some of the backends expect:
# - get_meshes returning mesh_x, mesh_y, mesh_z (2D np.arrays)
# - get_points an alias for get_meshes
is_contour = False
# Some of the backends expect:
# - get_meshes returning mesh_x, mesh_y, mesh_z (2D np.arrays)
# - get_points an alias for get_meshes
is_implicit = False
# Some of the backends expect:
# - get_meshes returning mesh_x (1D array), mesh_y(1D array,
# mesh_z (2D np.arrays)
# - get_points an alias for get_meshes
# Different from is_contour as the colormap in backend will be
# different
is_parametric = False
# The calculation of aesthetics expects:
# - get_parameter_points returning one or two np.arrays (1D or 2D)
# used for calculation aesthetics
def __init__(self):
super().__init__()
@property
def is_3D(self):
flags3D = [
self.is_3Dline,
self.is_3Dsurface
]
return any(flags3D)
@property
def is_line(self):
flagslines = [
self.is_2Dline,
self.is_3Dline
]
return any(flagslines)
### 2D lines
class Line2DBaseSeries(BaseSeries):
"""A base class for 2D lines.
- adding the label, steps and only_integers options
- making is_2Dline true
- defining get_segments and get_color_array
"""
is_2Dline = True
_dim = 2
def __init__(self):
super().__init__()
self.label = None
self.steps = False
self.only_integers = False
self.line_color = None
def get_data(self):
""" Return lists of coordinates for plotting the line.
Returns
=======
x : list
List of x-coordinates
y : list
List of y-coordinates
z : list
List of z-coordinates in case of Parametric3DLineSeries
"""
np = import_module('numpy')
points = self.get_points()
if self.steps is True:
if len(points) == 2:
x = np.array((points[0], points[0])).T.flatten()[1:]
y = np.array((points[1], points[1])).T.flatten()[:-1]
points = (x, y)
else:
x = np.repeat(points[0], 3)[2:]
y = np.repeat(points[1], 3)[:-2]
z = np.repeat(points[2], 3)[1:-1]
points = (x, y, z)
return points
def get_segments(self):
sympy_deprecation_warning(
"""
The Line2DBaseSeries.get_segments() method is deprecated.
Instead, use the MatplotlibBackend.get_segments() method, or use
The get_points() or get_data() methods.
""",
deprecated_since_version="1.9",
active_deprecations_target="deprecated-get-segments")
np = import_module('numpy')
points = type(self).get_data(self)
points = np.ma.array(points).T.reshape(-1, 1, self._dim)
return np.ma.concatenate([points[:-1], points[1:]], axis=1)
def get_color_array(self):
np = import_module('numpy')
c = self.line_color
if hasattr(c, '__call__'):
f = np.vectorize(c)
nargs = arity(c)
if nargs == 1 and self.is_parametric:
x = self.get_parameter_points()
return f(centers_of_segments(x))
else:
variables = list(map(centers_of_segments, self.get_points()))
if nargs == 1:
return f(variables[0])
elif nargs == 2:
return f(*variables[:2])
else: # only if the line is 3D (otherwise raises an error)
return f(*variables)
else:
return c*np.ones(self.nb_of_points)
class List2DSeries(Line2DBaseSeries):
"""Representation for a line consisting of list of points."""
def __init__(self, list_x, list_y):
np = import_module('numpy')
super().__init__()
self.list_x = np.array(list_x)
self.list_y = np.array(list_y)
self.label = 'list'
def __str__(self):
return 'list plot'
def get_points(self):
return (self.list_x, self.list_y)
class LineOver1DRangeSeries(Line2DBaseSeries):
"""Representation for a line consisting of a SymPy expression over a range."""
def __init__(self, expr, var_start_end, **kwargs):
super().__init__()
self.expr = sympify(expr)
self.label = kwargs.get('label', None) or self.expr
self.var = sympify(var_start_end[0])
self.start = float(var_start_end[1])
self.end = float(var_start_end[2])
self.nb_of_points = kwargs.get('nb_of_points', 300)
self.adaptive = kwargs.get('adaptive', True)
self.depth = kwargs.get('depth', 12)
self.line_color = kwargs.get('line_color', None)
self.xscale = kwargs.get('xscale', 'linear')
def __str__(self):
return 'cartesian line: %s for %s over %s' % (
str(self.expr), str(self.var), str((self.start, self.end)))
def get_points(self):
""" Return lists of coordinates for plotting. Depending on the
``adaptive`` option, this function will either use an adaptive algorithm
or it will uniformly sample the expression over the provided range.
Returns
=======
x : list
List of x-coordinates
y : list
List of y-coordinates
Explanation
===========
The adaptive sampling is done by recursively checking if three
points are almost collinear. If they are not collinear, then more
points are added between those points.
References
==========
.. [1] Adaptive polygonal approximation of parametric curves,
Luiz Henrique de Figueiredo.
"""
if self.only_integers or not self.adaptive:
return self._uniform_sampling()
else:
f = lambdify([self.var], self.expr)
x_coords = []
y_coords = []
np = import_module('numpy')
def sample(p, q, depth):
""" Samples recursively if three points are almost collinear.
For depth < 6, points are added irrespective of whether they
satisfy the collinearity condition or not. The maximum depth
allowed is 12.
"""
# Randomly sample to avoid aliasing.
random = 0.45 + np.random.rand() * 0.1
if self.xscale == 'log':
xnew = 10**(np.log10(p[0]) + random * (np.log10(q[0]) -
np.log10(p[0])))
else:
xnew = p[0] + random * (q[0] - p[0])
ynew = f(xnew)
new_point = np.array([xnew, ynew])
# Maximum depth
if depth > self.depth:
x_coords.append(q[0])
y_coords.append(q[1])
# Sample irrespective of whether the line is flat till the
# depth of 6. We are not using linspace to avoid aliasing.
elif depth < 6:
sample(p, new_point, depth + 1)
sample(new_point, q, depth + 1)
# Sample ten points if complex values are encountered
# at both ends. If there is a real value in between, then
# sample those points further.
elif p[1] is None and q[1] is None:
if self.xscale == 'log':
xarray = np.logspace(p[0], q[0], 10)
else:
xarray = np.linspace(p[0], q[0], 10)
yarray = list(map(f, xarray))
if not all(y is None for y in yarray):
for i in range(len(yarray) - 1):
if not (yarray[i] is None and yarray[i + 1] is None):
sample([xarray[i], yarray[i]],
[xarray[i + 1], yarray[i + 1]], depth + 1)
# Sample further if one of the end points in None (i.e. a
# complex value) or the three points are not almost collinear.
elif (p[1] is None or q[1] is None or new_point[1] is None
or not flat(p, new_point, q)):
sample(p, new_point, depth + 1)
sample(new_point, q, depth + 1)
else:
x_coords.append(q[0])
y_coords.append(q[1])
f_start = f(self.start)
f_end = f(self.end)
x_coords.append(self.start)
y_coords.append(f_start)
sample(np.array([self.start, f_start]),
np.array([self.end, f_end]), 0)
return (x_coords, y_coords)
def _uniform_sampling(self):
np = import_module('numpy')
if self.only_integers is True:
if self.xscale == 'log':
list_x = np.logspace(int(self.start), int(self.end),
num=int(self.end) - int(self.start) + 1)
else:
list_x = np.linspace(int(self.start), int(self.end),
num=int(self.end) - int(self.start) + 1)
else:
if self.xscale == 'log':
list_x = np.logspace(self.start, self.end, num=self.nb_of_points)
else:
list_x = np.linspace(self.start, self.end, num=self.nb_of_points)
f = vectorized_lambdify([self.var], self.expr)
list_y = f(list_x)
return (list_x, list_y)
class Parametric2DLineSeries(Line2DBaseSeries):
"""Representation for a line consisting of two parametric SymPy expressions
over a range."""
is_parametric = True
def __init__(self, expr_x, expr_y, var_start_end, **kwargs):
super().__init__()
self.expr_x = sympify(expr_x)
self.expr_y = sympify(expr_y)
self.label = kwargs.get('label', None) or \
Tuple(self.expr_x, self.expr_y)
self.var = sympify(var_start_end[0])
self.start = float(var_start_end[1])
self.end = float(var_start_end[2])
self.nb_of_points = kwargs.get('nb_of_points', 300)
self.adaptive = kwargs.get('adaptive', True)
self.depth = kwargs.get('depth', 12)
self.line_color = kwargs.get('line_color', None)
def __str__(self):
return 'parametric cartesian line: (%s, %s) for %s over %s' % (
str(self.expr_x), str(self.expr_y), str(self.var),
str((self.start, self.end)))
def get_parameter_points(self):
np = import_module('numpy')
return np.linspace(self.start, self.end, num=self.nb_of_points)
def _uniform_sampling(self):
param = self.get_parameter_points()
fx = vectorized_lambdify([self.var], self.expr_x)
fy = vectorized_lambdify([self.var], self.expr_y)
list_x = fx(param)
list_y = fy(param)
return (list_x, list_y)
def get_points(self):
""" Return lists of coordinates for plotting. Depending on the
``adaptive`` option, this function will either use an adaptive algorithm
or it will uniformly sample the expression over the provided range.
Returns
=======
x : list
List of x-coordinates
y : list
List of y-coordinates
Explanation
===========
The adaptive sampling is done by recursively checking if three
points are almost collinear. If they are not collinear, then more
points are added between those points.
References
==========
.. [1] Adaptive polygonal approximation of parametric curves,
Luiz Henrique de Figueiredo.
"""
if not self.adaptive:
return self._uniform_sampling()
f_x = lambdify([self.var], self.expr_x)
f_y = lambdify([self.var], self.expr_y)
x_coords = []
y_coords = []
def sample(param_p, param_q, p, q, depth):
""" Samples recursively if three points are almost collinear.
For depth < 6, points are added irrespective of whether they
satisfy the collinearity condition or not. The maximum depth
allowed is 12.
"""
# Randomly sample to avoid aliasing.
np = import_module('numpy')
random = 0.45 + np.random.rand() * 0.1
param_new = param_p + random * (param_q - param_p)
xnew = f_x(param_new)
ynew = f_y(param_new)
new_point = np.array([xnew, ynew])
# Maximum depth
if depth > self.depth:
x_coords.append(q[0])
y_coords.append(q[1])
# Sample irrespective of whether the line is flat till the
# depth of 6. We are not using linspace to avoid aliasing.
elif depth < 6:
sample(param_p, param_new, p, new_point, depth + 1)
sample(param_new, param_q, new_point, q, depth + 1)
# Sample ten points if complex values are encountered
# at both ends. If there is a real value in between, then
# sample those points further.
elif ((p[0] is None and q[1] is None) or
(p[1] is None and q[1] is None)):
param_array = np.linspace(param_p, param_q, 10)
x_array = list(map(f_x, param_array))
y_array = list(map(f_y, param_array))
if not all(x is None and y is None
for x, y in zip(x_array, y_array)):
for i in range(len(y_array) - 1):
if ((x_array[i] is not None and y_array[i] is not None) or
(x_array[i + 1] is not None and y_array[i + 1] is not None)):
point_a = [x_array[i], y_array[i]]
point_b = [x_array[i + 1], y_array[i + 1]]
sample(param_array[i], param_array[i], point_a,
point_b, depth + 1)
# Sample further if one of the end points in None (i.e. a complex
# value) or the three points are not almost collinear.
elif (p[0] is None or p[1] is None
or q[1] is None or q[0] is None
or not flat(p, new_point, q)):
sample(param_p, param_new, p, new_point, depth + 1)
sample(param_new, param_q, new_point, q, depth + 1)
else:
x_coords.append(q[0])
y_coords.append(q[1])
f_start_x = f_x(self.start)
f_start_y = f_y(self.start)
start = [f_start_x, f_start_y]
f_end_x = f_x(self.end)
f_end_y = f_y(self.end)
end = [f_end_x, f_end_y]
x_coords.append(f_start_x)
y_coords.append(f_start_y)
sample(self.start, self.end, start, end, 0)
return x_coords, y_coords
### 3D lines
class Line3DBaseSeries(Line2DBaseSeries):
"""A base class for 3D lines.
Most of the stuff is derived from Line2DBaseSeries."""
is_2Dline = False
is_3Dline = True
_dim = 3
def __init__(self):
super().__init__()
class Parametric3DLineSeries(Line3DBaseSeries):
"""Representation for a 3D line consisting of three parametric SymPy
expressions and a range."""
is_parametric = True
def __init__(self, expr_x, expr_y, expr_z, var_start_end, **kwargs):
super().__init__()
self.expr_x = sympify(expr_x)
self.expr_y = sympify(expr_y)
self.expr_z = sympify(expr_z)
self.label = kwargs.get('label', None) or \
Tuple(self.expr_x, self.expr_y)
self.var = sympify(var_start_end[0])
self.start = float(var_start_end[1])
self.end = float(var_start_end[2])
self.nb_of_points = kwargs.get('nb_of_points', 300)
self.line_color = kwargs.get('line_color', None)
self._xlim = None
self._ylim = None
self._zlim = None
def __str__(self):
return '3D parametric cartesian line: (%s, %s, %s) for %s over %s' % (
str(self.expr_x), str(self.expr_y), str(self.expr_z),
str(self.var), str((self.start, self.end)))
def get_parameter_points(self):
np = import_module('numpy')
return np.linspace(self.start, self.end, num=self.nb_of_points)
def get_points(self):
np = import_module('numpy')
param = self.get_parameter_points()
fx = vectorized_lambdify([self.var], self.expr_x)
fy = vectorized_lambdify([self.var], self.expr_y)
fz = vectorized_lambdify([self.var], self.expr_z)
list_x = fx(param)
list_y = fy(param)
list_z = fz(param)
list_x = np.array(list_x, dtype=np.float64)
list_y = np.array(list_y, dtype=np.float64)
list_z = np.array(list_z, dtype=np.float64)
list_x = np.ma.masked_invalid(list_x)
list_y = np.ma.masked_invalid(list_y)
list_z = np.ma.masked_invalid(list_z)
self._xlim = (np.amin(list_x), np.amax(list_x))
self._ylim = (np.amin(list_y), np.amax(list_y))
self._zlim = (np.amin(list_z), np.amax(list_z))
return list_x, list_y, list_z
### Surfaces
class SurfaceBaseSeries(BaseSeries):
"""A base class for 3D surfaces."""
is_3Dsurface = True
def __init__(self):
super().__init__()
self.surface_color = None
def get_color_array(self):
np = import_module('numpy')
c = self.surface_color
if isinstance(c, Callable):
f = np.vectorize(c)
nargs = arity(c)
if self.is_parametric:
variables = list(map(centers_of_faces, self.get_parameter_meshes()))
if nargs == 1:
return f(variables[0])
elif nargs == 2:
return f(*variables)
variables = list(map(centers_of_faces, self.get_meshes()))
if nargs == 1:
return f(variables[0])
elif nargs == 2:
return f(*variables[:2])
else:
return f(*variables)
else:
if isinstance(self, SurfaceOver2DRangeSeries):
return c*np.ones(min(self.nb_of_points_x, self.nb_of_points_y))
else:
return c*np.ones(min(self.nb_of_points_u, self.nb_of_points_v))
class SurfaceOver2DRangeSeries(SurfaceBaseSeries):
"""Representation for a 3D surface consisting of a SymPy expression and 2D
range."""
def __init__(self, expr, var_start_end_x, var_start_end_y, **kwargs):
super().__init__()
self.expr = sympify(expr)
self.var_x = sympify(var_start_end_x[0])
self.start_x = float(var_start_end_x[1])
self.end_x = float(var_start_end_x[2])
self.var_y = sympify(var_start_end_y[0])
self.start_y = float(var_start_end_y[1])
self.end_y = float(var_start_end_y[2])
self.nb_of_points_x = kwargs.get('nb_of_points_x', 50)
self.nb_of_points_y = kwargs.get('nb_of_points_y', 50)
self.surface_color = kwargs.get('surface_color', None)
self._xlim = (self.start_x, self.end_x)
self._ylim = (self.start_y, self.end_y)
def __str__(self):
return ('cartesian surface: %s for'
' %s over %s and %s over %s') % (
str(self.expr),
str(self.var_x),
str((self.start_x, self.end_x)),
str(self.var_y),
str((self.start_y, self.end_y)))
def get_meshes(self):
np = import_module('numpy')
mesh_x, mesh_y = np.meshgrid(np.linspace(self.start_x, self.end_x,
num=self.nb_of_points_x),
np.linspace(self.start_y, self.end_y,
num=self.nb_of_points_y))
f = vectorized_lambdify((self.var_x, self.var_y), self.expr)
mesh_z = f(mesh_x, mesh_y)
mesh_z = np.array(mesh_z, dtype=np.float64)
mesh_z = np.ma.masked_invalid(mesh_z)
self._zlim = (np.amin(mesh_z), np.amax(mesh_z))
return mesh_x, mesh_y, mesh_z
class ParametricSurfaceSeries(SurfaceBaseSeries):
"""Representation for a 3D surface consisting of three parametric SymPy
expressions and a range."""
is_parametric = True
def __init__(
self, expr_x, expr_y, expr_z, var_start_end_u, var_start_end_v,
**kwargs):
super().__init__()
self.expr_x = sympify(expr_x)
self.expr_y = sympify(expr_y)
self.expr_z = sympify(expr_z)
self.var_u = sympify(var_start_end_u[0])
self.start_u = float(var_start_end_u[1])
self.end_u = float(var_start_end_u[2])
self.var_v = sympify(var_start_end_v[0])
self.start_v = float(var_start_end_v[1])
self.end_v = float(var_start_end_v[2])
self.nb_of_points_u = kwargs.get('nb_of_points_u', 50)
self.nb_of_points_v = kwargs.get('nb_of_points_v', 50)
self.surface_color = kwargs.get('surface_color', None)
def __str__(self):
return ('parametric cartesian surface: (%s, %s, %s) for'
' %s over %s and %s over %s') % (
str(self.expr_x),
str(self.expr_y),
str(self.expr_z),
str(self.var_u),
str((self.start_u, self.end_u)),
str(self.var_v),
str((self.start_v, self.end_v)))
def get_parameter_meshes(self):
np = import_module('numpy')
return np.meshgrid(np.linspace(self.start_u, self.end_u,
num=self.nb_of_points_u),
np.linspace(self.start_v, self.end_v,
num=self.nb_of_points_v))
def get_meshes(self):
np = import_module('numpy')
mesh_u, mesh_v = self.get_parameter_meshes()
fx = vectorized_lambdify((self.var_u, self.var_v), self.expr_x)
fy = vectorized_lambdify((self.var_u, self.var_v), self.expr_y)
fz = vectorized_lambdify((self.var_u, self.var_v), self.expr_z)
mesh_x = fx(mesh_u, mesh_v)
mesh_y = fy(mesh_u, mesh_v)
mesh_z = fz(mesh_u, mesh_v)
mesh_x = np.array(mesh_x, dtype=np.float64)
mesh_y = np.array(mesh_y, dtype=np.float64)
mesh_z = np.array(mesh_z, dtype=np.float64)
mesh_x = np.ma.masked_invalid(mesh_x)
mesh_y = np.ma.masked_invalid(mesh_y)
mesh_z = np.ma.masked_invalid(mesh_z)
self._xlim = (np.amin(mesh_x), np.amax(mesh_x))
self._ylim = (np.amin(mesh_y), np.amax(mesh_y))
self._zlim = (np.amin(mesh_z), np.amax(mesh_z))
return mesh_x, mesh_y, mesh_z
### Contours
class ContourSeries(BaseSeries):
"""Representation for a contour plot."""
# The code is mostly repetition of SurfaceOver2DRange.
# Presently used in contour_plot function
is_contour = True
def __init__(self, expr, var_start_end_x, var_start_end_y):
super().__init__()
self.nb_of_points_x = 50
self.nb_of_points_y = 50
self.expr = sympify(expr)
self.var_x = sympify(var_start_end_x[0])
self.start_x = float(var_start_end_x[1])
self.end_x = float(var_start_end_x[2])
self.var_y = sympify(var_start_end_y[0])
self.start_y = float(var_start_end_y[1])
self.end_y = float(var_start_end_y[2])
self.get_points = self.get_meshes
self._xlim = (self.start_x, self.end_x)
self._ylim = (self.start_y, self.end_y)
def __str__(self):
return ('contour: %s for '
'%s over %s and %s over %s') % (
str(self.expr),
str(self.var_x),
str((self.start_x, self.end_x)),
str(self.var_y),
str((self.start_y, self.end_y)))
def get_meshes(self):
np = import_module('numpy')
mesh_x, mesh_y = np.meshgrid(np.linspace(self.start_x, self.end_x,
num=self.nb_of_points_x),
np.linspace(self.start_y, self.end_y,
num=self.nb_of_points_y))
f = vectorized_lambdify((self.var_x, self.var_y), self.expr)
return (mesh_x, mesh_y, f(mesh_x, mesh_y))
##############################################################################
# Backends
##############################################################################
class BaseBackend:
"""Base class for all backends. A backend represents the plotting library,
which implements the necessary functionalities in order to use SymPy
plotting functions.
How the plotting module works:
1. Whenever a plotting function is called, the provided expressions are
processed and a list of instances of the :class:`BaseSeries` class is
created, containing the necessary information to plot the expressions
(e.g. the expression, ranges, series name, ...). Eventually, these
objects will generate the numerical data to be plotted.
2. A :class:`~.Plot` object is instantiated, which stores the list of
series and the main attributes of the plot (e.g. axis labels, title, ...).
3. When the ``show`` command is executed, a new backend is instantiated,
which loops through each series object to generate and plot the
numerical data. The backend is also going to set the axis labels, title,
..., according to the values stored in the Plot instance.
The backend should check if it supports the data series that it is given
(e.g. :class:`TextBackend` supports only :class:`LineOver1DRangeSeries`).
It is the backend responsibility to know how to use the class of data series
that it's given. Note that the current implementation of the ``*Series``
classes is "matplotlib-centric": the numerical data returned by the
``get_points`` and ``get_meshes`` methods is meant to be used directly by
Matplotlib. Therefore, the new backend will have to pre-process the
numerical data to make it compatible with the chosen plotting library.
Keep in mind that future SymPy versions may improve the ``*Series`` classes
in order to return numerical data "non-matplotlib-centric", hence if you code
a new backend you have the responsibility to check if its working on each
SymPy release.
Please explore the :class:`MatplotlibBackend` source code to understand how a
backend should be coded.
Methods
=======
In order to be used by SymPy plotting functions, a backend must implement
the following methods:
* show(self): used to loop over the data series, generate the numerical
data, plot it and set the axis labels, title, ...
* save(self, path): used to save the current plot to the specified file
path.
* close(self): used to close the current plot backend (note: some plotting
library does not support this functionality. In that case, just raise a
warning).
See also
========
MatplotlibBackend
"""
def __init__(self, parent):
super().__init__()
self.parent = parent
def show(self):
raise NotImplementedError
def save(self, path):
raise NotImplementedError
def close(self):
raise NotImplementedError
# Don't have to check for the success of importing matplotlib in each case;
# we will only be using this backend if we can successfully import matploblib
class MatplotlibBackend(BaseBackend):
""" This class implements the functionalities to use Matplotlib with SymPy
plotting functions.
"""
def __init__(self, parent):
super().__init__(parent)
self.matplotlib = import_module('matplotlib',
import_kwargs={'fromlist': ['pyplot', 'cm', 'collections']},
min_module_version='1.1.0', catch=(RuntimeError,))
self.plt = self.matplotlib.pyplot
self.cm = self.matplotlib.cm
self.LineCollection = self.matplotlib.collections.LineCollection
aspect = getattr(self.parent, 'aspect_ratio', 'auto')
if aspect != 'auto':
aspect = float(aspect[1]) / aspect[0]
if isinstance(self.parent, Plot):
nrows, ncolumns = 1, 1
series_list = [self.parent._series]
elif isinstance(self.parent, PlotGrid):
nrows, ncolumns = self.parent.nrows, self.parent.ncolumns
series_list = self.parent._series
self.ax = []
self.fig = self.plt.figure(figsize=parent.size)
for i, series in enumerate(series_list):
are_3D = [s.is_3D for s in series]
if any(are_3D) and not all(are_3D):
raise ValueError('The matplotlib backend cannot mix 2D and 3D.')
elif all(are_3D):
# mpl_toolkits.mplot3d is necessary for
# projection='3d'
mpl_toolkits = import_module('mpl_toolkits', # noqa
import_kwargs={'fromlist': ['mplot3d']})
self.ax.append(self.fig.add_subplot(nrows, ncolumns, i + 1, projection='3d', aspect=aspect))
elif not any(are_3D):
self.ax.append(self.fig.add_subplot(nrows, ncolumns, i + 1, aspect=aspect))
self.ax[i].spines['left'].set_position('zero')
self.ax[i].spines['right'].set_color('none')
self.ax[i].spines['bottom'].set_position('zero')
self.ax[i].spines['top'].set_color('none')
self.ax[i].xaxis.set_ticks_position('bottom')
self.ax[i].yaxis.set_ticks_position('left')
@staticmethod
def get_segments(x, y, z=None):
""" Convert two list of coordinates to a list of segments to be used
with Matplotlib's :external:class:`~matplotlib.collections.LineCollection`.
Parameters
==========
x : list
List of x-coordinates
y : list
List of y-coordinates
z : list
List of z-coordinates for a 3D line.
"""
np = import_module('numpy')
if z is not None:
dim = 3
points = (x, y, z)
else:
dim = 2
points = (x, y)
points = np.ma.array(points).T.reshape(-1, 1, dim)
return np.ma.concatenate([points[:-1], points[1:]], axis=1)
def _process_series(self, series, ax, parent):
np = import_module('numpy')
mpl_toolkits = import_module(
'mpl_toolkits', import_kwargs={'fromlist': ['mplot3d']})
# XXX Workaround for matplotlib issue
# https://github.com/matplotlib/matplotlib/issues/17130
xlims, ylims, zlims = [], [], []
for s in series:
# Create the collections
if s.is_2Dline:
x, y = s.get_data()
if (isinstance(s.line_color, (int, float)) or
callable(s.line_color)):
segments = self.get_segments(x, y)
collection = self.LineCollection(segments)
collection.set_array(s.get_color_array())
ax.add_collection(collection)
else:
lbl = _str_or_latex(s.label)
line, = ax.plot(x, y, label=lbl, color=s.line_color)
elif s.is_contour:
ax.contour(*s.get_meshes())
elif s.is_3Dline:
x, y, z = s.get_data()
if (isinstance(s.line_color, (int, float)) or
callable(s.line_color)):
art3d = mpl_toolkits.mplot3d.art3d
segments = self.get_segments(x, y, z)
collection = art3d.Line3DCollection(segments)
collection.set_array(s.get_color_array())
ax.add_collection(collection)
else:
lbl = _str_or_latex(s.label)
ax.plot(x, y, z, label=lbl, color=s.line_color)
xlims.append(s._xlim)
ylims.append(s._ylim)
zlims.append(s._zlim)
elif s.is_3Dsurface:
x, y, z = s.get_meshes()
collection = ax.plot_surface(x, y, z,
cmap=getattr(self.cm, 'viridis', self.cm.jet),
rstride=1, cstride=1, linewidth=0.1)
if isinstance(s.surface_color, (float, int, Callable)):
color_array = s.get_color_array()
color_array = color_array.reshape(color_array.size)
collection.set_array(color_array)
else:
collection.set_color(s.surface_color)
xlims.append(s._xlim)
ylims.append(s._ylim)
zlims.append(s._zlim)
elif s.is_implicit:
points = s.get_raster()
if len(points) == 2:
# interval math plotting
x, y = _matplotlib_list(points[0])
ax.fill(x, y, facecolor=s.line_color, edgecolor='None')
else:
# use contourf or contour depending on whether it is
# an inequality or equality.
# XXX: ``contour`` plots multiple lines. Should be fixed.
ListedColormap = self.matplotlib.colors.ListedColormap
colormap = ListedColormap(["white", s.line_color])
xarray, yarray, zarray, plot_type = points
if plot_type == 'contour':
ax.contour(xarray, yarray, zarray, cmap=colormap)
else:
ax.contourf(xarray, yarray, zarray, cmap=colormap)
else:
raise NotImplementedError(
'{} is not supported in the SymPy plotting module '
'with matplotlib backend. Please report this issue.'
.format(ax))
Axes3D = mpl_toolkits.mplot3d.Axes3D
if not isinstance(ax, Axes3D):
ax.autoscale_view(
scalex=ax.get_autoscalex_on(),
scaley=ax.get_autoscaley_on())
else:
# XXX Workaround for matplotlib issue
# https://github.com/matplotlib/matplotlib/issues/17130
if xlims:
xlims = np.array(xlims)
xlim = (np.amin(xlims[:, 0]), np.amax(xlims[:, 1]))
ax.set_xlim(xlim)
else:
ax.set_xlim([0, 1])
if ylims:
ylims = np.array(ylims)
ylim = (np.amin(ylims[:, 0]), np.amax(ylims[:, 1]))
ax.set_ylim(ylim)
else:
ax.set_ylim([0, 1])
if zlims:
zlims = np.array(zlims)
zlim = (np.amin(zlims[:, 0]), np.amax(zlims[:, 1]))
ax.set_zlim(zlim)
else:
ax.set_zlim([0, 1])
# Set global options.
# TODO The 3D stuff
# XXX The order of those is important.
if parent.xscale and not isinstance(ax, Axes3D):
ax.set_xscale(parent.xscale)
if parent.yscale and not isinstance(ax, Axes3D):
ax.set_yscale(parent.yscale)
if not isinstance(ax, Axes3D) or self.matplotlib.__version__ >= '1.2.0': # XXX in the distant future remove this check
ax.set_autoscale_on(parent.autoscale)
if parent.axis_center:
val = parent.axis_center
if isinstance(ax, Axes3D):
pass
elif val == 'center':
ax.spines['left'].set_position('center')
ax.spines['bottom'].set_position('center')
elif val == 'auto':
xl, xh = ax.get_xlim()
yl, yh = ax.get_ylim()
pos_left = ('data', 0) if xl*xh <= 0 else 'center'
pos_bottom = ('data', 0) if yl*yh <= 0 else 'center'
ax.spines['left'].set_position(pos_left)
ax.spines['bottom'].set_position(pos_bottom)
else:
ax.spines['left'].set_position(('data', val[0]))
ax.spines['bottom'].set_position(('data', val[1]))
if not parent.axis:
ax.set_axis_off()
if parent.legend:
if ax.legend():
ax.legend_.set_visible(parent.legend)
if parent.margin:
ax.set_xmargin(parent.margin)
ax.set_ymargin(parent.margin)
if parent.title:
ax.set_title(parent.title)
if parent.xlabel:
xlbl = _str_or_latex(parent.xlabel)
ax.set_xlabel(xlbl, position=(1, 0))
if parent.ylabel:
ylbl = _str_or_latex(parent.ylabel)
ax.set_ylabel(ylbl, position=(0, 1))
if isinstance(ax, Axes3D) and parent.zlabel:
zlbl = _str_or_latex(parent.zlabel)
ax.set_zlabel(zlbl, position=(0, 1))
if parent.annotations:
for a in parent.annotations:
ax.annotate(**a)
if parent.markers:
for marker in parent.markers:
# make a copy of the marker dictionary
# so that it doesn't get altered
m = marker.copy()
args = m.pop('args')
ax.plot(*args, **m)
if parent.rectangles:
for r in parent.rectangles:
rect = self.matplotlib.patches.Rectangle(**r)
ax.add_patch(rect)
if parent.fill:
ax.fill_between(**parent.fill)
# xlim and ylim should always be set at last so that plot limits
# doesn't get altered during the process.
if parent.xlim:
ax.set_xlim(parent.xlim)
if parent.ylim:
ax.set_ylim(parent.ylim)
def process_series(self):
"""
Iterates over every ``Plot`` object and further calls
_process_series()
"""
parent = self.parent
if isinstance(parent, Plot):
series_list = [parent._series]
else:
series_list = parent._series
for i, (series, ax) in enumerate(zip(series_list, self.ax)):
if isinstance(self.parent, PlotGrid):
parent = self.parent.args[i]
self._process_series(series, ax, parent)
def show(self):
self.process_series()
#TODO after fixing https://github.com/ipython/ipython/issues/1255
# you can uncomment the next line and remove the pyplot.show() call
#self.fig.show()
if _show:
self.fig.tight_layout()
self.plt.show()
else:
self.close()
def save(self, path):
self.process_series()
self.fig.savefig(path)
def close(self):
self.plt.close(self.fig)
class TextBackend(BaseBackend):
def __init__(self, parent):
super().__init__(parent)
def show(self):
if not _show:
return
if len(self.parent._series) != 1:
raise ValueError(
'The TextBackend supports only one graph per Plot.')
elif not isinstance(self.parent._series[0], LineOver1DRangeSeries):
raise ValueError(
'The TextBackend supports only expressions over a 1D range')
else:
ser = self.parent._series[0]
textplot(ser.expr, ser.start, ser.end)
def close(self):
pass
class DefaultBackend(BaseBackend):
def __new__(cls, parent):
matplotlib = import_module('matplotlib', min_module_version='1.1.0', catch=(RuntimeError,))
if matplotlib:
return MatplotlibBackend(parent)
else:
return TextBackend(parent)
plot_backends = {
'matplotlib': MatplotlibBackend,
'text': TextBackend,
'default': DefaultBackend
}
##############################################################################
# Finding the centers of line segments or mesh faces
##############################################################################
def centers_of_segments(array):
np = import_module('numpy')
return np.mean(np.vstack((array[:-1], array[1:])), 0)
def centers_of_faces(array):
np = import_module('numpy')
return np.mean(np.dstack((array[:-1, :-1],
array[1:, :-1],
array[:-1, 1:],
array[:-1, :-1],
)), 2)
def flat(x, y, z, eps=1e-3):
"""Checks whether three points are almost collinear"""
np = import_module('numpy')
# Workaround plotting piecewise (#8577):
# workaround for `lambdify` in `.experimental_lambdify` fails
# to return numerical values in some cases. Lower-level fix
# in `lambdify` is possible.
vector_a = (x - y).astype(np.float64)
vector_b = (z - y).astype(np.float64)
dot_product = np.dot(vector_a, vector_b)
vector_a_norm = np.linalg.norm(vector_a)
vector_b_norm = np.linalg.norm(vector_b)
cos_theta = dot_product / (vector_a_norm * vector_b_norm)
return abs(cos_theta + 1) < eps
def _matplotlib_list(interval_list):
"""
Returns lists for matplotlib ``fill`` command from a list of bounding
rectangular intervals
"""
xlist = []
ylist = []
if len(interval_list):
for intervals in interval_list:
intervalx = intervals[0]
intervaly = intervals[1]
xlist.extend([intervalx.start, intervalx.start,
intervalx.end, intervalx.end, None])
ylist.extend([intervaly.start, intervaly.end,
intervaly.end, intervaly.start, None])
else:
#XXX Ugly hack. Matplotlib does not accept empty lists for ``fill``
xlist.extend((None, None, None, None))
ylist.extend((None, None, None, None))
return xlist, ylist
####New API for plotting module ####
# TODO: Add color arrays for plots.
# TODO: Add more plotting options for 3d plots.
# TODO: Adaptive sampling for 3D plots.
def plot(*args, show=True, **kwargs):
"""Plots a function of a single variable as a curve.
Parameters
==========
args :
The first argument is the expression representing the function
of single variable to be plotted.
The last argument is a 3-tuple denoting the range of the free
variable. e.g. ``(x, 0, 5)``
Typical usage examples are in the following:
- Plotting a single expression with a single range.
``plot(expr, range, **kwargs)``
- Plotting a single expression with the default range (-10, 10).
``plot(expr, **kwargs)``
- Plotting multiple expressions with a single range.
``plot(expr1, expr2, ..., range, **kwargs)``
- Plotting multiple expressions with multiple ranges.
``plot((expr1, range1), (expr2, range2), ..., **kwargs)``
It is best practice to specify range explicitly because default
range may change in the future if a more advanced default range
detection algorithm is implemented.
show : bool, optional
The default value is set to ``True``. Set show to ``False`` and
the function will not display the plot. The returned instance of
the ``Plot`` class can then be used to save or display the plot
by calling the ``save()`` and ``show()`` methods respectively.
line_color : string, or float, or function, optional
Specifies the color for the plot.
See ``Plot`` to see how to set color for the plots.
Note that by setting ``line_color``, it would be applied simultaneously
to all the series.
title : str, optional
Title of the plot. It is set to the latex representation of
the expression, if the plot has only one expression.
label : str, optional
The label of the expression in the plot. It will be used when
called with ``legend``. Default is the name of the expression.
e.g. ``sin(x)``
xlabel : str or expression, optional
Label for the x-axis.
ylabel : str or expression, optional
Label for the y-axis.
xscale : 'linear' or 'log', optional
Sets the scaling of the x-axis.
yscale : 'linear' or 'log', optional
Sets the scaling of the y-axis.
axis_center : (float, float), optional
Tuple of two floats denoting the coordinates of the center or
{'center', 'auto'}
xlim : (float, float), optional
Denotes the x-axis limits, ``(min, max)```.
ylim : (float, float), optional
Denotes the y-axis limits, ``(min, max)```.
annotations : list, optional
A list of dictionaries specifying the type of annotation
required. The keys in the dictionary should be equivalent
to the arguments of the :external:mod:`matplotlib`'s
:external:meth:`~matplotlib.axes.Axes.annotate` method.
markers : list, optional
A list of dictionaries specifying the type the markers required.
The keys in the dictionary should be equivalent to the arguments
of the :external:mod:`matplotlib`'s :external:func:`~matplotlib.pyplot.plot()` function
along with the marker related keyworded arguments.
rectangles : list, optional
A list of dictionaries specifying the dimensions of the
rectangles to be plotted. The keys in the dictionary should be
equivalent to the arguments of the :external:mod:`matplotlib`'s
:external:class:`~matplotlib.patches.Rectangle` class.
fill : dict, optional
A dictionary specifying the type of color filling required in
the plot. The keys in the dictionary should be equivalent to the
arguments of the :external:mod:`matplotlib`'s
:external:meth:`~matplotlib.axes.Axes.fill_between` method.
adaptive : bool, optional
The default value is set to ``True``. Set adaptive to ``False``
and specify ``nb_of_points`` if uniform sampling is required.
The plotting uses an adaptive algorithm which samples
recursively to accurately plot. The adaptive algorithm uses a
random point near the midpoint of two points that has to be
further sampled. Hence the same plots can appear slightly
different.
depth : int, optional
Recursion depth of the adaptive algorithm. A depth of value
`n` samples a maximum of `2^{n}` points.
If the ``adaptive`` flag is set to ``False``, this will be
ignored.
nb_of_points : int, optional
Used when the ``adaptive`` is set to ``False``. The function
is uniformly sampled at ``nb_of_points`` number of points.
If the ``adaptive`` flag is set to ``True``, this will be
ignored.
size : (float, float), optional
A tuple in the form (width, height) in inches to specify the size of
the overall figure. The default value is set to ``None``, meaning
the size will be set by the default backend.
Examples
========
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> from sympy import symbols
>>> from sympy.plotting import plot
>>> x = symbols('x')
Single Plot
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> plot(x**2, (x, -5, 5))
Plot object containing:
[0]: cartesian line: x**2 for x over (-5.0, 5.0)
Multiple plots with single range.
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> plot(x, x**2, x**3, (x, -5, 5))
Plot object containing:
[0]: cartesian line: x for x over (-5.0, 5.0)
[1]: cartesian line: x**2 for x over (-5.0, 5.0)
[2]: cartesian line: x**3 for x over (-5.0, 5.0)
Multiple plots with different ranges.
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> plot((x**2, (x, -6, 6)), (x, (x, -5, 5)))
Plot object containing:
[0]: cartesian line: x**2 for x over (-6.0, 6.0)
[1]: cartesian line: x for x over (-5.0, 5.0)
No adaptive sampling.
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> plot(x**2, adaptive=False, nb_of_points=400)
Plot object containing:
[0]: cartesian line: x**2 for x over (-10.0, 10.0)
See Also
========
Plot, LineOver1DRangeSeries
"""
args = list(map(sympify, args))
free = set()
for a in args:
if isinstance(a, Expr):
free |= a.free_symbols
if len(free) > 1:
raise ValueError(
'The same variable should be used in all '
'univariate expressions being plotted.')
x = free.pop() if free else Symbol('x')
kwargs.setdefault('xlabel', x)
kwargs.setdefault('ylabel', Function('f')(x))
series = []
plot_expr = check_arguments(args, 1, 1)
series = [LineOver1DRangeSeries(*arg, **kwargs) for arg in plot_expr]
plots = Plot(*series, **kwargs)
if show:
plots.show()
return plots
def plot_parametric(*args, show=True, **kwargs):
"""
Plots a 2D parametric curve.
Parameters
==========
args
Common specifications are:
- Plotting a single parametric curve with a range
``plot_parametric((expr_x, expr_y), range)``
- Plotting multiple parametric curves with the same range
``plot_parametric((expr_x, expr_y), ..., range)``
- Plotting multiple parametric curves with different ranges
``plot_parametric((expr_x, expr_y, range), ...)``
``expr_x`` is the expression representing $x$ component of the
parametric function.
``expr_y`` is the expression representing $y$ component of the
parametric function.
``range`` is a 3-tuple denoting the parameter symbol, start and
stop. For example, ``(u, 0, 5)``.
If the range is not specified, then a default range of (-10, 10)
is used.
However, if the arguments are specified as
``(expr_x, expr_y, range), ...``, you must specify the ranges
for each expressions manually.
Default range may change in the future if a more advanced
algorithm is implemented.
adaptive : bool, optional
Specifies whether to use the adaptive sampling or not.
The default value is set to ``True``. Set adaptive to ``False``
and specify ``nb_of_points`` if uniform sampling is required.
depth : int, optional
The recursion depth of the adaptive algorithm. A depth of
value $n$ samples a maximum of $2^n$ points.
nb_of_points : int, optional
Used when the ``adaptive`` flag is set to ``False``.
Specifies the number of the points used for the uniform
sampling.
line_color : string, or float, or function, optional
Specifies the color for the plot.
See ``Plot`` to see how to set color for the plots.
Note that by setting ``line_color``, it would be applied simultaneously
to all the series.
label : str, optional
The label of the expression in the plot. It will be used when
called with ``legend``. Default is the name of the expression.
e.g. ``sin(x)``
xlabel : str, optional
Label for the x-axis.
ylabel : str, optional
Label for the y-axis.
xscale : 'linear' or 'log', optional
Sets the scaling of the x-axis.
yscale : 'linear' or 'log', optional
Sets the scaling of the y-axis.
axis_center : (float, float), optional
Tuple of two floats denoting the coordinates of the center or
{'center', 'auto'}
xlim : (float, float), optional
Denotes the x-axis limits, ``(min, max)```.
ylim : (float, float), optional
Denotes the y-axis limits, ``(min, max)```.
size : (float, float), optional
A tuple in the form (width, height) in inches to specify the size of
the overall figure. The default value is set to ``None``, meaning
the size will be set by the default backend.
Examples
========
.. plot::
:context: reset
:format: doctest
:include-source: True
>>> from sympy import plot_parametric, symbols, cos, sin
>>> u = symbols('u')
A parametric plot with a single expression:
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> plot_parametric((cos(u), sin(u)), (u, -5, 5))
Plot object containing:
[0]: parametric cartesian line: (cos(u), sin(u)) for u over (-5.0, 5.0)
A parametric plot with multiple expressions with the same range:
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> plot_parametric((cos(u), sin(u)), (u, cos(u)), (u, -10, 10))
Plot object containing:
[0]: parametric cartesian line: (cos(u), sin(u)) for u over (-10.0, 10.0)
[1]: parametric cartesian line: (u, cos(u)) for u over (-10.0, 10.0)
A parametric plot with multiple expressions with different ranges
for each curve:
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> plot_parametric((cos(u), sin(u), (u, -5, 5)),
... (cos(u), u, (u, -5, 5)))
Plot object containing:
[0]: parametric cartesian line: (cos(u), sin(u)) for u over (-5.0, 5.0)
[1]: parametric cartesian line: (cos(u), u) for u over (-5.0, 5.0)
Notes
=====
The plotting uses an adaptive algorithm which samples recursively to
accurately plot the curve. The adaptive algorithm uses a random point
near the midpoint of two points that has to be further sampled.
Hence, repeating the same plot command can give slightly different
results because of the random sampling.
If there are multiple plots, then the same optional arguments are
applied to all the plots drawn in the same canvas. If you want to
set these options separately, you can index the returned ``Plot``
object and set it.
For example, when you specify ``line_color`` once, it would be
applied simultaneously to both series.
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> from sympy import pi
>>> expr1 = (u, cos(2*pi*u)/2 + 1/2)
>>> expr2 = (u, sin(2*pi*u)/2 + 1/2)
>>> p = plot_parametric(expr1, expr2, (u, 0, 1), line_color='blue')
If you want to specify the line color for the specific series, you
should index each item and apply the property manually.
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> p[0].line_color = 'red'
>>> p.show()
See Also
========
Plot, Parametric2DLineSeries
"""
args = list(map(sympify, args))
series = []
plot_expr = check_arguments(args, 2, 1)
series = [Parametric2DLineSeries(*arg, **kwargs) for arg in plot_expr]
plots = Plot(*series, **kwargs)
if show:
plots.show()
return plots
def plot3d_parametric_line(*args, show=True, **kwargs):
"""
Plots a 3D parametric line plot.
Usage
=====
Single plot:
``plot3d_parametric_line(expr_x, expr_y, expr_z, range, **kwargs)``
If the range is not specified, then a default range of (-10, 10) is used.
Multiple plots.
``plot3d_parametric_line((expr_x, expr_y, expr_z, range), ..., **kwargs)``
Ranges have to be specified for every expression.
Default range may change in the future if a more advanced default range
detection algorithm is implemented.
Arguments
=========
expr_x : Expression representing the function along x.
expr_y : Expression representing the function along y.
expr_z : Expression representing the function along z.
range : (:class:`~.Symbol`, float, float)
A 3-tuple denoting the range of the parameter variable, e.g., (u, 0, 5).
Keyword Arguments
=================
Arguments for ``Parametric3DLineSeries`` class.
nb_of_points : The range is uniformly sampled at ``nb_of_points``
number of points.
Aesthetics:
line_color : string, or float, or function, optional
Specifies the color for the plot.
See ``Plot`` to see how to set color for the plots.
Note that by setting ``line_color``, it would be applied simultaneously
to all the series.
label : str
The label to the plot. It will be used when called with ``legend=True``
to denote the function with the given label in the plot.
If there are multiple plots, then the same series arguments are applied to
all the plots. If you want to set these options separately, you can index
the returned ``Plot`` object and set it.
Arguments for ``Plot`` class.
title : str
Title of the plot.
size : (float, float), optional
A tuple in the form (width, height) in inches to specify the size of
the overall figure. The default value is set to ``None``, meaning
the size will be set by the default backend.
Examples
========
.. plot::
:context: reset
:format: doctest
:include-source: True
>>> from sympy import symbols, cos, sin
>>> from sympy.plotting import plot3d_parametric_line
>>> u = symbols('u')
Single plot.
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> plot3d_parametric_line(cos(u), sin(u), u, (u, -5, 5))
Plot object containing:
[0]: 3D parametric cartesian line: (cos(u), sin(u), u) for u over (-5.0, 5.0)
Multiple plots.
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> plot3d_parametric_line((cos(u), sin(u), u, (u, -5, 5)),
... (sin(u), u**2, u, (u, -5, 5)))
Plot object containing:
[0]: 3D parametric cartesian line: (cos(u), sin(u), u) for u over (-5.0, 5.0)
[1]: 3D parametric cartesian line: (sin(u), u**2, u) for u over (-5.0, 5.0)
See Also
========
Plot, Parametric3DLineSeries
"""
args = list(map(sympify, args))
series = []
plot_expr = check_arguments(args, 3, 1)
series = [Parametric3DLineSeries(*arg, **kwargs) for arg in plot_expr]
kwargs.setdefault("xlabel", "x")
kwargs.setdefault("ylabel", "y")
kwargs.setdefault("zlabel", "z")
plots = Plot(*series, **kwargs)
if show:
plots.show()
return plots
def plot3d(*args, show=True, **kwargs):
"""
Plots a 3D surface plot.
Usage
=====
Single plot
``plot3d(expr, range_x, range_y, **kwargs)``
If the ranges are not specified, then a default range of (-10, 10) is used.
Multiple plot with the same range.
``plot3d(expr1, expr2, range_x, range_y, **kwargs)``
If the ranges are not specified, then a default range of (-10, 10) is used.
Multiple plots with different ranges.
``plot3d((expr1, range_x, range_y), (expr2, range_x, range_y), ..., **kwargs)``
Ranges have to be specified for every expression.
Default range may change in the future if a more advanced default range
detection algorithm is implemented.
Arguments
=========
expr : Expression representing the function along x.
range_x : (:class:`~.Symbol`, float, float)
A 3-tuple denoting the range of the x variable, e.g. (x, 0, 5).
range_y : (:class:`~.Symbol`, float, float)
A 3-tuple denoting the range of the y variable, e.g. (y, 0, 5).
Keyword Arguments
=================
Arguments for ``SurfaceOver2DRangeSeries`` class:
nb_of_points_x : int
The x range is sampled uniformly at ``nb_of_points_x`` of points.
nb_of_points_y : int
The y range is sampled uniformly at ``nb_of_points_y`` of points.
Aesthetics:
surface_color : Function which returns a float
Specifies the color for the surface of the plot.
See :class:`~.Plot` for more details.
If there are multiple plots, then the same series arguments are applied to
all the plots. If you want to set these options separately, you can index
the returned ``Plot`` object and set it.
Arguments for ``Plot`` class:
title : str
Title of the plot.
size : (float, float), optional
A tuple in the form (width, height) in inches to specify the size of the
overall figure. The default value is set to ``None``, meaning the size will
be set by the default backend.
Examples
========
.. plot::
:context: reset
:format: doctest
:include-source: True
>>> from sympy import symbols
>>> from sympy.plotting import plot3d
>>> x, y = symbols('x y')
Single plot
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> plot3d(x*y, (x, -5, 5), (y, -5, 5))
Plot object containing:
[0]: cartesian surface: x*y for x over (-5.0, 5.0) and y over (-5.0, 5.0)
Multiple plots with same range
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> plot3d(x*y, -x*y, (x, -5, 5), (y, -5, 5))
Plot object containing:
[0]: cartesian surface: x*y for x over (-5.0, 5.0) and y over (-5.0, 5.0)
[1]: cartesian surface: -x*y for x over (-5.0, 5.0) and y over (-5.0, 5.0)
Multiple plots with different ranges.
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> plot3d((x**2 + y**2, (x, -5, 5), (y, -5, 5)),
... (x*y, (x, -3, 3), (y, -3, 3)))
Plot object containing:
[0]: cartesian surface: x**2 + y**2 for x over (-5.0, 5.0) and y over (-5.0, 5.0)
[1]: cartesian surface: x*y for x over (-3.0, 3.0) and y over (-3.0, 3.0)
See Also
========
Plot, SurfaceOver2DRangeSeries
"""
args = list(map(sympify, args))
series = []
plot_expr = check_arguments(args, 1, 2)
series = [SurfaceOver2DRangeSeries(*arg, **kwargs) for arg in plot_expr]
kwargs.setdefault("xlabel", series[0].var_x)
kwargs.setdefault("ylabel", series[0].var_y)
kwargs.setdefault("zlabel", Function('f')(series[0].var_x, series[0].var_y))
plots = Plot(*series, **kwargs)
if show:
plots.show()
return plots
def plot3d_parametric_surface(*args, show=True, **kwargs):
"""
Plots a 3D parametric surface plot.
Explanation
===========
Single plot.
``plot3d_parametric_surface(expr_x, expr_y, expr_z, range_u, range_v, **kwargs)``
If the ranges is not specified, then a default range of (-10, 10) is used.
Multiple plots.
``plot3d_parametric_surface((expr_x, expr_y, expr_z, range_u, range_v), ..., **kwargs)``
Ranges have to be specified for every expression.
Default range may change in the future if a more advanced default range
detection algorithm is implemented.
Arguments
=========
expr_x : Expression representing the function along ``x``.
expr_y : Expression representing the function along ``y``.
expr_z : Expression representing the function along ``z``.
range_u : (:class:`~.Symbol`, float, float)
A 3-tuple denoting the range of the u variable, e.g. (u, 0, 5).
range_v : (:class:`~.Symbol`, float, float)
A 3-tuple denoting the range of the v variable, e.g. (v, 0, 5).
Keyword Arguments
=================
Arguments for ``ParametricSurfaceSeries`` class:
nb_of_points_u : int
The ``u`` range is sampled uniformly at ``nb_of_points_v`` of points
nb_of_points_y : int
The ``v`` range is sampled uniformly at ``nb_of_points_y`` of points
Aesthetics:
surface_color : Function which returns a float
Specifies the color for the surface of the plot. See
:class:`~Plot` for more details.
If there are multiple plots, then the same series arguments are applied for
all the plots. If you want to set these options separately, you can index
the returned ``Plot`` object and set it.
Arguments for ``Plot`` class:
title : str
Title of the plot.
size : (float, float), optional
A tuple in the form (width, height) in inches to specify the size of the
overall figure. The default value is set to ``None``, meaning the size will
be set by the default backend.
Examples
========
.. plot::
:context: reset
:format: doctest
:include-source: True
>>> from sympy import symbols, cos, sin
>>> from sympy.plotting import plot3d_parametric_surface
>>> u, v = symbols('u v')
Single plot.
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> plot3d_parametric_surface(cos(u + v), sin(u - v), u - v,
... (u, -5, 5), (v, -5, 5))
Plot object containing:
[0]: parametric cartesian surface: (cos(u + v), sin(u - v), u - v) for u over (-5.0, 5.0) and v over (-5.0, 5.0)
See Also
========
Plot, ParametricSurfaceSeries
"""
args = list(map(sympify, args))
series = []
plot_expr = check_arguments(args, 3, 2)
series = [ParametricSurfaceSeries(*arg, **kwargs) for arg in plot_expr]
kwargs.setdefault("xlabel", "x")
kwargs.setdefault("ylabel", "y")
kwargs.setdefault("zlabel", "z")
plots = Plot(*series, **kwargs)
if show:
plots.show()
return plots
def plot_contour(*args, show=True, **kwargs):
"""
Draws contour plot of a function
Usage
=====
Single plot
``plot_contour(expr, range_x, range_y, **kwargs)``
If the ranges are not specified, then a default range of (-10, 10) is used.
Multiple plot with the same range.
``plot_contour(expr1, expr2, range_x, range_y, **kwargs)``
If the ranges are not specified, then a default range of (-10, 10) is used.
Multiple plots with different ranges.
``plot_contour((expr1, range_x, range_y), (expr2, range_x, range_y), ..., **kwargs)``
Ranges have to be specified for every expression.
Default range may change in the future if a more advanced default range
detection algorithm is implemented.
Arguments
=========
expr : Expression representing the function along x.
range_x : (:class:`Symbol`, float, float)
A 3-tuple denoting the range of the x variable, e.g. (x, 0, 5).
range_y : (:class:`Symbol`, float, float)
A 3-tuple denoting the range of the y variable, e.g. (y, 0, 5).
Keyword Arguments
=================
Arguments for ``ContourSeries`` class:
nb_of_points_x : int
The x range is sampled uniformly at ``nb_of_points_x`` of points.
nb_of_points_y : int
The y range is sampled uniformly at ``nb_of_points_y`` of points.
Aesthetics:
surface_color : Function which returns a float
Specifies the color for the surface of the plot. See
:class:`sympy.plotting.Plot` for more details.
If there are multiple plots, then the same series arguments are applied to
all the plots. If you want to set these options separately, you can index
the returned ``Plot`` object and set it.
Arguments for ``Plot`` class:
title : str
Title of the plot.
size : (float, float), optional
A tuple in the form (width, height) in inches to specify the size of
the overall figure. The default value is set to ``None``, meaning
the size will be set by the default backend.
See Also
========
Plot, ContourSeries
"""
args = list(map(sympify, args))
plot_expr = check_arguments(args, 1, 2)
series = [ContourSeries(*arg) for arg in plot_expr]
plot_contours = Plot(*series, **kwargs)
if len(plot_expr[0].free_symbols) > 2:
raise ValueError('Contour Plot cannot Plot for more than two variables.')
if show:
plot_contours.show()
return plot_contours
def check_arguments(args, expr_len, nb_of_free_symbols):
"""
Checks the arguments and converts into tuples of the
form (exprs, ranges).
Examples
========
.. plot::
:context: reset
:format: doctest
:include-source: True
>>> from sympy import cos, sin, symbols
>>> from sympy.plotting.plot import check_arguments
>>> x = symbols('x')
>>> check_arguments([cos(x), sin(x)], 2, 1)
[(cos(x), sin(x), (x, -10, 10))]
>>> check_arguments([x, x**2], 1, 1)
[(x, (x, -10, 10)), (x**2, (x, -10, 10))]
"""
if not args:
return []
if expr_len > 1 and isinstance(args[0], Expr):
# Multiple expressions same range.
# The arguments are tuples when the expression length is
# greater than 1.
if len(args) < expr_len:
raise ValueError("len(args) should not be less than expr_len")
for i in range(len(args)):
if isinstance(args[i], Tuple):
break
else:
i = len(args) + 1
exprs = Tuple(*args[:i])
free_symbols = list(set().union(*[e.free_symbols for e in exprs]))
if len(args) == expr_len + nb_of_free_symbols:
#Ranges given
plots = [exprs + Tuple(*args[expr_len:])]
else:
default_range = Tuple(-10, 10)
ranges = []
for symbol in free_symbols:
ranges.append(Tuple(symbol) + default_range)
for i in range(len(free_symbols) - nb_of_free_symbols):
ranges.append(Tuple(Dummy()) + default_range)
plots = [exprs + Tuple(*ranges)]
return plots
if isinstance(args[0], Expr) or (isinstance(args[0], Tuple) and
len(args[0]) == expr_len and
expr_len != 3):
# Cannot handle expressions with number of expression = 3. It is
# not possible to differentiate between expressions and ranges.
#Series of plots with same range
for i in range(len(args)):
if isinstance(args[i], Tuple) and len(args[i]) != expr_len:
break
if not isinstance(args[i], Tuple):
args[i] = Tuple(args[i])
else:
i = len(args) + 1
exprs = args[:i]
assert all(isinstance(e, Expr) for expr in exprs for e in expr)
free_symbols = list(set().union(*[e.free_symbols for expr in exprs
for e in expr]))
if len(free_symbols) > nb_of_free_symbols:
raise ValueError("The number of free_symbols in the expression "
"is greater than %d" % nb_of_free_symbols)
if len(args) == i + nb_of_free_symbols and isinstance(args[i], Tuple):
ranges = Tuple(*list(args[
i:i + nb_of_free_symbols]))
plots = [expr + ranges for expr in exprs]
return plots
else:
# Use default ranges.
default_range = Tuple(-10, 10)
ranges = []
for symbol in free_symbols:
ranges.append(Tuple(symbol) + default_range)
for i in range(nb_of_free_symbols - len(free_symbols)):
ranges.append(Tuple(Dummy()) + default_range)
ranges = Tuple(*ranges)
plots = [expr + ranges for expr in exprs]
return plots
elif isinstance(args[0], Tuple) and len(args[0]) == expr_len + nb_of_free_symbols:
# Multiple plots with different ranges.
for arg in args:
for i in range(expr_len):
if not isinstance(arg[i], Expr):
raise ValueError("Expected an expression, given %s" %
str(arg[i]))
for i in range(nb_of_free_symbols):
if not len(arg[i + expr_len]) == 3:
raise ValueError("The ranges should be a tuple of "
"length 3, got %s" % str(arg[i + expr_len]))
return args
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