r"""
Graph Plotting

*(For LaTeX drawings of graphs, see the* :mod:`~sage.graphs.graph_latex` *module.)*

All graphs have an associated Sage graphics object, which you can display::

    sage: G = graphs.WheelGraph(15)
    sage: P = G.plot()
    sage: P.show() # long time

If you create a graph in Sage using the ``Graph`` command, then plot that graph,
the positioning of nodes is determined using the spring-layout algorithm. For
the special graph constructors, which you get using ``graphs.[tab]``, the
positions are preset. For example, consider the Petersen graph with default node
positioning vs. the Petersen graph constructed by this database::

    sage: petersen_spring = Graph(':I`ES@obGkqegW~')
    sage: petersen_spring.show() # long time
    sage: petersen_database = graphs.PetersenGraph()
    sage: petersen_database.show() # long time

For all the constructors in this database (except the octahedral, dodecahedral,
random and empty graphs), the position dictionary is filled in, instead of using
the spring-layout algorithm.

**Plot options**

Here is the list of options accepted by
:meth:`~sage.graphs.generic_graph.GenericGraph.plot` and the constructor of
:class:`GraphPlot`. Those two functions also accept all options of
:meth:`sage.plot.graphics.Graphics.show`.

.. csv-table::
    :class: contentstable
    :widths: 30, 70
    :delim: |

{PLOT_OPTIONS_TABLE}

**Default options**

This module defines two dictionaries containing default options for the
:meth:`~sage.graphs.generic_graph.GenericGraph.plot` and
:meth:`~sage.graphs.generic_graph.GenericGraph.show` methods. These two dictionaries are
``sage.graphs.graph_plot.DEFAULT_PLOT_OPTIONS`` and
``sage.graphs.graph_plot.DEFAULT_SHOW_OPTIONS``, respectively.

Obviously, these values are overruled when arguments are given explicitly.

Here is how to define the default size of a graph drawing to be ``[6,6]``. The
first two calls to :meth:`~sage.graphs.generic_graph.GenericGraph.show` use this
option, while the third does not (a value for ``figsize`` is explicitly given)::

    sage: sage.graphs.graph_plot.DEFAULT_SHOW_OPTIONS['figsize'] = [6,6]
    sage: graphs.PetersenGraph().show() # long time
    sage: graphs.ChvatalGraph().show()  # long time
    sage: graphs.PetersenGraph().show(figsize=[4,4]) # long time

We can now reset the default to its initial value, and now display graphs as
previously::

    sage: sage.graphs.graph_plot.DEFAULT_SHOW_OPTIONS['figsize'] = [4,4]
    sage: graphs.PetersenGraph().show() # long time
    sage: graphs.ChvatalGraph().show()  # long time

.. NOTE::

    * While ``DEFAULT_PLOT_OPTIONS`` affects both ``G.show()`` and ``G.plot()``,
      settings from ``DEFAULT_SHOW_OPTIONS`` only affects ``G.show()``.

    * In order to define a default value permanently, you can add a couple of
      lines to `Sage's startup scripts <../../../repl/startup.html>`_. Example ::

       sage: import sage.graphs.graph_plot
       sage: sage.graphs.graph_plot.DEFAULT_SHOW_OPTIONS['figsize'] = [4,4]

**Index of methods and functions**

.. csv-table::
    :class: contentstable
    :widths: 30, 70
    :delim: |

    :meth:`GraphPlot.set_pos` | Sets the position plotting parameters for this GraphPlot.
    :meth:`GraphPlot.set_vertices` | Sets the vertex plotting parameters for this GraphPlot.
    :meth:`GraphPlot.set_edges` | Sets the edge (or arrow) plotting parameters for the GraphPlot object.
    :meth:`GraphPlot.show` | Shows the (Di)Graph associated with this GraphPlot object.
    :meth:`GraphPlot.plot` | Returns a graphics object representing the (di)graph.
    :meth:`GraphPlot.layout_tree` | Compute a nice layout of a tree.
    :meth:`~sage.graphs.graph_plot._circle_embedding` | Sets some vertices on a circle in the embedding of a graph G.
    :meth:`~sage.graphs.graph_plot._line_embedding` | Sets some vertices on a line in the embedding of a graph G.

Methods and classes
-------------------
.. autofunction:: _circle_embedding
.. autofunction:: _line_embedding
"""

layout_options =   {
                    'layout': 'A layout algorithm -- one of : "acyclic", "circular" (plots the graph with vertices evenly distributed on a circle), "ranked", "graphviz", "planar", "spring" (traditional spring layout, using the graph\'s current positions as initial positions), or "tree" (the tree will be plotted in levels, depending on minimum distance for the root).',
                    'iterations': 'The number of times to execute the spring layout algorithm.',
                    'heights': 'A dictionary mapping heights to the list of vertices at this height.',
                    'spring': 'Use spring layout to finalize the current layout.',
                    'tree_root': 'A vertex designation for drawing trees. A vertex of the tree to be used as the root for the ``layout=\'tree\'`` option. If no root is specified, then one is chosen close to the center of the tree. Ignored unless ``layout=\'tree\'``',
                    'tree_orientation': 'The direction of tree branches -- \'up\', \'down\', \'left\' or \'right\'.',
                    'save_pos': 'Whether or not to save the computed position for the graph.',
                    'dim': 'The dimension of the layout -- 2 or 3.',
                    'prog': 'Which graphviz layout program to use -- one of "circo", "dot", "fdp", "neato", or "twopi".',
                    'by_component': 'Whether to do the spring layout by connected component -- a boolean.',
                    }

graphplot_options = layout_options.copy()

graphplot_options.update(
                   {'pos': 'The position dictionary of vertices',
                    'vertex_labels': 'Whether or not to draw vertex labels.',
                    'vertex_colors': 'Dictionary of vertex coloring : each '
                        'key is a color recognizable by matplotlib, and each '
                        'corresponding entry is a list of vertices. '
                        'If a vertex is not listed, it looks invisible on '
                        'the resulting plot (it does not get drawn).',
                    'vertex_size': 'The size to draw the vertices.',
                    'vertex_shape': 'The shape to draw the vertices. '
                        'Currently unavailable for Multi-edged DiGraphs.',
                    'edge_labels': 'Whether or not to draw edge labels.',
                    'edge_style': 'The linestyle of the edges. It should be '
                        'one of "solid", "dashed", "dotted", dashdot", or '
                        '"-", "--", ":", "-.", respectively. '
                        'This currently only works for directed graphs, '
                        'since we pass off the undirected graph to networkx.',
                    'edge_color': 'The default color for edges.',
                    'edge_colors': 'a dictionary specifying edge colors: each '
                        'key is a color recognized by matplotlib, and each '
                        'entry is a list of edges.',
                    'color_by_label': 'Whether to color the edges according '
                        'to their labels. This also accepts a function or '
                        'dictionary mapping labels to colors.',
                    'partition': 'A partition of the vertex set. If specified, '
                        'plot will show each cell in a different color. '
                        'vertex_colors takes precedence.',
                    'loop_size': 'The radius of the smallest loop.',
                    'dist': 'The distance between multiedges.',
                    'max_dist': 'The max distance range to allow multiedges.',
                    'talk': 'Whether to display the vertices in talk mode '
                        '(larger and white).',
                    'graph_border': 'Whether or not to draw a frame around the graph.'})

_PLOT_OPTIONS_TABLE = ""
for key, value in graphplot_options.iteritems():
    _PLOT_OPTIONS_TABLE += "    ``"+str(key)+"`` | "+str(value)+"\n"
__doc__ = __doc__.format(PLOT_OPTIONS_TABLE=_PLOT_OPTIONS_TABLE)


#*****************************************************************************
#      Copyright (C) 2009   Emily Kirkman
#                    2009   Robert L. Miller <rlmillster@gmail.com>
#
#  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.structure.sage_object import SageObject
from sage.plot.all import Graphics, scatter_plot, bezier_path, line, arrow, text, circle
from sage.misc.decorators import options
from math import sqrt, cos, sin, atan, pi

DEFAULT_SHOW_OPTIONS = {
    "figsize"             : [4,4]
    }

DEFAULT_PLOT_OPTIONS = {
    "vertex_size"         : 200,
    "vertex_labels"       : True,
    "layout"              : None,
    "edge_style"          : 'solid',
    "edge_color"          : 'black',
    "edge_colors"         : None,
    "edge_labels"         : False,
    "iterations"          : 50,
    "tree_orientation"    : 'down',
    "heights"             : None,
    "graph_border"        : False,
    "talk"                : False,
    "color_by_label"      : False,
    "partition"           : None,
    "dist"                : .075,
    "max_dist"            : 1.5,
    "loop_size"           : .075
    }

class GraphPlot(SageObject):
    def __init__(self, graph, options):
        """
        Returns a ``GraphPlot`` object, which stores all the parameters needed for
        plotting (Di)Graphs.  A ``GraphPlot`` has a plot and show function, as well
        as some functions to set parameters for vertices and edges.  This constructor
        assumes default options are set.  Defaults are shown in the example below.

        EXAMPLE::

            sage: from sage.graphs.graph_plot import GraphPlot
            sage: options = {
            ...     'vertex_size':200,
            ...     'vertex_labels':True,
            ...     'layout':None,
            ...     'edge_style':'solid',
            ...     'edge_color':'black',
            ...     'edge_colors':None,
            ...     'edge_labels':False,
            ...     'iterations':50,
            ...     'tree_orientation':'down',
            ...     'heights':None,
            ...     'graph_border':False,
            ...     'talk':False,
            ...     'color_by_label':False,
            ...     'partition':None,
            ...     'dist':.075,
            ...     'max_dist':1.5,
            ...     'loop_size':.075}
            sage: g = Graph({0:[1,2], 2:[3], 4:[0,1]})
            sage: GP = GraphPlot(g, options)

        """
        # Setting the default values if needed
        for k,value in DEFAULT_PLOT_OPTIONS.iteritems():
            if k not in options:
                options[k] = value
        self._plot_components = {}
        self._nodelist = graph.vertices()
        self._graph = graph
        self._options = options # contains both plot and show options
        self.set_pos()
        self._arcs = self._graph.has_multiple_edges(to_undirected=True)
        self._loops = self._graph.has_loops()
        if self._graph.is_directed() and self._arcs:
            self._arcdigraph = True
        else:
            self._arcdigraph = False

        self.set_vertices()
        self.set_edges()

    def _repr_(self):
        """
        Returns a string representation of a ``GraphPlot`` object.

        EXAMPLE:

        This function is called implicitly by the code below::

            sage: g = Graph({0:[1,2], 2:[3], 4:[0,1]})
            sage: g.graphplot() # indirect doctest
            GraphPlot object for Graph on 5 vertices
        """
        return "GraphPlot object for %s"%self._graph

    def set_pos(self):
        """
        Sets the position plotting parameters for this GraphPlot.

        EXAMPLES:

        This function is called implicitly by the code below::

            sage: g = Graph({0:[1,2], 2:[3], 4:[0,1]})
            sage: g.graphplot(save_pos=True, layout='circular') # indirect doctest
            GraphPlot object for Graph on 5 vertices

        The following illustrates the format of a position dictionary,
        but due to numerical noise we do not check the values themselves::

            sage: g.get_pos()
            {0: [...e-17, 1.0],
             1: [-0.951..., 0.309...],
             2: [-0.587..., -0.809...],
             3: [0.587..., -0.809...],
             4: [0.951..., 0.309...]}

        ::

            sage: T = list(graphs.trees(7))
            sage: t = T[3]
            sage: t.plot(heights={0:[0], 1:[4,5,1], 2:[2], 3:[3,6]})
            Graphics object consisting of 14 graphics primitives

        TESTS:

        Make sure that vertex locations are floats.  Not being floats
        isn't a bug in itself but makes it too easy to accidentally
        introduce a bug elsewhere, such as in :meth:`set_edges` (:trac:`10124`),
        via silent truncating division of integers::

            sage: g = graphs.FruchtGraph()
            sage: gp = g.graphplot()
            sage: set(map(type, flatten(gp._pos.values())))
            {<type 'float'>}
            sage: g = graphs.BullGraph()
            sage: gp = g.graphplot(save_pos=True)
            sage: set(map(type, flatten(gp._pos.values())))
            {<type 'float'>}

        """
        self._pos = self._graph.layout(**self._options)
        # make sure the positions are floats (trac #10124)
        self._pos = dict((k,(float(v[0]), float(v[1]))) for k,v in self._pos.iteritems())

    def set_vertices(self, **vertex_options):
        """
        Sets the vertex plotting parameters for this ``GraphPlot``.  This function
        is called by the constructor but can also be called to make updates to
        the vertex options of an existing ``GraphPlot`` object.  Note that the
        changes are cumulative.

        EXAMPLES::

            sage: g = Graph({}, loops=True, multiedges=True, sparse=True)
            sage: g.add_edges([(0,0,'a'),(0,0,'b'),(0,1,'c'),(0,1,'d'),
            ...     (0,1,'e'),(0,1,'f'),(0,1,'f'),(2,1,'g'),(2,2,'h')])
            sage: GP = g.graphplot(vertex_size=100, edge_labels=True, color_by_label=True, edge_style='dashed')
            sage: GP.set_vertices(talk=True)
            sage: GP.plot()
            Graphics object consisting of 22 graphics primitives
            sage: GP.set_vertices(vertex_colors='pink', vertex_shape='^')
            sage: GP.plot()
            Graphics object consisting of 22 graphics primitives
        """
        # Handle base vertex options
        voptions = {}

        for arg in vertex_options:
            self._options[arg] = vertex_options[arg]

        # First set defaults for styles
        vertex_colors = None
        if self._options['talk']:
            voptions['markersize'] = 500
            if self._options['partition'] is None:
                vertex_colors = '#ffffff'
        else:
            voptions['markersize'] = self._options['vertex_size']

        if 'vertex_colors' not in self._options or self._options['vertex_colors'] is None:
            if self._options['partition'] is not None:
                from sage.plot.colors import rainbow,rgbcolor
                partition = self._options['partition']
                l = len(partition)
                R = rainbow(l)
                vertex_colors = {}
                for i in range(l):
                    vertex_colors[R[i]] = partition[i]
            elif len(self._graph._boundary) != 0:
                vertex_colors = {}
                bdy_verts = []
                int_verts = []
                for v in self._graph.vertex_iterator():
                    if v in self._graph._boundary:
                        bdy_verts.append(v)
                    else:
                        int_verts.append(v)
                vertex_colors['#fec7b8'] = int_verts
                vertex_colors['#b3e8ff'] = bdy_verts
            elif not vertex_colors:
                vertex_colors='#fec7b8'
        else:
            vertex_colors = self._options['vertex_colors']

        if 'vertex_shape' in self._options:
            voptions['marker'] = self._options['vertex_shape']

        if self._graph.is_directed():
            self._vertex_radius = sqrt(voptions['markersize']/pi)
            self._arrowshorten = 2*self._vertex_radius
            if self._arcdigraph:
                self._vertex_radius = sqrt(voptions['markersize']/(20500*pi))

        voptions['zorder'] = 7

        if not isinstance(vertex_colors, dict):
            voptions['facecolor'] = vertex_colors
            if self._arcdigraph:
                self._plot_components['vertices'] = [circle(center,
                    self._vertex_radius, fill=True, facecolor=vertex_colors, edgecolor='black', clip=False)
                    for center in self._pos.values()]
            else:
                self._plot_components['vertices'] = scatter_plot(
                    self._pos.values(), clip=False, **voptions)
        else:
            # Color list must be ordered:
            pos = []
            colors = []
            for i in vertex_colors:
                pos += [self._pos[j] for j in vertex_colors[i]]
                colors += [i]*len(vertex_colors[i])

            # If all the vertices have not been assigned a color
            if len(self._pos)!=len(pos):
                from sage.plot.colors import rainbow,rgbcolor
                vertex_colors_rgb=[rgbcolor(c) for c in vertex_colors]
                for c in rainbow(len(vertex_colors)+1):
                    if rgbcolor(c) not in vertex_colors_rgb:
                        break
                leftovers=[j for j in self._pos.values() if j not in pos]
                pos+=leftovers
                colors+=[c]*len(leftovers)

            if self._arcdigraph:
                self._plot_components['vertices'] = [circle(pos[i],
                    self._vertex_radius, fill=True, facecolor=colors[i], edgecolor='black', clip=False)
                    for i in range(len(pos))]
            else:
                self._plot_components['vertices'] = scatter_plot(pos,
                    facecolor=colors, clip=False, **voptions)

        if self._options['vertex_labels']:
            self._plot_components['vertex_labels'] = []
            # TODO: allow text options
            for v in self._nodelist:
                self._plot_components['vertex_labels'].append(text(str(v),
                    self._pos[v], rgbcolor=(0,0,0), zorder=8))

    def set_edges(self, **edge_options):
        """
        Sets the edge (or arrow) plotting parameters for the ``GraphPlot`` object.

        This function is called by the constructor but can also be called to make
        updates to the vertex options of an existing ``GraphPlot`` object.  Note
        that the changes are cumulative.

        EXAMPLES::

            sage: g = Graph({}, loops=True, multiedges=True, sparse=True)
            sage: g.add_edges([(0,0,'a'),(0,0,'b'),(0,1,'c'),(0,1,'d'),
            ...     (0,1,'e'),(0,1,'f'),(0,1,'f'),(2,1,'g'),(2,2,'h')])
            sage: GP = g.graphplot(vertex_size=100, edge_labels=True, color_by_label=True, edge_style='dashed')
            sage: GP.set_edges(edge_style='solid')
            sage: GP.plot()
            Graphics object consisting of 22 graphics primitives
            sage: GP.set_edges(edge_color='black')
            sage: GP.plot()
            Graphics object consisting of 22 graphics primitives

            sage: d = DiGraph({}, loops=True, multiedges=True, sparse=True)
            sage: d.add_edges([(0,0,'a'),(0,0,'b'),(0,1,'c'),(0,1,'d'),
            ...     (0,1,'e'),(0,1,'f'),(0,1,'f'),(2,1,'g'),(2,2,'h')])
            sage: GP = d.graphplot(vertex_size=100, edge_labels=True, color_by_label=True, edge_style='dashed')
            sage: GP.set_edges(edge_style='solid')
            sage: GP.plot()
            Graphics object consisting of 24 graphics primitives
            sage: GP.set_edges(edge_color='black')
            sage: GP.plot()
            Graphics object consisting of 24 graphics primitives

        TESTS::

            sage: G = Graph("Fooba")
            sage: G.show(edge_colors={'red':[(3,6),(2,5)]})

        Verify that default edge labels are pretty close to being between the vertices
        in some cases where they weren't due to truncating division (:trac:`10124`)::

            sage: test_graphs = graphs.FruchtGraph(), graphs.BullGraph()
            sage: tol = 0.001
            sage: for G in test_graphs:
            ...       E=G.edges()
            ...       for e0, e1, elab in E:
            ...           G.set_edge_label(e0, e1, '%d %d' % (e0, e1))
            ...       gp = G.graphplot(save_pos=True,edge_labels=True)
            ...       vx = gp._plot_components['vertices'][0].xdata
            ...       vy = gp._plot_components['vertices'][0].ydata
            ...       for elab in gp._plot_components['edge_labels']:
            ...           textobj = elab[0]
            ...           x, y, s = textobj.x, textobj.y, textobj.string
            ...           v0, v1 = map(int, s.split())
            ...           vn = vector(((x-(vx[v0]+vx[v1])/2.),y-(vy[v0]+vy[v1])/2.)).norm()
            ...           assert vn < tol

        """
        for arg in edge_options:
            self._options[arg] = edge_options[arg]
        if 'edge_colors' in edge_options: self._options['color_by_label'] = False

        # Handle base edge options: thickness, linestyle
        eoptions={}
        if 'edge_style' in self._options:
            from sage.plot.misc import get_matplotlib_linestyle
            eoptions['linestyle'] = get_matplotlib_linestyle(
                                        self._options['edge_style'],
                                        return_type='long')
        if 'thickness' in self._options:
            eoptions['thickness'] = self._options['thickness']

        # Set labels param to add labels on the fly
        labels = False
        if self._options['edge_labels']:
            labels = True
            self._plot_components['edge_labels'] = []

        # Make dict collection of all edges (keep label and edge color)
        edges_to_draw = {}
        if self._options['color_by_label'] or isinstance(self._options['edge_colors'], dict):
            if self._options['color_by_label']:
                edge_colors = self._graph._color_by_label(format=self._options['color_by_label'])
            else: edge_colors = self._options['edge_colors']
            for color in edge_colors:
                for edge in edge_colors[color]:
                    key = tuple(sorted([edge[0],edge[1]]))
                    if key == (edge[0],edge[1]): head = 1
                    else: head = 0

                    if len(edge) < 3:
                        label = self._graph.edge_label(edge[0],edge[1])
                        if isinstance(label, list):
                            if key in edges_to_draw:
                                edges_to_draw[key].append((label[-1], color, head))
                            else:
                                edges_to_draw[key] = [(label[-1], color, head)]
                            for i in range(len(label)-1):
                                edges_to_draw[key].append((label[-1], color, head))
                    else:
                        label = edge[2]

                    if key in edges_to_draw:
                        edges_to_draw[key].append((label, color, head))
                    else:
                        edges_to_draw[key] = [(label, color, head)]
            # add unspecified edges in (default color black)
            for edge in self._graph.edge_iterator():
                key = tuple(sorted([edge[0],edge[1]]))
                label = edge[2]
                specified = False
                if key in edges_to_draw:
                    for old_label, old_color, old_head in edges_to_draw[key]:
                        if label == old_label:
                            specified = True
                            break
                if not specified:
                    if key == (edge[0],edge[1]): head = 1
                    else: head = 0
                    edges_to_draw[key] = [(label, 'black', head)]
        else:
            for edge in self._graph.edges(sort=True):
                key = tuple(sorted([edge[0],edge[1]]))
                if key == (edge[0],edge[1]): head = 1
                else: head = 0
                if key in edges_to_draw:
                    edges_to_draw[key].append((edge[2], self._options['edge_color'], head))
                else:
                    edges_to_draw[key] = [(edge[2], self._options['edge_color'], head)]

        if edges_to_draw:
            self._plot_components['edges'] = []
        else:
            return

        # Check for multi-edges or loops
        if self._arcs or self._loops:
            tmp = edges_to_draw.copy()
            dist = self._options['dist']*2.
            loop_size = self._options['loop_size']
            max_dist = self._options['max_dist']
            from sage.functions.all import sqrt
            for (a,b) in tmp:
                if a == b:
                    # Loops
                    distance = dist
                    local_labels = edges_to_draw.pop((a,b))
                    if len(local_labels)*dist > max_dist:
                        distance = float(max_dist)/len(local_labels)
                    curr_loop_size = loop_size
                    for i in range(len(local_labels)):
                        self._plot_components['edges'].append(circle((self._pos[a][0],self._pos[a][1]-curr_loop_size), curr_loop_size, rgbcolor=local_labels[i][1], **eoptions))
                        if labels:
                            self._plot_components['edge_labels'].append(text(local_labels[i][0], (self._pos[a][0], self._pos[a][1]-2*curr_loop_size)))
                        curr_loop_size += distance/4
                elif len(edges_to_draw[(a,b)]) > 1:
                    # Multi-edge
                    local_labels = edges_to_draw.pop((a,b))

                    # Compute perpendicular bisector
                    p1 = self._pos[a]
                    p2 = self._pos[b]
                    M = ((p1[0]+p2[0])/2., (p1[1]+p2[1])/2.) # midpoint
                    if not p1[1] == p2[1]:
                        S = float(p1[0]-p2[0])/(p2[1]-p1[1]) # perp slope
                        y = lambda x : S*x-S*M[0]+M[1] # perp bisector line

                        # f,g are functions of distance d to determine x values
                        # on line y at d from point M
                        f = lambda d : sqrt(d**2/(1.+S**2)) + M[0]
                        g = lambda d : -sqrt(d**2/(1.+S**2)) + M[0]

                        odd_x = f
                        even_x = g
                        if p1[0] == p2[0]:
                            odd_y = lambda d : M[1]
                            even_y = odd_y
                        else:
                            odd_y = lambda x : y(f(x))
                            even_y = lambda x : y(g(x))
                    else:
                        odd_x = lambda d : M[0]
                        even_x = odd_x
                        odd_y = lambda d : M[1] + d
                        even_y = lambda d : M[1] - d

                    # We now have the control points for each bezier curve
                    # in terms of distance parameter d.
                    # Also note that the label for each edge should be drawn at d/2.
                    # (This is because we're using the perp bisectors).
                    distance = dist
                    if len(local_labels)*dist > max_dist:
                        distance = float(max_dist)/len(local_labels)
                    for i in range(len(local_labels)//2):
                        k = (i+1.0)*distance
                        if self._arcdigraph:
                            odd_start = self._polar_hack_for_multidigraph(p1, [odd_x(k),odd_y(k)], self._vertex_radius)[0]
                            odd_end = self._polar_hack_for_multidigraph([odd_x(k),odd_y(k)], p2, self._vertex_radius)[1]
                            even_start = self._polar_hack_for_multidigraph(p1, [even_x(k),even_y(k)], self._vertex_radius)[0]
                            even_end = self._polar_hack_for_multidigraph([even_x(k),even_y(k)], p2, self._vertex_radius)[1]
                            self._plot_components['edges'].append(arrow(path=[[odd_start,[odd_x(k),odd_y(k)],odd_end]], head=local_labels[2*i][2], zorder=1, rgbcolor=local_labels[2*i][1], **eoptions))
                            self._plot_components['edges'].append(arrow(path=[[even_start,[even_x(k),even_y(k)],even_end]], head=local_labels[2*i+1][2], zorder=1, rgbcolor=local_labels[2*i+1][1], **eoptions))
                        else:
                            self._plot_components['edges'].append(bezier_path([[p1,[odd_x(k),odd_y(k)],p2]],zorder=1, rgbcolor=local_labels[2*i][1], **eoptions))
                            self._plot_components['edges'].append(bezier_path([[p1,[even_x(k),even_y(k)],p2]],zorder=1, rgbcolor=local_labels[2*i+1][1], **eoptions))
                        if labels:
                            j = k/2.0
                            self._plot_components['edge_labels'].append(text(local_labels[2*i][0],[odd_x(j),odd_y(j)]))
                            self._plot_components['edge_labels'].append(text(local_labels[2*i+1][0],[even_x(j),even_y(j)]))
                    if len(local_labels)%2 == 1:
                        edges_to_draw[(a,b)] = [local_labels[-1]] # draw line for last odd

        dir = self._graph.is_directed()
        for (a,b) in edges_to_draw:
            if self._arcdigraph:
                C,D = self._polar_hack_for_multidigraph(self._pos[a], self._pos[b], self._vertex_radius)
                self._plot_components['edges'].append(arrow(C,D, rgbcolor=edges_to_draw[(a,b)][0][1], head=edges_to_draw[(a,b)][0][2], **eoptions))
                if labels:
                    self._plot_components['edge_labels'].append(text(str(edges_to_draw[(a,b)][0][0]),[(C[0]+D[0])/2., (C[1]+D[1])/2.]))
            elif dir:
                self._plot_components['edges'].append(arrow(self._pos[a],self._pos[b], rgbcolor=edges_to_draw[(a,b)][0][1], arrowshorten=self._arrowshorten, head=edges_to_draw[(a,b)][0][2], **eoptions))
            else:
                self._plot_components['edges'].append(line([self._pos[a],self._pos[b]], rgbcolor=edges_to_draw[(a,b)][0][1], **eoptions))
            if labels and not self._arcdigraph:
                self._plot_components['edge_labels'].append(text(str(edges_to_draw[(a,b)][0][0]),[(self._pos[a][0]+self._pos[b][0])/2., (self._pos[a][1]+self._pos[b][1])/2.]))

    def _polar_hack_for_multidigraph(self, A, B, VR):
        """
        Helper function to quickly compute the two points of intersection of a line
        segment from A to B (xy tuples) and circles centered at A and B, both with
        radius VR.  Returns a tuple of xy tuples representing the two points.

        EXAMPLE::

            sage: d = DiGraph({}, loops=True, multiedges=True, sparse=True)
            sage: d.add_edges([(0,0,'a'),(0,0,'b'),(0,1,'c'),(0,1,'d'),
            ...     (0,1,'e'),(0,1,'f'),(0,1,'f'),(2,1,'g'),(2,2,'h')])
            sage: GP = d.graphplot(vertex_size=100, edge_labels=True, color_by_label=True, edge_style='dashed')
            sage: GP._polar_hack_for_multidigraph((0,1),(1,1),.1)
            ([0.10..., 1.00...], [0.90..., 1.00...])

        TESTS:

        Make sure that Python ints are acceptable arguments (:trac:`10124`)::

            sage: GP = DiGraph().graphplot()
            sage: GP._polar_hack_for_multidigraph((0, 1), (2, 2), .1)
            ([0.08..., 1.04...], [1.91..., 1.95...])
            sage: GP._polar_hack_for_multidigraph((int(0),int(1)),(int(2),int(2)),.1)
            ([0.08..., 1.04...], [1.91..., 1.95...])

        """
        D = [float(B[i]-A[i]) for i in range(2)]
        R = sqrt(D[0]**2+D[1]**2)
        theta = 3*pi/2
        if D[0] > 0:
            theta = atan(D[1]/D[0])
            if D[1] < 0:
                theta += 2*pi
        elif D[0] < 0:
            theta = atan(D[1]/D[0]) + pi
        elif D[1] > 0:
            theta = pi/2
        return ([VR*cos(theta)+A[0], VR*sin(theta)+A[1]], [(R-VR)*cos(theta)+A[0], (R-VR)*sin(theta)+A[1]])

    def show(self, **kwds):
        """
        Shows the (Di)Graph associated with this ``GraphPlot`` object.

        INPUT:

        This method accepts all parameters of
        :meth:`sage.plot.graphics.Graphics.show`.

        .. NOTE::

            - See :mod:`the module's documentation <sage.graphs.graph_plot>` for
              information on default values of this method.

            - Any options not used by plot will be passed on to the
              :meth:`~sage.plot.graphics.Graphics.show` method.

        EXAMPLE::

            sage: C = graphs.CubeGraph(8)
            sage: P = C.graphplot(vertex_labels=False, vertex_size=0, graph_border=True)
            sage: P.show()
        """
        # Setting the default values if needed
        for k,value in DEFAULT_SHOW_OPTIONS.iteritems():
            if k not in kwds:
                kwds[k] = value

        self.plot().show(**kwds)

    def plot(self, **kwds):
        """
        Returns a graphics object representing the (di)graph.

        INPUT:

        The options accepted by this method are to be found in the documentation
        of the :mod:`sage.graphs.graph_plot` module, and the
        :meth:`~sage.plot.graphics.Graphics.show` method.

        .. NOTE::

            See :mod:`the module's documentation <sage.graphs.graph_plot>` for
            information on default values of this method.

        We can specify some pretty precise plotting of familiar graphs::

            sage: from math import sin, cos, pi
            sage: P = graphs.PetersenGraph()
            sage: d = {'#FF0000':[0,5], '#FF9900':[1,6], '#FFFF00':[2,7], '#00FF00':[3,8], '#0000FF':[4,9]}
            sage: pos_dict = {}
            sage: for i in range(5):
            ...    x = float(cos(pi/2 + ((2*pi)/5)*i))
            ...    y = float(sin(pi/2 + ((2*pi)/5)*i))
            ...    pos_dict[i] = [x,y]
            ...
            sage: for i in range(10)[5:]:
            ...    x = float(0.5*cos(pi/2 + ((2*pi)/5)*i))
            ...    y = float(0.5*sin(pi/2 + ((2*pi)/5)*i))
            ...    pos_dict[i] = [x,y]
            ...
            sage: pl = P.graphplot(pos=pos_dict, vertex_colors=d)
            sage: pl.show()

        Here are some more common graphs with typical options::

            sage: C = graphs.CubeGraph(8)
            sage: P = C.graphplot(vertex_labels=False, vertex_size=0, graph_border=True)
            sage: P.show()

            sage: G = graphs.HeawoodGraph().copy(sparse=True)
            sage: for u,v,l in G.edges():
            ...    G.set_edge_label(u,v,'(' + str(u) + ',' + str(v) + ')')
            sage: G.graphplot(edge_labels=True).show()

        The options for plotting also work with directed graphs::

            sage: D = DiGraph( { 0: [1, 10, 19], 1: [8, 2], 2: [3, 6], 3: [19, 4], 4: [17, 5], 5: [6, 15], 6: [7], 7: [8, 14], 8: [9], 9: [10, 13], 10: [11], 11: [12, 18], 12: [16, 13], 13: [14], 14: [15], 15: [16], 16: [17], 17: [18], 18: [19], 19: []})
            sage: for u,v,l in D.edges():
            ...    D.set_edge_label(u,v,'(' + str(u) + ',' + str(v) + ')')
            sage: D.graphplot(edge_labels=True, layout='circular').show()

        This example shows off the coloring of edges::

            sage: from sage.plot.colors import rainbow
            sage: C = graphs.CubeGraph(5)
            sage: R = rainbow(5)
            sage: edge_colors = {}
            sage: for i in range(5):
            ...    edge_colors[R[i]] = []
            sage: for u,v,l in C.edges():
            ...    for i in range(5):
            ...        if u[i] != v[i]:
            ...            edge_colors[R[i]].append((u,v,l))
            sage: C.graphplot(vertex_labels=False, vertex_size=0, edge_colors=edge_colors).show()


        With the ``partition`` option, we can separate out same-color groups
        of vertices::

            sage: D = graphs.DodecahedralGraph()
            sage: Pi = [[6,5,15,14,7],[16,13,8,2,4],[12,17,9,3,1],[0,19,18,10,11]]
            sage: D.show(partition=Pi)

        Loops are also plotted correctly::

            sage: G = graphs.PetersenGraph()
            sage: G.allow_loops(True)
            sage: G.add_edge(0,0)
            sage: G.show()

        ::

            sage: D = DiGraph({0:[0,1], 1:[2], 2:[3]}, loops=True)
            sage: D.show()
            sage: D.show(edge_colors={(0,1,0):[(0,1,None),(1,2,None)],(0,0,0):[(2,3,None)]})

        More options::

            sage: pos = {0:[0.0, 1.5], 1:[-0.8, 0.3], 2:[-0.6, -0.8], 3:[0.6, -0.8], 4:[0.8, 0.3]}
            sage: g = Graph({0:[1], 1:[2], 2:[3], 3:[4], 4:[0]})
            sage: g.graphplot(pos=pos, layout='spring', iterations=0).plot()
            Graphics object consisting of 11 graphics primitives

            sage: G = Graph()
            sage: P = G.graphplot().plot()
            sage: P.axes()
            False
            sage: G = DiGraph()
            sage: P = G.graphplot().plot()
            sage: P.axes()
            False

        We can plot multiple graphs::

            sage: T = list(graphs.trees(7))
            sage: t = T[3]
            sage: t.graphplot(heights={0:[0], 1:[4,5,1], 2:[2], 3:[3,6]}).plot()
            Graphics object consisting of 14 graphics primitives

        ::

            sage: T = list(graphs.trees(7))
            sage: t = T[3]
            sage: t.graphplot(heights={0:[0], 1:[4,5,1], 2:[2], 3:[3,6]}).plot()
            Graphics object consisting of 14 graphics primitives
            sage: t.set_edge_label(0,1,-7)
            sage: t.set_edge_label(0,5,3)
            sage: t.set_edge_label(0,5,99)
            sage: t.set_edge_label(1,2,1000)
            sage: t.set_edge_label(3,2,'spam')
            sage: t.set_edge_label(2,6,3/2)
            sage: t.set_edge_label(0,4,66)
            sage: t.graphplot(heights={0:[0], 1:[4,5,1], 2:[2], 3:[3,6]}, edge_labels=True).plot()
            Graphics object consisting of 20 graphics primitives

        ::

            sage: T = list(graphs.trees(7))
            sage: t = T[3]
            sage: t.graphplot(layout='tree').show()

        The tree layout is also useful::

            sage: t = DiGraph('JCC???@A??GO??CO??GO??')
            sage: t.graphplot(layout='tree', tree_root=0, tree_orientation="up").show()

        More examples::

            sage: D = DiGraph({0:[1,2,3], 2:[1,4], 3:[0]})
            sage: D.graphplot().show()

            sage: D = DiGraph(multiedges=True, sparse=True)
            sage: for i in range(5):
            ...     D.add_edge((i,i+1,'a'))
            ...     D.add_edge((i,i-1,'b'))
            sage: D.graphplot(edge_labels=True,edge_colors=D._color_by_label()).plot()
            Graphics object consisting of 34 graphics primitives

            sage: g = Graph({}, loops=True, multiedges=True, sparse=True)
            sage: g.add_edges([(0,0,'a'),(0,0,'b'),(0,1,'c'),(0,1,'d'),
            ...     (0,1,'e'),(0,1,'f'),(0,1,'f'),(2,1,'g'),(2,2,'h')])
            sage: g.graphplot(edge_labels=True, color_by_label=True, edge_style='dashed').plot()
            Graphics object consisting of 22 graphics primitives

        The ``edge_style`` option may be provided in the short format too::

            sage: g.graphplot(edge_labels=True, color_by_label=True, edge_style='--').plot()
            Graphics object consisting of 22 graphics primitives

        TESTS:

        Make sure that show options work with plot also::

            sage: g = Graph({})
            sage: g.plot(title='empty graph', axes=True)
            Graphics object consisting of 0 graphics primitives

        Check for invalid inputs::

            sage: p = graphs.PetersenGraph().plot(egabrag='garbage')
            Traceback (most recent call last):
            ...
            ValueError: Invalid input 'egabrag=garbage'

        Make sure that no graphics primitive is clipped::

            sage: tadpole = Graph({0:[0,1]}).plot()
            sage: bbox = tadpole.get_minmax_data()
            sage: for part in tadpole:
            ....:      part_bbox = part.get_minmax_data()
            ....:      assert bbox['xmin'] <= part_bbox['xmin'] <= part_bbox['xmax'] <= bbox['xmax']
            ....:      assert bbox['ymin'] <= part_bbox['ymin'] <= part_bbox['ymax'] <= bbox['ymax']
        """
        G = Graphics()
        options = self._options.copy()
        options.update(kwds)
        G._set_extra_kwds(Graphics._extract_kwds_for_show(options))

        # Check the arguments
        for o in options:
            if o not in graphplot_options and o not in G._extra_kwds:
                raise ValueError("Invalid input '{}={}'".format(o, options[o]))

        for comp in self._plot_components.values():
            if not isinstance(comp, list):
                G += comp
            else:
                for item in comp:
                    G += item

        if self._options['graph_border']:
            xmin = G.xmin()
            xmax = G.xmax()
            ymin = G.ymin()
            ymax = G.ymax()
            dx = (xmax-xmin)/10.0
            dy = (ymax-ymin)/10.0
            border = (line([( xmin - dx, ymin - dy), ( xmin - dx, ymax + dy ), ( xmax + dx, ymax + dy ), ( xmax + dx, ymin - dy ), ( xmin - dx, ymin - dy )], thickness=1.3))
            border.axes_range(xmin = (xmin - dx), xmax = (xmax + dx), ymin = (ymin - dy), ymax = (ymax + dy))
            G += border
        G.set_aspect_ratio(1)
        G.axes(False)
        return G

    def layout_tree(self,root,orientation):
        """
        Compute a nice layout of a tree.

        INPUT:

        - ``root`` -- the root vertex.

        - ``orientation`` -- Whether to place the root
          at the top or at the bottom :

            - ``orientation="down"`` -- children are placed below
              their parent
            - ``orientation="top"`` -- children are placed above
              their parent

        EXAMPLES::

            sage: T = graphs.RandomLobster(25,0.3,0.3)
            sage: T.show(layout='tree',tree_orientation='up') # indirect doctest

            sage: from sage.graphs.graph_plot import GraphPlot
            sage: G = graphs.HoffmanSingletonGraph()
            sage: T = Graph()
            sage: T.add_edges(G.min_spanning_tree(starting_vertex=0))
            sage: T.show(layout='tree',tree_root=0) # indirect doctest

        """

        T = self._graph

        if not self._graph.is_tree():
            raise RuntimeError("Cannot use tree layout on this graph: self.is_tree() returns False.")

        children = {root:T.neighbors(root)}

        #always make a copy of the children because they get eaten
        stack = [[u for u in children[root]]]
        stick = [root]
        parent = dict([(u,root) for u in children[root]])
        pos = {}
        obstruction = [0.0]*T.num_verts()
        if orientation == 'down':
            o = -1
        else:
            o = 1

        def slide(v,dx):
            """

            shift the vertex v and its descendants to the right by dx

            Precondition: v and its descendents have already had their
            positions computed.

            """

            level = [v]
            while level:
                nextlevel = []
                for u in level:
                    x,y = pos[u]
                    x+= dx
                    obstruction[y] = max(x+1, obstruction[y])
                    pos[u] = x,y
                    nextlevel += children[u]

                level = nextlevel

        while stack:
            C = stack[-1]
            if len(C) == 0:
                p = stick.pop()
                stack.pop()
                cp = children[p]
                y = o*len(stack)
                if len(cp) == 0:
                    x = obstruction[y]
                    pos[p] = x,y
                else:
                    x = sum([pos[c][0] for c in cp])/(float(len(cp)))
                    pos[p] = x,y
                    ox = obstruction[y]
                    if x < ox:
                        slide(p,ox-x)
                        x = ox
                obstruction[y] = x+1
                continue

            t = C.pop()
            pt = parent[t]

            ct = [u for u in T.neighbors(t) if u != pt]
            for c in ct:
                parent[c] = t
            children[t] = ct

            stack.append([c for c in ct])
            stick.append(t)

        return pos

####################
# Helper functions #
####################

def _circle_embedding(g, vertices, center=(0, 0), radius=1, shift=0):
    r"""
    Sets some vertices on a circle in the embedding of a graph G.

    This method modifies the graph's embedding so that the vertices
    listed in ``vertices`` appear in this ordering on a circle of given
    radius and center. The ``shift`` parameter is actually a rotation of
    the circle. A value of ``shift=1`` will replace in the drawing the
    `i`-th element of the list by the `(i-1)`-th. Non-integer values are
    admissible, and a value of `\alpha` corresponds to a rotation of the
    circle by an angle of `\alpha 2\pi/n` (where `n` is the number of
    vertices set on the circle).

    EXAMPLE::

        sage: from sage.graphs.graph_plot import _circle_embedding
        sage: g = graphs.CycleGraph(5)
        sage: _circle_embedding(g, [0, 2, 4, 1, 3], radius=2, shift=.5)
        sage: g.show()
    """
    c_x, c_y = center
    n = len(vertices)
    d = g.get_pos()
    if d is None:
        d = {}

    for i,v in enumerate(vertices):
        i += shift
        v_x = c_x + radius * cos(2*i*pi / n)
        v_y = c_y + radius * sin(2*i*pi / n)
        d[v] = (v_x, v_y)

    g.set_pos(d)

def _line_embedding(g, vertices, first=(0, 0), last=(0, 1)):
    r"""
    Sets some vertices on a line in the embedding of a graph G.

    This method modifies the graph's embedding so that the vertices of
    ``vertices`` appear on a line, where the position of ``vertices[0]``
    is the pair ``first`` and the position of ``vertices[-1]`` is
    ``last``. The vertices are evenly spaced.

    EXAMPLE::

        sage: from sage.graphs.graph_plot import _line_embedding
        sage: g = graphs.PathGraph(5)
        sage: _line_embedding(g, [0, 2, 4, 1, 3], first=(-1, -1), last=(1, 1))
        sage: g.show()
    """
    n = len(vertices) - 1.

    fx, fy = first
    dx = (last[0] - first[0])/n
    dy = (last[1] - first[1])/n

    d = g.get_pos()
    if d is None:
        d = {}

    for v in vertices:
        d[v] = (fx, fy)
        fx += dx
        fy += dy
