contour_plot.py

"""
Contour Plots
"""

#*****************************************************************************
#       Copyright (C) 2006 Alex Clemesha <[email protected]>,
#                          William Stein <[email protected]>,
#                     2008 Mike Hansen <[email protected]>,
#
#  Distributed under the terms of the GNU General Public License (GPL)
#
#    This code is distributed in the hope that it will be useful,
#    but WITHOUT ANY WARRANTY; without even the implied warranty of
#    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
#    General Public License for more details.
#
#  The full text of the GPL is available at:
#
#                  http://www.gnu.org/licenses/
#*****************************************************************************
from sage.plot.primitive import GraphicPrimitive
from sage.misc.decorators import options, suboptions
from sage.plot.colors import rgbcolor, get_cmap
from sage.misc.misc import xsrange
import operator

class ContourPlot(GraphicPrimitive):
    """
    Primitive class for the contour plot graphics type.  See
    ``contour_plot?`` for help actually doing contour plots.

    INPUT:

    - ``xy_data_array`` - list of lists giving evaluated values of the function on the grid

    - ``xrange`` - tuple of 2 floats indicating range for horizontal direction

    - ``yrange`` - tuple of 2 floats indicating range for vertical direction

    - ``options`` - dict of valid plot options to pass to constructor

    EXAMPLES:

    Note this should normally be used indirectly via ``contour_plot``::

        sage: from sage.plot.contour_plot import ContourPlot
        sage: C = ContourPlot([[1,3],[2,4]],(1,2),(2,3),options={})
        sage: C
        ContourPlot defined by a 2 x 2 data grid
        sage: C.xrange
        (1, 2)

    TESTS:

    We test creating a contour plot::

        sage: x,y = var('x,y')
        sage: contour_plot(x^2-y^3+10*sin(x*y), (x, -4, 4), (y, -4, 4),plot_points=121,cmap='hsv')
    """
    def __init__(self, xy_data_array, xrange, yrange, options):
        """
        Initializes base class ContourPlot.

        EXAMPLES::

            sage: x,y = var('x,y')
            sage: C = contour_plot(x^2-y^3+10*sin(x*y), (x, -4, 4), (y, -4, 4),plot_points=121,cmap='hsv')
            sage: C[0].xrange
            (-4.0, 4.0)
            sage: C[0].options()['plot_points']
            121
        """
        self.xrange = xrange
        self.yrange = yrange
        self.xy_data_array = xy_data_array
        self.xy_array_row = len(xy_data_array)
        self.xy_array_col = len(xy_data_array[0])
        GraphicPrimitive.__init__(self, options)

    def get_minmax_data(self):
        """
        Returns a dictionary with the bounding box data.

        EXAMPLES::

            sage: x,y = var('x,y')
            sage: f(x,y) = x^2 + y^2
            sage: d = contour_plot(f, (3, 6), (3, 6))[0].get_minmax_data()
            sage: d['xmin']
            3.0
            sage: d['ymin']
            3.0
        """
        from sage.plot.plot import minmax_data
        return minmax_data(self.xrange, self.yrange, dict=True)

    def _allowed_options(self):
        """
        Return the allowed options for the ContourPlot class.

        EXAMPLES::

            sage: x,y = var('x,y')
            sage: C = contour_plot(x^2-y^2,(x,-2,2),(y,-2,2))
            sage: isinstance(C[0]._allowed_options(),dict)
            True
        """
        return {'plot_points':'How many points to use for plotting precision',
                'cmap':"""the name of a predefined colormap,
                        a list of colors, or an instance of a
                        matplotlib Colormap. Type: import matplotlib.cm; matplotlib.cm.datad.keys()
                        for available colormap names.""",
                'colorbar': "Include a colorbar indicating the levels",
                'colorbar_options': "a dictionary of options for colorbars",
                'fill':'Fill contours or not',
                'legend_label':'The label for this item in the legend.',
                'contours':"""Either an integer specifying the number of
                        contour levels, or a sequence of numbers giving
                        the actual contours to use.""",
                'linewidths':'the width of the lines to be plotted',
                'linestyles':'the style of the lines to be plotted',
                'labels':'show line labels or not',
                'label_options':'a dictionary of options for the labels',
                'zorder':'The layer level in which to draw'}

    def _repr_(self):
        """
        String representation of ContourPlot primitive.

        EXAMPLES::

            sage: x,y = var('x,y')
            sage: C = contour_plot(x^2-y^2,(x,-2,2),(y,-2,2))
            sage: c = C[0]; c
            ContourPlot defined by a 100 x 100 data grid
        """
        return "ContourPlot defined by a %s x %s data grid"%(self.xy_array_row, self.xy_array_col)

    def _render_on_subplot(self, subplot):
        """
        TESTS:

        A somewhat random plot, but fun to look at::

            sage: x,y = var('x,y')
            sage: contour_plot(x^2-y^3+10*sin(x*y), (x, -4, 4), (y, -4, 4),plot_points=121,cmap='hsv')
        """
        from sage.rings.integer import Integer
        options = self.options()
        fill = options['fill']
        contours = options['contours']
        if options.has_key('cmap'):
            cmap = get_cmap(options['cmap'])
        elif fill or contours is None:
            cmap = get_cmap('gray')
        else:
            if isinstance(contours, (int, Integer)):
                cmap = get_cmap([(i,i,i) for i in xsrange(0,1,1/contours)])
            else:
                l = Integer(len(contours))
                cmap = get_cmap([(i,i,i) for i in xsrange(0,1,1/l)])

        x0,x1 = float(self.xrange[0]), float(self.xrange[1])
        y0,y1 = float(self.yrange[0]), float(self.yrange[1])

        if isinstance(contours, (int, Integer)):
            contours = int(contours)

        CSF=None
        if fill:
            if contours is None:
                CSF=subplot.contourf(self.xy_data_array, cmap=cmap, extent=(x0,x1,y0,y1), label=options['legend_label'])
            else:
                CSF=subplot.contourf(self.xy_data_array, contours, cmap=cmap, extent=(x0,x1,y0,y1),extend='both', label=options['legend_label'])

        linewidths = options.get('linewidths',None)
        if isinstance(linewidths, (int, Integer)):
            linewidths = int(linewidths)
        elif isinstance(linewidths, (list, tuple)):
            linewidths = tuple(int(x) for x in linewidths)

        from sage.plot.misc import get_matplotlib_linestyle
        linestyles = options.get('linestyles', None)
        if isinstance(linestyles, (list, tuple)):
            linestyles = [get_matplotlib_linestyle(l, 'long') for l in linestyles]
        else:
            linestyles = get_matplotlib_linestyle(linestyles, 'long')
        if contours is None:
            CS = subplot.contour(self.xy_data_array, cmap=cmap, extent=(x0,x1,y0,y1),
                                 linewidths=linewidths, linestyles=linestyles, label=options['legend_label'])
        else:
            CS = subplot.contour(self.xy_data_array, contours, cmap=cmap, extent=(x0,x1,y0,y1),
                            linewidths=linewidths, linestyles=linestyles, label=options['legend_label'])
        if options.get('labels', False):
            label_options = options['label_options']
            label_options['fontsize'] = int(label_options['fontsize'])
            if fill and label_options is None:
                label_options['inline']=False
            subplot.clabel(CS, **label_options)
        if options.get('colorbar', False):
            colorbar_options = options['colorbar_options']
            from matplotlib import colorbar
            cax,kwds=colorbar.make_axes_gridspec(subplot,**colorbar_options)
            if CSF is None:
                cb=colorbar.Colorbar(cax,CS, **kwds)
            else:
                cb=colorbar.Colorbar(cax,CSF, **kwds)
                cb.add_lines(CS)

@suboptions('colorbar', orientation='vertical', format=None, spacing=None)
@suboptions('label', fontsize=9, colors='blue', inline=None, inline_spacing=3, fmt="%1.2f")
@options(plot_points=100, fill=True, contours=None, linewidths=None, linestyles=None, labels=False, frame=True, axes=False, colorbar=False, legend_label=None, aspect_ratio=1, region=None)
def contour_plot(f, xrange, yrange, **options):
    r"""
    ``contour_plot`` takes a function of two variables, `f(x,y)`
    and plots contour lines of the function over the specified
    ``xrange`` and ``yrange`` as demonstrated below.

    ``contour_plot(f, (xmin, xmax), (ymin, ymax), ...)``

    INPUT:

    - ``f`` -- a function of two variables

    - ``(xmin, xmax)`` -- 2-tuple, the range of ``x`` values OR 3-tuple
      ``(x,xmin,xmax)``

    - ``(ymin, ymax)`` -- 2-tuple, the range of ``y`` values OR 3-tuple
      ``(y,ymin,ymax)``

    The following inputs must all be passed in as named parameters:

    - ``plot_points``  -- integer (default: 100); number of points to plot
      in each direction of the grid.  For old computers, 25 is fine, but
      should not be used to verify specific intersection points.

    - ``fill`` -- bool (default: ``True``), whether to color in the area
      between contour lines

    - ``cmap`` -- a colormap (default: ``'gray'``), the name of
      a predefined colormap, a list of colors or an instance of a matplotlib
      Colormap. Type: ``import matplotlib.cm; matplotlib.cm.datad.keys()``
      for available colormap names.

    - ``contours`` -- integer or list of numbers (default: ``None``):
      If a list of numbers is given, then this specifies the contour levels
      to use.  If an integer is given, then this many contour lines are
      used, but the exact levels are determined automatically. If ``None``
      is passed (or the option is not given), then the number of contour
      lines is determined automatically, and is usually about 5.

    - ``linewidths`` -- integer or list of integer (default: None), if
      a single integer all levels will be of the width given,
      otherwise the levels will be plotted with the width in the order
      given.  If the list is shorter than the number of contours, then
      the widths will be repeated cyclically.

    - ``linestyles`` -- string or list of strings (default: None), the
      style of the lines to be plotted, one of: ``"solid"``, ``"dashed"``,
      ``"dashdot"``, ``"dotted"``, respectively ``"-"``, ``"--"``,
      ``"-."``, ``":"``.  If the list is shorter than the number of
      contours, then the styles will be repeated cyclically.

    - ``labels`` -- boolean (default: False) Show level labels or not.

      The following options are to adjust the style and placement of
      labels, they have no effect if no labels are shown.

      - ``label_fontsize`` -- integer (default: 9), the font size of the labels.

      - ``label_colors`` -- string or sequence of colors (default:
        None) If a string, gives the name of a single color with which
        to draw all labels.  If a sequence, gives the colors of the
        labels.  A color is a string giving the name of one or a
        3-tuple of floats.

      - ``label_inline`` -- boolean (default: False if fill is True,
        otherwise True), controls whether the underlying contour is
        removed or not.

      - ``label_inline_spacing`` -- integer (default: 3), When inline,
        this is the amount of contour that is removed from each side,
        in pixels.

      - ``label_fmt`` -- a format string (default: "%1.2f"), this is
        used to get the label text from the level.  This can also be a
        dictionary with the contour levels as keys and corresponding
        text string labels as values.  It can also be any callable which
        returns a string when called with a numeric contour level.

    - ``colorbar`` -- boolean (default: False) Show a colorbar or not.

      The following options are to adjust the style and placement of
      colorbars.  They have no effect if a colorbar is not shown.

      - ``colorbar_orientation`` -- string (default: 'vertical'),
        controls placement of the colorbar, can be either 'vertical'
        or 'horizontal'

      - ``colorbar_format`` -- a format string, this is used to format
        the colorbar labels.

      - ``colorbar_spacing`` -- string (default: 'proportional').  If
        'proportional', make the contour divisions proportional to
        values.  If 'uniform', space the colorbar divisions uniformly,
        without regard for numeric values.

    - ``legend_label`` -- the label for this item in the legend

    -  ``region`` - (default: None) If region is given, it must be a function
        of two variables. Only segments of the surface where region(x,y) returns a
        number >0 will be included in the plot.

    EXAMPLES:

    Here we plot a simple function of two variables.  Note that
    since the input function is an expression, we need to explicitly
    declare the variables in 3-tuples for the range::

        sage: x,y = var('x,y')
        sage: contour_plot(cos(x^2+y^2), (x, -4, 4), (y, -4, 4))

    Here we change the ranges and add some options::

        sage: x,y = var('x,y')
        sage: contour_plot((x^2)*cos(x*y), (x, -10, 5), (y, -5, 5), fill=False, plot_points=150)

    An even more complicated plot::

        sage: x,y = var('x,y')
        sage: contour_plot(sin(x^2 + y^2)*cos(x)*sin(y), (x, -4, 4), (y, -4, 4),plot_points=150)

    Some elliptic curves, but with symbolic endpoints.  In the first
    example, the plot is rotated 90 degrees because we switch the
    variables `x`, `y`::

        sage: x,y = var('x,y')
        sage: contour_plot(y^2 + 1 - x^3 - x, (y,-pi,pi), (x,-pi,pi))

    ::

        sage: contour_plot(y^2 + 1 - x^3 - x, (x,-pi,pi), (y,-pi,pi))

    We can play with the contour levels::

        sage: x,y = var('x,y')
        sage: f(x,y) = x^2 + y^2
        sage: contour_plot(f, (-2, 2), (-2, 2))

    ::

        sage: contour_plot(f, (-2, 2), (-2, 2), contours=2, cmap=[(1,0,0), (0,1,0), (0,0,1)])

    ::

        sage: contour_plot(f, (-2, 2), (-2, 2), contours=(0.1, 1.0, 1.2, 1.4), cmap='hsv')

    ::

        sage: contour_plot(f, (-2, 2), (-2, 2), contours=(1.0,), fill=False)

    ::

        sage: contour_plot(x-y^2,(x,-5,5),(y,-3,3),contours=[-4,0,1])

    We can change the style of the lines::

        sage: contour_plot(f, (-2,2), (-2,2), fill=False, linewidths=10)

    ::

        sage: contour_plot(f, (-2,2), (-2,2), fill=False, linestyles='dashdot')

    ::

        sage: P=contour_plot(x^2-y^2,(x,-3,3),(y,-3,3),contours=[0,1,2,3,4],\
        ...    linewidths=[1,5],linestyles=['solid','dashed'],fill=False)
        sage: P

    ::

        sage: P=contour_plot(x^2-y^2,(x,-3,3),(y,-3,3),contours=[0,1,2,3,4],\
        ...    linewidths=[1,5],linestyles=['solid','dashed'])
        sage: P

        sage: P=contour_plot(x^2-y^2,(x,-3,3),(y,-3,3),contours=[0,1,2,3,4],\
        ...    linewidths=[1,5],linestyles=['-',':'])
        sage: P

    We can add labels and play with them::

        sage: contour_plot(y^2 + 1 - x^3 - x, (x,-pi,pi), (y,-pi,pi),  fill=False, cmap='hsv', labels=True)

    ::

        sage: P=contour_plot(y^2 + 1 - x^3 - x, (x,-pi,pi), (y,-pi,pi), fill=False, cmap='hsv',\
        ...     labels=True, label_fmt="%1.0f", label_colors='black')
        sage: P

    ::

        sage: P=contour_plot(y^2 + 1 - x^3 - x, (x,-pi,pi), (y,-pi,pi), fill=False, cmap='hsv',labels=True,\
        ...    contours=[-4,0,4],  label_fmt={-4:"low", 0:"medium", 4: "hi"}, label_colors='black')
        sage: P

    ::

        sage: P=contour_plot(y^2 + 1 - x^3 - x, (x,-pi,pi), (y,-pi,pi), fill=False, cmap='hsv',labels=True,\
        ...    contours=[-4,0,4],  label_fmt=lambda x: "$z=%s$"%x, label_colors='black', label_inline=True, \
        ...    label_fontsize=12)
        sage: P

    ::

        sage: P=contour_plot(y^2 + 1 - x^3 - x, (x,-pi,pi), (y,-pi,pi), \
        ...    fill=False, cmap='hsv', labels=True, label_fontsize=18)
        sage: P

    ::

        sage: P=contour_plot(y^2 + 1 - x^3 - x, (x,-pi,pi), (y,-pi,pi), \
        ...    fill=False, cmap='hsv', labels=True, label_inline_spacing=1)
        sage: P

    ::

        sage: P= contour_plot(y^2 + 1 - x^3 - x, (x,-pi,pi), (y,-pi,pi), \
        ...    fill=False, cmap='hsv', labels=True, label_inline=False)
        sage: P

    We can change the color of the labels if so desired::

        sage: contour_plot(f, (-2,2), (-2,2), labels=True, label_colors='red')

    We can add a colorbar as well::

        sage: f(x,y)=x^2-y^2
        sage: contour_plot(f, (x,-3,3), (y,-3,3), colorbar=True)

    ::

        sage: contour_plot(f, (x,-3,3), (y,-3,3), colorbar=True,colorbar_orientation='horizontal')

    ::

        sage: contour_plot(f, (x,-3,3), (y,-3,3), contours=[-2,-1,4],colorbar=True)

    ::

        sage: contour_plot(f, (x,-3,3), (y,-3,3), contours=[-2,-1,4],colorbar=True,colorbar_spacing='uniform')

    ::

        sage: contour_plot(f, (x,-3,3), (y,-3,3), contours=[0,2,3,6],colorbar=True,colorbar_format='%.3f')

    ::

        sage: contour_plot(f, (x,-3,3), (y,-3,3), labels=True,label_colors='red',contours=[0,2,3,6],colorbar=True)

    ::

        sage: contour_plot(f, (x,-3,3), (y,-3,3), cmap='winter', contours=20, fill=False, colorbar=True)

    This should plot concentric circles centered at the origin::

        sage: x,y = var('x,y')
        sage: contour_plot(x^2+y^2-2,(x,-1,1), (y,-1,1))

    Extra options will get passed on to show(), as long as they are valid::

        sage: f(x, y) = cos(x) + sin(y)
        sage: contour_plot(f, (0, pi), (0, pi), axes=True)

    One can also plot over a reduced region::

        sage: contour_plot(x**2-y**2, (x,-2, 2), (y,-2, 2),region=x-y,plot_points=300)

    ::

        sage: contour_plot(f, (0, pi), (0, pi)).show(axes=True) # These are equivalent

    Note that with ``fill=False`` and grayscale contours, there is the
    possibility of confusion between the contours and the axes, so use
    ``fill=False`` together with ``axes=True`` with caution::

        sage: contour_plot(f, (-pi, pi), (-pi, pi), fill=False, axes=True)

    TESTS:

    To check that ticket 5221 is fixed, note that this has three curves, not two::

        sage: x,y = var('x,y')
        sage: contour_plot(x-y^2,(x,-5,5),(y,-3,3),contours=[-4,-2,0], fill=False)
    """
    from sage.plot.all import Graphics
    from sage.plot.misc import setup_for_eval_on_grid

    region = options.pop('region')
    ev = [f] if region is None else [f,region]

    F, ranges = setup_for_eval_on_grid(ev, [xrange, yrange], options['plot_points'])
    g = F[0]
    xrange,yrange=[r[:2] for r in ranges]

    xy_data_array = [[g(x, y) for x in xsrange(*ranges[0], include_endpoint=True)]
                              for y in xsrange(*ranges[1], include_endpoint=True)]

    if region is not None:
        import numpy

        xy_data_array = numpy.ma.asarray(xy_data_array,dtype=float)

        m = F[1]

        mask = numpy.asarray([[m(x, y)<=0 for x in xsrange(*ranges[0], include_endpoint=True)]
                                          for y in xsrange(*ranges[1], include_endpoint=True)],dtype=bool)

        xy_data_array[mask] = numpy.ma.masked

    g = Graphics()

    # Reset aspect_ratio to 'automatic' in case scale is 'semilog[xy]'.
    # Otherwise matplotlib complains.
    scale = options.get('scale', None)
    if isinstance(scale, (list, tuple)):
        scale = scale[0]
    if scale == 'semilogy' or scale == 'semilogx':
        options['aspect_ratio'] = 'automatic'

    g._set_extra_kwds(Graphics._extract_kwds_for_show(options, ignore=['xmin', 'xmax']))
    g.add_primitive(ContourPlot(xy_data_array, xrange, yrange, options))
    return g

@options(plot_points=150, contours=(0,0), fill=False, cmap=["blue"])
def implicit_plot(f, xrange, yrange, **options):
    r"""
    ``implicit_plot`` takes a function of two variables, `f(x,y)`
    and plots the curve `f(x,y) = 0` over the specified
    ``xrange`` and ``yrange`` as demonstrated below.

    ``implicit_plot(f, (xmin, xmax), (ymin, ymax), ...)``

    ``implicit_plot(f, (x, xmin, xmax), (y, ymin, ymax), ...)``

    INPUT:

    - ``f`` -- a function of two variables or equation in two variables

    - ``(xmin, xmax)`` -- 2-tuple, the range of ``x`` values or ``(x,xmin,xmax)``

    - ``(ymin, ymax)`` -- 2-tuple, the range of ``y`` values or ``(y,ymin,ymax)``

    The following inputs must all be passed in as named parameters:

    - ``plot_points`` -- integer (default: 150); number of points to plot
      in each direction of the grid

    - ``fill`` -- boolean (default: ``False``); if ``True``, fill the region
      `f(x,y) < 0`.

    - ``linewidth`` -- integer (default: None), if a single integer all levels
      will be of the width given, otherwise the levels will be plotted with the
      widths in the order given.

    - ``linestyle`` -- string (default: None), the style of the line to be
      plotted, one of: ``"solid"``, ``"dashed"``, ``"dashdot"`` or
      ``"dotted"``, respectively ``"-"``, ``"--"``, ``"-."``, or ``":"``.

    - ``color`` -- string (default: ``blue``), the color of the plot. Colors are
      defined in :mod:`sage.plot.colors`; try ``colors?`` to see them all.

    - ``legend_label`` -- the label for this item in the legend

    - ``base`` - (default: 10) the base of the logarithm if
      a logarithmic scale is set. This must be greater than 1. The base
      can be also given as a list or tuple ``(basex, basey)``.
      ``basex`` sets the base of the logarithm along the horizontal
      axis and ``basey`` sets the base along the vertical axis.

    - ``scale`` -- (default: ``"linear"``) string. The scale of the axes.
      Possible values are ``"linear"``, ``"loglog"``, ``"semilogx"``,
      ``"semilogy"``.

      The scale can be also be given as single argument that is a list
      or tuple ``(scale, base)`` or ``(scale, basex, basey)``.

      The ``"loglog"`` scale sets both the horizontal and vertical axes to
      logarithmic scale. The ``"semilogx"`` scale sets the horizontal axis
      to logarithmic scale. The ``"semilogy"`` scale sets the vertical axis
      to logarithmic scale. The ``"linear"`` scale is the default value
      when :class:`~sage.plot.graphics.Graphics` is initialized.

    EXAMPLES:

    A simple circle with a radius of 2. Note that
    since the input function is an expression, we need to explicitly
    declare the variables in 3-tuples for the range::

        sage: var("x y")
        (x, y)
        sage: implicit_plot(x^2+y^2-2, (x,-3,3), (y,-3,3))

    I can do the same thing, but using a callable function so I don't need
    to explicitly define the variables in the ranges, and filling the inside::

        sage: f(x,y) = x^2 + y^2 - 2
        sage: implicit_plot(f, (-3, 3), (-3, 3),fill=True)

    The same circle but with a different line width::

        sage: implicit_plot(f, (-3,3), (-3,3), linewidth=6)

    And again the same circle but this time with a dashdot border::

        sage: implicit_plot(f, (-3,3), (-3,3), linestyle='dashdot')

    You can also plot an equation::

        sage: var("x y")
        (x, y)
        sage: implicit_plot(x^2+y^2 == 2, (x,-3,3), (y,-3,3))

    You can even change the color of the plot::

        sage: implicit_plot(x^2+y^2 == 2, (x,-3,3), (y,-3,3), color="red")

    Here is a beautiful (and long) example which also tests that all
    colors work with this::

        sage: G = Graphics()
        sage: counter = 0
        sage: for col in colors.keys(): # long time
        ...       G += implicit_plot(x^2+y^2==1+counter*.1, (x,-4,4),(y,-4,4),color=col)
        ...       counter += 1
        sage: G.show(frame=False)

    We can define a level-`n` approximation of the boundary of the
    Mandelbrot set::

        sage: def mandel(n):
        ...       c = polygen(CDF, 'c')
        ...       z = 0
        ...       for i in range(n):
        ...           z = z*z + c
        ...       def f(x, y):
        ...           val = z(CDF(x, y))
        ...           return val.norm() - 4
        ...       return f

    The first-level approximation is just a circle::

        sage: implicit_plot(mandel(1), (-3, 3), (-3, 3))

    A third-level approximation starts to get interesting::

        sage: implicit_plot(mandel(3), (-2, 1), (-1.5, 1.5))

    The seventh-level approximation is a degree 64 polynomial, and
    ``implicit_plot`` does a pretty good job on this part of the curve.
    (``plot_points=200`` looks even better, but it takes over a second.)

    ::

        sage: implicit_plot(mandel(7), (-0.3, 0.05), (-1.15, -0.9),plot_points=50)

    When making a filled implicit plot using a python function rather than a
    symbolic expression the user should increase the number of plot points to
    avoid artifacts::

        sage: implicit_plot(lambda x,y: x^2+y^2-2, (x,-3,3), (y,-3,3), fill=True, plot_points=500) # long time

    An example of an implicit plot on 'loglog' scale::

        sage: implicit_plot(x^2+y^2 == 200, (x,1,200), (y,1,200), scale='loglog')

    TESTS::

        sage: f(x,y) = x^2 + y^2 - 2
        sage: implicit_plot(f, (-3, 3), (-3, 3),fill=5)
        Traceback (most recent call last):
        ...
        ValueError: fill=5 is not supported
    """
    from sage.symbolic.expression import is_SymbolicEquation
    if is_SymbolicEquation(f):
        if f.operator() != operator.eq:
            raise ValueError, "input to implicit plot must be function or equation"
        f = f.lhs() - f.rhs()
    linewidths = options.pop('linewidth', None)
    linestyles = options.pop('linestyle', None)

    if 'color' in options:
        options['cmap']=[options.pop('color', None)]

    if options['fill'] is True:
        options.pop('fill')
        options.pop('contours',None)
        options.pop('cmap',None)
        from sage.symbolic.expression import is_Expression
        if not is_Expression(f):
            return region_plot(lambda x,y: f(x,y)<0, xrange, yrange,
                               borderwidth=linewidths, borderstyle=linestyles,
                               **options)
        else:
            return region_plot(f<0, xrange, yrange, borderwidth=linewidths,
                               borderstyle=linestyles, **options)
    elif options['fill'] is False:
        return contour_plot(f, xrange, yrange, linewidths=linewidths,
                            linestyles=linestyles, **options)
    else:
        raise ValueError("fill=%s is not supported" % options['fill'])


@options(plot_points=100, incol='blue', outcol='white', bordercol=None, borderstyle=None, borderwidth=None,frame=False,axes=True, legend_label=None, aspect_ratio=1)
def region_plot(f, xrange, yrange, plot_points, incol, outcol, bordercol, borderstyle, borderwidth,**options):
    r"""
    ``region_plot`` takes a boolean function of two variables, `f(x,y)`
    and plots the region where f is True over the specified
    ``xrange`` and ``yrange`` as demonstrated below.

    ``region_plot(f, (xmin, xmax), (ymin, ymax), ...)``

    INPUT:

    - ``f`` -- a boolean function of two variables

    - ``(xmin, xmax)`` -- 2-tuple, the range of ``x`` values OR 3-tuple
      ``(x,xmin,xmax)``

    - ``(ymin, ymax)`` -- 2-tuple, the range of ``y`` values OR 3-tuple
      ``(y,ymin,ymax)``

    - ``plot_points``  -- integer (default: 100); number of points to plot
      in each direction of the grid

    - ``incol`` -- a color (default: ``'blue'``), the color inside the region

    - ``outcol`` -- a color (default: ``'white'``), the color of the outside
      of the region

    If any of these options are specified, the border will be shown as indicated,
    otherwise it is only implicit (with color ``incol``) as the border of the
    inside of the region.

     - ``bordercol`` -- a color (default: ``None``), the color of the border
       (``'black'`` if ``borderwidth`` or ``borderstyle`` is specified but not ``bordercol``)

    - ``borderstyle``  -- string (default: 'solid'), one of ``'solid'``,
      ``'dashed'``, ``'dotted'``, ``'dashdot'``, respectively ``'-'``,
      ``'--'``, ``':'``, ``'-.'``.

    - ``borderwidth``  -- integer (default: None), the width of the border in pixels

    - ``legend_label`` -- the label for this item in the legend

    - ``base`` - (default: 10) the base of the logarithm if
      a logarithmic scale is set. This must be greater than 1. The base
      can be also given as a list or tuple ``(basex, basey)``.
      ``basex`` sets the base of the logarithm along the horizontal
      axis and ``basey`` sets the base along the vertical axis.

    - ``scale`` -- (default: ``"linear"``) string. The scale of the axes.
      Possible values are ``"linear"``, ``"loglog"``, ``"semilogx"``,
      ``"semilogy"``.

      The scale can be also be given as single argument that is a list
      or tuple ``(scale, base)`` or ``(scale, basex, basey)``.

      The ``"loglog"`` scale sets both the horizontal and vertical axes to
      logarithmic scale. The ``"semilogx"`` scale sets the horizontal axis
      to logarithmic scale. The ``"semilogy"`` scale sets the vertical axis
      to logarithmic scale. The ``"linear"`` scale is the default value
      when :class:`~sage.plot.graphics.Graphics` is initialized.


    EXAMPLES:

    Here we plot a simple function of two variables::

        sage: x,y = var('x,y')
        sage: region_plot(cos(x^2+y^2) <= 0, (x, -3, 3), (y, -3, 3))

    Here we play with the colors::

        sage: region_plot(x^2+y^3 < 2, (x, -2, 2), (y, -2, 2), incol='lightblue', bordercol='gray')

    An even more complicated plot, with dashed borders::

        sage: region_plot(sin(x)*sin(y) >= 1/4, (x,-10,10), (y,-10,10), incol='yellow', bordercol='black', borderstyle='dashed', plot_points=250)

    A disk centered at the origin::

        sage: region_plot(x^2+y^2<1, (x,-1,1), (y,-1,1))

    A plot with more than one condition (all conditions must be true for the statement to be true)::

        sage: region_plot([x^2+y^2<1, x<y], (x,-2,2), (y,-2,2))

    Since it doesn't look very good, let's increase ``plot_points``::

        sage: region_plot([x^2+y^2<1, x<y], (x,-2,2), (y,-2,2), plot_points=400)

    To get plots where only one condition needs to be true, use a function.
    Using lambda functions, we definitely need the extra ``plot_points``::

        sage: region_plot(lambda x,y: x^2+y^2<1 or x<y, (x,-2,2), (y,-2,2), plot_points=400)

    The first quadrant of the unit circle::

        sage: region_plot([y>0, x>0, x^2+y^2<1], (x,-1.1, 1.1), (y,-1.1, 1.1), plot_points = 400)

    Here is another plot, with a huge border::

        sage: region_plot(x*(x-1)*(x+1)+y^2<0, (x, -3, 2), (y, -3, 3), incol='lightblue', bordercol='gray', borderwidth=10, plot_points=50)

    If we want to keep only the region where x is positive::

        sage: region_plot([x*(x-1)*(x+1)+y^2<0, x>-1], (x, -3, 2), (y, -3, 3), incol='lightblue', plot_points=50)

    Here we have a cut circle::

        sage: region_plot([x^2+y^2<4, x>-1], (x, -2, 2), (y, -2, 2), incol='lightblue', bordercol='gray', plot_points=200)

    The first variable range corresponds to the horizontal axis and
    the second variable range corresponds to the vertical axis::

        sage: s,t=var('s,t')
        sage: region_plot(s>0,(t,-2,2),(s,-2,2))

    ::

        sage: region_plot(s>0,(s,-2,2),(t,-2,2))

    An example of a region plot in 'loglog' scale::

        sage: region_plot(x^2+y^2<100, (x,1,10), (y,1,10), scale='loglog')

    """

    from sage.plot.all import Graphics
    from sage.plot.misc import setup_for_eval_on_grid
    import numpy

    if not isinstance(f, (list, tuple)):
        f = [f]

    f = [equify(g) for g in f]

    g, ranges = setup_for_eval_on_grid(f, [xrange, yrange], plot_points)
    xrange,yrange=[r[:2] for r in ranges]

    xy_data_arrays = numpy.asarray([[[func(x, y) for x in xsrange(*ranges[0], include_endpoint=True)]
                                     for y in xsrange(*ranges[1], include_endpoint=True)]
                                    for func in g],dtype=float)
    xy_data_array=numpy.abs(xy_data_arrays.prod(axis=0))
    # Now we need to set entries to negative iff all
    # functions were negative at that point.
    neg_indices = (xy_data_arrays<0).all(axis=0)
    xy_data_array[neg_indices]=-xy_data_array[neg_indices]

    from matplotlib.colors import ListedColormap
    incol = rgbcolor(incol)
    outcol = rgbcolor(outcol)
    cmap = ListedColormap([incol, outcol])
    cmap.set_over(outcol)
    cmap.set_under(incol)

    g = Graphics()

    # Reset aspect_ratio to 'automatic' in case scale is 'semilog[xy]'.
    # Otherwise matplotlib complains.
    scale = options.get('scale', None)
    if isinstance(scale, (list, tuple)):
        scale = scale[0]
    if scale == 'semilogy' or scale == 'semilogx':
        options['aspect_ratio'] = 'automatic'

    g._set_extra_kwds(Graphics._extract_kwds_for_show(options, ignore=['xmin', 'xmax']))
    g.add_primitive(ContourPlot(xy_data_array, xrange,yrange,
                                dict(contours=[-1e307, 0, 1e307], cmap=cmap, fill=True, **options)))

    if bordercol or borderstyle or borderwidth:
        cmap = [rgbcolor(bordercol)] if bordercol else ['black']
        linestyles = [borderstyle] if borderstyle else None
        linewidths = [borderwidth] if borderwidth else None
        g.add_primitive(ContourPlot(xy_data_array, xrange, yrange,
                                    dict(linestyles=linestyles, linewidths=linewidths,
                                         contours=[0], cmap=[bordercol], fill=False, **options)))

    return g

def equify(f):
    """
    Returns the equation rewritten as a symbolic function to give
    negative values when True, positive when False.

    EXAMPLES::

        sage: from sage.plot.contour_plot import equify
        sage: var('x, y')
        (x, y)
        sage: equify(x^2 < 2)
        x^2 - 2
        sage: equify(x^2 > 2)
        -x^2 + 2
        sage: equify(x*y > 1)
        -x*y + 1
        sage: equify(y > 0)
        -y
        sage: f=equify(lambda x,y: x>y)
        sage: f(1,2)
        1
        sage: f(2,1)
        -1
    """
    import operator
    from sage.calculus.all import symbolic_expression
    from sage.symbolic.expression import is_Expression
    if not is_Expression(f):
        return lambda x,y: -1 if f(x,y) else 1

    op = f.operator()
    if op is operator.gt or op is operator.ge:
        return symbolic_expression(f.rhs() - f.lhs())
    else:
        return symbolic_expression(f.lhs() - f.rhs())