"""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_segments``, ``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. Don't 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.
"""


import warnings

from sympy import sympify, Expr, Tuple, Dummy, Symbol
from sympy.external import import_module
from sympy.core.function import arity
from sympy.core.compatibility import Callable
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

##############################################################################
# The public interface
##############################################################################


class Plot:
    """The central class of the plotting module.

    For interactive work the function ``plot`` is better suited.

    This class permits the plotting of sympy expressions using numerous
    backends (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 (eg title, xlabel, scale, etc) and
    per-data series options (eg name) and aesthetics (eg. 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
    - ylabel : str
    - 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:

    ListSeries, LineOver1DRangeSeries, Parametric2DLineSeries,
    Parametric3DLineSeries support the following:

    Aesthetics:

    - line_color : function which returns a float.

    options:

    - label : str
    - steps : bool
    - integers_only : bool

    SurfaceOver2DRangeSeries, ParametricSurfaceSeries support the following:

    aesthetics:

    - surface_color : function which returns a float.
    """

    def __init__(self, *args,
        title=None, xlabel=None, ylabel=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.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.

    The backend should check if it supports the data series that it's given.
    (eg TextBackend supports only LineOver1DRange).
    It's the backend responsibility to know how to use the class of
    data series that it's 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 (eg. The LineOver1DRange 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_segments returning np.array (done in Line2DBaseSeries)
    #  - 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_segments returning np.array (done in Line2DBaseSeries)
    #  - 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_segments(self):
        np = import_module('numpy')
        points = self.get_points()
        if self.steps is True:
            x = np.array((points[0], points[0])).T.flatten()[1:]
            y = np.array((points[1], points[1])).T.flatten()[:-1]
            points = (x, y)
        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 str(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_segments(self):
        """
        Adaptively gets segments for plotting.

        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 super().get_segments()
        else:
            f = lambdify([self.var], self.expr)
            list_segments = []
            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:
                    list_segments.append([p, q])

                # 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 any(y is not None for y in yarray):
                        for i in range(len(yarray) - 1):
                            if yarray[i] is not None or yarray[i + 1] is not 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:
                    list_segments.append([p, q])

            f_start = f(self.start)
            f_end = f(self.end)
            sample(np.array([self.start, f_start]),
                   np.array([self.end, f_end]), 0)

            return list_segments

    def get_points(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 \
                            "(%s, %s)" % (str(self.expr_x), str(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 get_points(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_segments(self):
        """
        Adaptively gets segments for plotting.

        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 super().get_segments()

        f_x = lambdify([self.var], self.expr_x)
        f_y = lambdify([self.var], self.expr_y)
        list_segments = []

        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:
                list_segments.append([p, q])

            # 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 any(x is not None and y is not 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:
                list_segments.append([p, q])

        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]
        sample(self.start, self.end, start, end, 0)
        return list_segments


### 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 two parametric sympy
    expressions and a range."""

    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 \
                        "(%s, %s)" % (str(self.expr_x), str(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)

    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:
            return c*np.ones(self.nb_of_points)


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(object):
    """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 `BaseSeries` class is created,
        containing the necessary information to plot the expressions (eg the
        expression, ranges, series name, ...). Eventually, these objects will
        generate the numerical data to be plotted.
    2. A Plot object is instantiated, which stores the list of series and the
        main attributes of the plot (eg 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's given
    (eg TextBackend supports only LineOver1DRange).

    It's 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 `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 doesn't support this functionality. In that case, just raise a
        warning).

    See also
    ========

    MatplotlibBackend
    """
    def __init__(self, parent):
        super(BaseBackend, self).__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 can not 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')

    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:
                collection = self.LineCollection(s.get_segments())
                ax.add_collection(collection)
            elif s.is_contour:
                ax.contour(*s.get_meshes())
            elif s.is_3Dline:
                # TODO too complicated, I blame matplotlib
                art3d = mpl_toolkits.mplot3d.art3d
                collection = art3d.Line3DCollection(s.get_segments())
                ax.add_collection(collection)
                x, y, z = s.get_points()
                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)
                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))

            # Customise the collections with the corresponding per-series
            # options.
            if hasattr(s, 'label'):
                collection.set_label(s.label)
            if s.is_line and s.line_color:
                if isinstance(s.line_color, (float, int)) or isinstance(s.line_color, Callable):
                    color_array = s.get_color_array()
                    collection.set_array(color_array)
                else:
                    collection.set_color(s.line_color)
            if s.is_3Dsurface and s.surface_color:
                if self.matplotlib.__version__ < "1.2.0":  # TODO in the distant future remove this check
                    warnings.warn('The version of matplotlib is too old to use surface coloring.')
                elif isinstance(s.surface_color, (float, int)) or isinstance(s.surface_color, 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)

        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:
            ax.set_xlabel(parent.xlabel, position=(1, 0))
        if parent.ylabel:
            ax.set_ylabel(parent.ylabel, 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 shoulld 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.float)
    vector_b = (z - y).astype(np.float)
    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 followings:

        - 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 : float, optional
        Specifies the color for the plot.
        See ``Plot`` to see how to set color for the plots.

        If there are multiple plots, then the same series series are
        applied to all the plots. If you want to set these options
        separately, you can index the ``Plot`` object returned and set
        it.

    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, 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)```.

    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 matplotlib's annotate() function.

    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 matplotlib's 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 matplotlib's
        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 matplotlib's fill_between() function.

    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.name)
    kwargs.setdefault('ylabel', 'f(%s)' % x.name)
    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 : function
        A function which returns a float.

        Specifies the color of the plot.

        See :class:`Plot` for more details.

    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 symbols, cos, sin
       >>> from sympy.plotting import plot_parametric
       >>> 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``: ``(u, 0, 5)``, A 3-tuple denoting the range of the parameter
    variable.

    Keyword Arguments
    =================

    Arguments for ``Parametric3DLineSeries`` class.

    ``nb_of_points``: The range is uniformly sampled at ``nb_of_points``
    number of points.

    Aesthetics:

    ``line_color``: function which returns a float. Specifies the color for the
    plot. See ``sympy.plotting.Plot`` for more details.

    ``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]
    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``: (x, 0, 5), A 3-tuple denoting the range of the x
    variable.

    ``range_y``: (y, 0, 5), A 3-tuple denoting the range of the y
     variable.

    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 ``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.

    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]
    plots = Plot(*series, **kwargs)
    if show:
        plots.show()
    return plots


def plot3d_parametric_surface(*args, show=True, **kwargs):
    """
    Plots a 3D parametric surface plot.

    Usage
    =====

    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``: ``(u, 0, 5)``,  A 3-tuple denoting the range of the ``u``
    variable.

    ``range_v``: ``(v, 0, 5)``,  A 3-tuple denoting the range of the v
    variable.

    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 ``sympy.plotting.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]
    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``: (x, 0, 5), A 3-tuple denoting the range of the x
    variable.

    ``range_y``: (y, 0, 5), A 3-tuple denoting the range of the y
     variable.

    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 ``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(*[range_expr for range_expr in 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
