text: r"""
  Helper functions for plotting the geometric representation of matroids
  
  
  AUTHORS:
  
  - Jayant Apte (2014-06-06): initial version
  
  .. NOTE::
  
      This file provides functions that are called by ``show()`` and ``plot()``
      methods of abstract matroids class. The basic idea is to first decide
      the placement of points in $\mathbb{R}^2$ and then draw lines in geometric
      representation through these points. Point placement procedures such as
      ``addtripts``, ``addnontripts`` together produce ``(x,y)`` tuples
      corresponding to ground set of the matroid in a dictionary.
      These methods provide simple but rigid point placement algorithm.
      Alternatively, one can build the point placement dictionary manually or
      via an optimization that gives aesthetically pleasing point placement (in
      some sense. This is not yet implemented).    One can then use
      ``createline`` function to produce sequence of ``100`` points on a smooth
      curve containing the points in the specified line which inturn uses
      ``scipy.interpolate.splprep`` and ``scipy.interpolate.splev``.  Then one
      can use sage's graphics primitives ``line``, ``point``, ``text`` and
      ``points`` to produce graphics object containing points (ground set
      elements) and lines (for a rank 3 matroid, these are flats of rank 2 of
      size greater than equal to 3) of the geometric representation of the
      matroid. Loops and parallel elements are added as per conventions in
      [Oxl2011]_ using function ``addlp``. The priority order for point placement
      methods used inside plot() and show() is as follows:
  
      1. User Specified points dictionary and lineorders
      2. cached point placement dictionary and line orders (a list of ordered
         lists) in M._cached_info (a dictionary)
      3. Internal point placement and orders deciding heuristics
         If a custom point placement and/or line orders is desired, then user
         can simply specify the custom points dictionary as::
  
              M.cached info = {'plot_positions':<dictionary_of _points>,
                               'plot_lineorders':<list of lists>}
  
  
  
  REFERENCES
  ==========
  
  - [Oxl2011]_ James Oxley, "Matroid Theory, Second Edition". Oxford University
    Press, 2011.
  
  EXAMPLES::
  
      sage: from sage.matroids import matroids_plot_helpers
      sage: M1=Matroid(ring=GF(2), matrix=[[1, 0, 0, 0, 1, 1, 1,0,1,0,1],
      ....: [0, 1, 0, 1, 0, 1, 1,0,0,1,0], [0, 0, 1, 1, 1, 0, 1,0,0,0,0]])
      sage: pos_dict= {0: (0, 0),  1: (2, 0),  2: (1, 2),  3: (1.5, 1.0),
      ....: 4: (0.5, 1.0),  5: (1.0, 0.0), 6: (1.0, 0.666666666666667),
      ....: 7: (3,3), 8: (4,0), 9: (-1,1), 10: (-2,-2)}
      sage: M1._cached_info={'plot_positions': pos_dict, 'plot_lineorders': None}
      sage: matroids_plot_helpers.geomrep(M1, sp=True)
      Graphics object consisting of 22 graphics primitives
  
  """
  # *****************************************************************************
  #       Copyright (C) 2013 Jayant Apte <jayant91089@gmail.com>
  #
  # This program is free software: you can redistribute it and/or modify
  # it under the terms of the GNU General Public License as published by
  # the Free Software Foundation, either version 3 of the License, or
  # (at your option) any later version.
  #                  https://www.gnu.org/licenses/
  # *****************************************************************************
  
  import scipy
  import scipy.interpolate
  import numpy as np
  from sage.misc.lazy_import import lazy_import
  lazy_import("sage.plot.all", ["Graphics", "line", "text", "polygon2d", "point", "points"])
  lazy_import("sage.plot.colors", "Color")
  from sage.sets.set import Set
  from sage.matroids.advanced import newlabel
  
  
  def it(M, B1, nB1, lps):
      r"""
      Return points on and off the triangle and lines to be drawn for a rank 3
      matroid.
  
      INPUT:
  
      - ``M`` -- A matroid.
      - ``B1``-- A list of groundset elements of ``M`` that corresponds to a
        basis of matroid ``M``.
      - ``nB1``-- A list of elements in the ground set of M that corresponds to
        ``M.simplify.groundset() \ B1``.
      - ``lps``-- A list of elements in the ground set of matroid M that are
        loops.
  
      OUTPUT:
  
      A tuple containing 4 elements in this order:
  
      1. A dictionary containing 2-tuple (x,y) coordinates with
         ``M.simplify.groundset()`` elements that can be placed on the sides of
         the triangle as keys.
      2. A list of 3 lists of elements of ``M.simplify.groundset()`` that can
         be placed on the 3 sides of the triangle.
      3. A list of elements of `M.simplify.groundset()`` that cane be placed
         inside the triangle in the geometric representation.
      4. A list of lists of elements of ``M.simplify.groundset()`` that
         correspond to lines in the geometric representation other than the sides
         of the triangle.
  
      EXAMPLES::
  
          sage: from sage.matroids import matroids_plot_helpers as mph
          sage: M = Matroid(ring=GF(2), matrix=[[1, 0, 0, 0, 1, 1, 1,0],
          ....: [0, 1, 0, 1, 0, 1, 1,0],[0, 0, 1, 1, 1, 0, 1,0]])
          sage: N = M.simplify()
          sage: B1 = list(N.basis())
          sage: nB1 = list(set(M.simplify().groundset())-set(B1))
          sage: pts,trilines,nontripts,curvedlines = mph.it(M,
          ....: B1,nB1,M.loops())
          sage: pts
          {1: (1.0, 0.0), 2: (1.5, 1.0), 3: (0.5, 1.0),
           4: (0, 0), 5: (1, 2), 6: (2, 0)}
          sage: trilines
          [[3, 4, 5], [2, 5, 6], [1, 4, 6]]
          sage: nontripts
          [0]
          sage: curvedlines
          [[0, 1, 5], [0, 2, 4], [0, 3, 6], [1, 2, 3], [1, 4, 6], [2, 5, 6],
           [3, 4, 5]]
  
      .. NOTE::
  
          This method does NOT do any checks.
      """
      tripts = [(0, 0), (1, 2), (2, 0)]
      pts = {}
      j = 0
      for i in B1:
          pts[i] = tripts[j]
          j = j + 1
      pairs = [[0, 1], [1, 2], [0, 2]]
      L1 = []
      L2 = []
      L3 = []
      for i in nB1:
          if M.is_dependent([i, B1[pairs[0][0]], B1[pairs[0][1]]]):
              # Add to L1
              L1.append(i)
          elif M.is_dependent([i, B1[pairs[1][0]], B1[pairs[1][1]]]):
              # Add to L2
              L2.append(i)
          elif M.is_dependent([i, B1[pairs[2][0]], B1[pairs[2][1]]]):
              # Add to L3
              L3.append(i)
      L = [L1, L2, L3]  # megalist
      lines = []  # the list of lines
      for i in range(1, len(L)+1):
          lines.append([B1[pairs[i-1][0]]])
          lines[i-1].extend(L[i-1])
          lines[i-1].extend([B1[pairs[i-1][1]]])
      # place triangle and L1,L2,L3
      for i in L:  # loop over megalist
          interval = 1/float(len(i)+1)
          pt1 = list(tripts[pairs[L.index(i)][0]])
          pt2 = list(tripts[pairs[L.index(i)][1]])
          for j in range(1, len(i)+1):
              # loop over L1,L2,L3
              cc = interval*j
              pts[i[j-1]] = (cc*pt1[0]+(1-cc)*pt2[0], cc*pt1[1]+(1-cc)*pt2[1])
      trilines = [list(set(x)) for x in lines if len(x) >= 3]
      curvedlines = [list(set(list(x)).difference(set(lps)))
                     for x in M.flats(2) if set(list(x)) not in trilines if
                     len(list(x)) >= 3]
      nontripts = [i for i in nB1 if i not in pts.keys()]
      return pts, trilines, nontripts, curvedlines
  
  
  def trigrid(tripts):
      """
      Return a grid of 4 points inside given 3 points as a list.
  
      INPUT:
  
      - ``tripts`` -- A list of 3 lists of the form [x,y] where x and y are the
        Cartesian coordinates of a point.
  
      OUTPUT:
  
      A list of lists containing 4 points in following order:
  
      - 1. Barycenter of 3 input points.
      - 2,3,4. Barycenters of 1. with 3 different 2-subsets of input points
        respectively.
  
      EXAMPLES::
  
          sage: from sage.matroids import matroids_plot_helpers
          sage: points = matroids_plot_helpers.trigrid([[2,1],[4,5],[5,2]])
          sage: points
          [[3.6666666666666665, 2.6666666666666665],
           [3.222222222222222, 2.888888888888889],
           [4.222222222222222, 3.222222222222222],
           [3.5555555555555554, 1.8888888888888886]]
  
      .. NOTE::
  
              This method does NOT do any checks.
  
      """
      pairs = [[0, 1], [1, 2], [0, 2]]
      cpt = list((float(tripts[0][0]+tripts[1][0]+tripts[2][0])/3,
                 float(tripts[0][1]+tripts[1][1]+tripts[2][1])/3))
      grid = [cpt]
      for p in pairs:
          pt = list((float(tripts[p[0]][0]+tripts[p[1]][0]+cpt[0])/3,
                    float(tripts[p[0]][1]+tripts[p[1]][1]+cpt[1])/3))
          grid.append(pt)
      return grid
  
  
  def addnontripts(tripts_labels, nontripts_labels, ptsdict):
      """
      Return modified ``ptsdict`` with additional keys and values corresponding
      to ``nontripts``.
  
      INPUT:
  
      - ``tripts`` -- A list of 3 ground set elements that are to be placed on
        vertices of the triangle.
      - ``ptsdict`` -- A dictionary (at least) containing ground set elements in
        ``tripts`` as keys and their (x,y) position as values.
      - ``nontripts``-- A list of ground set elements whose corresponding points
        are to be placed inside the triangle.
  
      OUTPUT:
  
      A dictionary containing ground set elements in ``tripts`` as keys and
      their (x,y) position as values along with all keys and respective values
      in ``ptsdict``.
  
      EXAMPLES::
  
          sage: from sage.matroids import matroids_plot_helpers
          sage: from sage.matroids.advanced import setprint
          sage: ptsdict={'a':(0,0),'b':(1,2),'c':(2,0)}
          sage: ptsdict_1=matroids_plot_helpers.addnontripts(['a','b','c'],
          ....:         ['d','e','f'],ptsdict)
          sage: setprint(ptsdict_1)
          {'a': [0, 0], 'b': [1, 2], 'c': [0, 2], 'd': [0.6666666666666666, 1.0],
          'e': [0.6666666666666666, 0.8888888888888888],
          'f': [0.8888888888888888, 1.3333333333333333]}
          sage: ptsdict_2=matroids_plot_helpers.addnontripts(['a','b','c'],
          ....:         ['d','e','f','g','h'],ptsdict)
          sage: setprint(ptsdict_2)
          {'a': [0, 0], 'b': [1, 2], 'c': [0, 2], 'd': [0.6666666666666666, 1.0],
          'e': [0.6666666666666666, 0.8888888888888888],
          'f': [0.8888888888888888, 1.3333333333333333],
          'g': [0.2222222222222222, 1.0],
          'h': [0.5185185185185185, 0.5555555555555555]}
  
      .. NOTE::
  
              This method does NOT do any checks.
  
      """
      tripts = [list(ptsdict[p]) for p in tripts_labels]
      pairs = [[0, 1], [1, 2], [0, 2]]
      q = [tripts]
      num = len(nontripts_labels)
      gridpts = [[float((tripts[0][0]+tripts[1][0]+tripts[2][0])/3),
                 float(tripts[0][1]+tripts[1][1]+tripts[2][1])/3]]
      n = 0
      while n < num+1:
          g = trigrid(q[0])
          q.extend([[g[0], q[0][pairs[0][0]], q[0][pairs[0][1]]],
                    [g[0], q[0][pairs[1][0]], q[0][pairs[1][1]]],
                    [g[0], q[0][pairs[2][0]], q[0][pairs[2][1]]]])
          q.remove(q[0])
          gridpts.extend(g[1:])
          if n == 0:
              n = n + 4
          else:
              n = n + 3
      for j, p in enumerate(nontripts_labels):
          ptsdict[p] = tuple(gridpts[j])
      return ptsdict
  
  
  def createline(ptsdict, ll, lineorders2=None):
      """
      Return ordered lists of coordinates of points to be traversed to draw a
      2D line.
  
      INPUT:
  
      - ``ptsdict`` -- A dictionary containing keys and their (x,y) position as
        values.
      - ``ll`` -- A list of keys in ``ptsdict`` through which a line is to be
        drawn.
      - ``lineorders2``-- (optional) A list of ordered lists of keys in
        ``ptsdict`` such that if ll is setwise same as any of these then points
        corresponding to values of the keys will be traversed in that order thus
        overriding internal order deciding heuristic.
  
      OUTPUT:
  
      A tuple containing 4 elements in this order:
  
      1. Ordered list of x-coordinates of values of keys in ``ll`` specified in
         ptsdict.
      2. Ordered list of y-coordinates of values of keys in ``ll`` specified
         in ptsdict.
      3. Ordered list of interpolated x-coordinates of points through which a
         line can be drawn.
      4. Ordered list of interpolated y-coordinates of points through which a
         line can be drawn.
  
      EXAMPLES::
  
          sage: from sage.matroids import matroids_plot_helpers
          sage: ptsdict={'a':(1,3),'b':(2,1),'c':(4,5),'d':(5,2)}
          sage: x,y,x_i,y_i=matroids_plot_helpers.createline(ptsdict,
          ....: ['a','b','c','d'])
          sage: [len(x), len(y), len(x_i), len(y_i)]
          [4, 4, 100, 100]
          sage: G = line(zip(x_i, y_i),color='black',thickness=3,zorder=1)
          sage: G+=points(zip(x, y), color='black', size=300,zorder=2)
          sage: G.show()
          sage: x,y,x_i,y_i=matroids_plot_helpers.createline(ptsdict,
          ....: ['a','b','c','d'],lineorders2=[['b','a','c','d'],
          ....: ['p','q','r','s']])
          sage: [len(x), len(y), len(x_i), len(y_i)]
          [4, 4, 100, 100]
          sage: G = line(zip(x_i, y_i),color='black',thickness=3,zorder=1)
          sage: G+=points(zip(x, y), color='black', size=300,zorder=2)
          sage: G.show()
  
      .. NOTE::
  
              This method does NOT do any checks.
  
      """
      x, lo = line_hasorder(ll, lineorders2)
      flip = False
      if not x:
          # convert dictionary to list of lists
          linepts = [list(ptsdict[i]) for i in ll]
          xpts = [xx[0] for xx in linepts]
          ypts = [yy[1] for yy in linepts]
          xdim = (float(max(xpts))-float(min(xpts)))
          ydim = (float(max(ypts))-float(min(ypts)))
          if xdim > ydim:
              sortedind = sorted(range(len(xpts)), key=lambda k: float(xpts[k]))
          else:
              sortedind = sorted(range(len(ypts)), key=lambda k: float(ypts[k]))
              flip = True
          sortedlinepts = [linepts[i] for i in sortedind]
          sortedx = [k[0] for k in sortedlinepts]
          sortedy = [k[1] for k in sortedlinepts]
      else:
          linepts = [list(ptsdict[i]) for i in lo]
          sortedx = [k[0] for k in linepts]
          sortedy = [k[1] for k in linepts]
  
      if flip:
          tck, u = scipy.interpolate.splprep([sortedy, sortedx], s=0.0, k=2)
          y_i, x_i = scipy.interpolate.splev(np.linspace(0, 1, 100), tck)
      else:
          tck, u = scipy.interpolate.splprep([sortedx, sortedy], s=0.0, k=2)
          x_i, y_i = scipy.interpolate.splev(np.linspace(0, 1, 100), tck)
      return sortedx, sortedy, x_i, y_i
  
  
  def slp(M1, pos_dict=None, B=None):
      """
      Return simple matroid, loops and parallel elements of given matroid.
  
      INPUT:
  
      - ``M1`` -- A matroid.
      - ``pos_dict`` -- (optional) A dictionary containing non loopy elements of
        ``M`` as keys and their (x,y) positions.
        as keys. While simplifying the matroid, all except one element in a
        parallel class that is also specified in ``pos_dict`` will be retained.
      - ``B`` -- (optional) A basis of M1 that has been chosen for placement on
        vertices of triangle.
  
      OUTPUT:
  
      A tuple containing 3 elements in this order:
  
      1. Simple matroid corresponding to ``M1``.
      2. Loops of matroid ``M1``.
      3. Elements that are in `M1.groundset()` but not in ground set of 1 or
         in 2
  
      EXAMPLES::
  
          sage: from sage.matroids import matroids_plot_helpers
          sage: from sage.matroids.advanced import setprint
          sage: M1=Matroid(ring=GF(2), matrix=[[1, 0, 0, 0, 1, 1, 1,0,1,0,1],
          ....: [0, 1, 0, 1, 0, 1, 1,0,0,1,0],[0, 0, 1, 1, 1, 0, 1,0,0,0,0]])
          sage: [M,L,P]=matroids_plot_helpers.slp(M1)
          sage: M.is_simple()
          True
          sage: setprint([L,P])
          [{10, 8, 9}, {7}]
          sage: M1=Matroid(ring=GF(2), matrix=[[1, 0, 0, 0, 1, 1, 1,0,1,0,1],
          ....: [0, 1, 0, 1, 0, 1, 1,0,0,1,0],[0, 0, 1, 1, 1, 0, 1,0,0,0,0]])
          sage: posdict= {8: (0, 0),  1: (2, 0),  2: (1, 2),  3: (1.5, 1.0),
          ....: 4: (0.5, 1.0),  5: (1.0, 0.0), 6: (1.0, 0.6666666666666666)}
          sage: [M,L,P]=matroids_plot_helpers.slp(M1,pos_dict=posdict)
          sage: M.is_simple()
          True
          sage: setprint([L,P])
          [{0, 10, 9}, {7}]
  
      .. NOTE::
  
              This method does NOT do any checks.
  
      """
      L = set(M1.loops())
      nP = L | set(M1.simplify().groundset())
      P = set(M1.groundset())-nP
      if P:
          if pos_dict is not None:
              pcls = list(set([frozenset(set(M1.closure([p])) - L)
                               for p in list(P)]))
              newP = []
              for pcl in pcls:
                  pcl_in_dict = [p for p in list(pcl) if p in pos_dict.keys()]
                  newP.extend(list(pcl-set([pcl_in_dict[0]])))
              return [M1.delete(L | set(newP)), L, set(newP)]
          elif B is not None:
              pcls = list(set([frozenset(set(M1.closure([p])) - L)
                               for p in list(P)]))
              newP = []
              for pcl in pcls:
                  pcl_list = list(pcl)
                  pcl_in_basis = [p for p in pcl_list if p in B]
                  if pcl_in_basis:
                      newP.extend(list(pcl - set([pcl_in_basis[0]])))
                  else:
                      newP.extend(list(pcl - set([pcl_list[0]])))
              return [M1.delete(L | set(newP)), L, set(newP)]
          else:
              return [M1.delete(L | P), L, P]
      else:
          return [M1.delete(L | P), L, P]
  
  
  def addlp(M, M1, L, P, ptsdict, G=None, limits=None):
      """
      Return a graphics object containing loops (in inset) and parallel elements
      of matroid.
  
      INPUT:
  
      - ``M`` -- A matroid.
      - ``M1`` -- A simple matroid corresponding to ``M``.
      - ``L`` -- List of elements in ``M.groundset()`` that are loops of matroid
        ``M``.
      - ``P`` -- List of elements in ``M.groundset()`` not in
        ``M.simplify.groundset()`` or ``L``.
      - ``ptsdict`` -- A dictionary containing elements in ``M.groundset()`` not
        necessarily containing elements of ``L``.
      - ``G`` -- (optional) A sage graphics object to which loops and parallel
        elements of matroid `M` added .
      - ``limits``-- (optional) Current axes limits [xmin,xmax,ymin,ymax].
  
      OUTPUT:
  
      A 2-tuple containing:
  
      1. A sage graphics object containing loops and parallel elements of
         matroid ``M``
      2. axes limits array
  
      EXAMPLES::
  
          sage: from sage.matroids import matroids_plot_helpers
          sage: M=Matroid(ring=GF(2), matrix=[[1, 0, 0, 0, 1, 1, 1,0,1],
          ....: [0, 1, 0, 1, 0, 1, 1,0,0],[0, 0, 1, 1, 1, 0, 1,0,0]])
          sage: [M1,L,P]=matroids_plot_helpers.slp(M)
          sage: G,lims=matroids_plot_helpers.addlp(M,M1,L,P,{0:(0,0)})
          sage: G.show(axes=False)
  
      .. NOTE::
  
              This method does NOT do any checks.
  
      """
      if G is None:
          G = Graphics()
      # deal with loops
      if L:
          loops = L
          looptext = ", ".join(str(l) for l in loops)
          if limits is None:
              rectx = -1
              recty = -1
          else:
              rectx = limits[0]
              recty = limits[2]-1
          rectw = 0.5 + 0.4*len(loops) + 0.5  # controlled based on len(loops)
          recth = 0.6
          G += polygon2d([[rectx, recty], [rectx, recty+recth],
                          [rectx+rectw, recty+recth], [rectx+rectw, recty]],
                         color='black', fill=False, thickness=4)
          G += text(looptext, (rectx+0.5, recty+0.3), color='black',
                    fontsize=13)
          G += point((rectx+0.2, recty+0.3), color=Color('#BDBDBD'), size=300,
                     zorder=2)
          G += text('Loop(s)', (rectx+0.5+0.4*len(loops)+0.1, recty+0.3),
                    fontsize=13, color='black')
          limits = tracklims(limits, [rectx, rectx+rectw], [recty, recty+recth])
      # deal with parallel elements
      if P:
          # create list of lists where inner lists are parallel classes
          pcls = []
          gnd = sorted(M1.groundset())
          for g in gnd:
              pcl = [g]
              for p in P:
                  if M.rank([g, p]) == 1:
                      pcl.extend([p])
              pcls.append(pcl)
          ext_gnd = list(M.groundset())
          for pcl in pcls:
              if len(pcl) > 1:
                  basept = list(ptsdict[pcl[0]])
                  if len(pcl) <= 2:
                      # add side by side
                      ptsdict[pcl[1]] = (basept[0], basept[1]-0.13)
                      G += points(zip([basept[0]], [basept[1]-0.13]),
                                  color=Color('#BDBDBD'), size=300, zorder=2)
                      G += text(pcl[0], (float(basept[0]),
                                float(basept[1])), color='black',
                                fontsize=13)
                      G += text(pcl[1], (float(basept[0]),
                                float(basept[1])-0.13), color='black',
                                fontsize=13)
                      limits = tracklims(limits, [basept[0]], [basept[1]-0.13])
                  else:
                      # add in a bracket
                      pce = sorted([str(kk) for kk in pcl])
                      l = newlabel(set(ext_gnd))
                      ext_gnd.append(l)
                      G += text(l+'={ '+", ".join(pce)+' }', (float(basept[0]),
                                float(basept[1]-0.2)-0.034), color='black',
                                fontsize=13)
                      G += text(l, (float(basept[0]),
                                float(basept[1])), color='black',
                                fontsize=13)
                      limits = tracklims(limits, [basept[0]],
                                         [(basept[1]-0.2)-0.034])
      return G, limits
  
  
  def line_hasorder(l, lodrs=None):
      """
      Determine if an order is specified for a line
  
      INPUT:
  
      - ``l`` -- A line specified as a list of ground set elements.
      - ``lordrs`` -- (optional) A list of lists each specifying an order on
        a subset of ground set elements that may or may not correspond to a
        line in geometric representation.
  
      OUTPUT:
  
      A tuple containing 2 elements in this order:
  
      1. A boolean indicating whether there is any list in ``lordrs`` that is
         setwise equal to ``l``.
      2. A list specifying an order on ``set(l)`` if 1. is True, otherwise
         an empty list.
  
      EXAMPLES::
  
          sage: from sage.matroids import matroids_plot_helpers
          sage: matroids_plot_helpers.line_hasorder(['a','b','c','d'],
          ....: [['a','c','d','b'],['p','q','r']])
          (True, ['a', 'c', 'd', 'b'])
          sage: matroids_plot_helpers.line_hasorder(['a','b','c','d'],
          ....: [['p','q','r'],['l','m','n','o']])
          (False, [])
  
      .. NOTE::
  
              This method does NOT do any checks.
      """
      if lodrs is not None:
          set_l = Set(l)
          if lodrs:
              for i in lodrs:
                  if Set(i) == set_l:
                      return True, i
      return False, []
  
  
  def lineorders_union(lineorders1, lineorders2):
      """
      Return a list of ordered lists of ground set elements that corresponds to
      union of two sets of ordered lists of ground set elements in a sense.
  
      INPUT:
  
      - ``lineorders1`` -- A list of ordered lists specifying orders on subsets
        of ground set.
      - ``lineorders2`` -- A list of ordered lists specifying orders subsets of
        ground set.
  
      OUTPUT:
  
      A list of ordered lists of ground set elements that are (setwise) in only
      one of ``lineorders1`` or ``lineorders2`` along with the ones in
      lineorder2 that are setwise equal to any list in lineorders1.
  
      EXAMPLES::
  
          sage: from sage.matroids import matroids_plot_helpers
          sage: matroids_plot_helpers.lineorders_union([['a','b','c'],
          ....: ['p','q','r'],['i','j','k','l']],[['r','p','q']])
          [['a', 'b', 'c'], ['p', 'q', 'r'], ['i', 'j', 'k', 'l']]
  
      """
      if lineorders1 is not None and lineorders2 is not None:
          lineorders = lineorders1
          for order in lineorders2:
              x, lo = line_hasorder(order, lineorders1)
              if not x:
                  lineorders.append(order)
                  lineorders.remove(lo)
          return lineorders
      elif lineorders1 is None and lineorders2 is not None:
          return lineorders2
      elif lineorders1 is not None:
          return lineorders1
      else:
          return None
  
  
  def posdict_is_sane(M1, pos_dict):
      """
      Return a boolean establishing sanity of ``posdict`` wrt matroid ``M``.
  
      INPUT:
  
      - ``M1`` -- A matroid.
      - ``posdict`` -- A dictionary mapping ground set elements to (x,y)
        positions.
  
      OUTPUT:
  
      A boolean that is ``True`` if posdict indeed has all the required elements
      to plot the geometric elements, otherwise ``False``.
  
      EXAMPLES::
  
          sage: from sage.matroids import matroids_plot_helpers
          sage: M1=Matroid(ring=GF(2), matrix=[[1, 0, 0, 0, 1, 1, 1,0,1,0,1],
          ....: [0, 1, 0, 1, 0, 1, 1,0,0,1,0],[0, 0, 1, 1, 1, 0, 1,0,0,0,0]])
          sage: pos_dict= {0: (0, 0),  1: (2, 0),  2: (1, 2),  3: (1.5, 1.0),
          ....: 4: (0.5, 1.0),  5: (1.0, 0.0), 6: (1.0, 0.6666666666666666)}
          sage: matroids_plot_helpers.posdict_is_sane(M1,pos_dict)
          True
          sage: pos_dict= {1: (2, 0),  2: (1, 2),  3: (1.5, 1.0),
          ....: 4: (0.5, 1.0), 5: (1.0, 0.0), 6: (1.0, 0.6666666666666666)}
          sage: matroids_plot_helpers.posdict_is_sane(M1,pos_dict)
          False
  
      .. NOTE::
  
              This method does NOT do any checks. ``M1`` is assumed to be a
              matroid and ``posdict`` is assumed to be a dictionary.
      """
      L = set(M1.loops())
      nP = L | set(M1.simplify().groundset())
      P = set(M1.groundset())-nP
      pcls = list(set([frozenset(set(M1.closure([p])) - L) for p in list(P)]))
      for pcl in pcls:
          pcl_list = list(pcl)
          if not any(x in pos_dict for x in pcl_list):
              return False
      allP = []
      for pcl in pcls:
          allP.extend(list(pcl))
      return all(x in pos_dict
                 for x in list(set(M1.groundset()) - (L | set(allP))))
  
  
  def tracklims(lims, x_i=[], y_i=[]):
      """
      Return modified limits list.
  
      INPUT:
  
      - ``lims`` -- A list with 4 elements ``[xmin,xmax,ymin,ymax]``
      - ``x_i`` -- New x values to track
      - ``y_i`` -- New y values to track
  
      OUTPUT:
  
      A list with 4 elements ``[xmin,xmax,ymin,ymax]``
  
      EXAMPLES::
  
          sage: from sage.matroids import matroids_plot_helpers
          sage: matroids_plot_helpers.tracklims([0,5,-1,7],[1,2,3,6,-1],
          ....: [-1,2,3,6])
          [-1, 6, -1, 7]
  
      .. NOTE::
  
              This method does NOT do any checks.
      """
      if lims is not None and lims[0] is not None and lims[1] is not None and \
         lims[2] is not None and lims[3] is not None:
          lims = [min(min(x_i), lims[0]), max(max(x_i), lims[1]),
                  min(min(y_i), lims[2]), max(max(y_i), lims[3])]
      else:
          lims = [min(x_i), max(x_i), min(y_i), max(y_i)]
      return lims
  
  
  def geomrep(M1, B1=None, lineorders1=None, pd=None, sp=False):
      """
      Return a sage graphics object containing geometric representation of
      matroid M1.
  
      INPUT:
  
      - ``M1`` -- A matroid.
      - ``B1`` -- (optional) A list of elements in ``M1.groundset()`` that
        correspond to a basis of ``M1`` and will be placed as vertices of the
        triangle in the geometric representation of ``M1``.
      - ``lineorders1`` -- (optional) A list of ordered lists of elements of
        ``M1.groundset()`` such that if a line in geometric representation is
        setwise same as any of these then points contained will be traversed in
        that order thus overriding internal order deciding heuristic.
      - ``pd`` - (optional) A dictionary mapping ground set elements to their
        (x,y) positions.
      - ``sp`` -- (optional) If True, a positioning dictionary and line orders
        will be placed in ``M._cached_info``.
  
      OUTPUT:
  
      A sage graphics object of type <class 'sage.plot.graphics.Graphics'> that
      corresponds to the geometric representation of the matroid.
  
      EXAMPLES::
  
          sage: from sage.matroids import matroids_plot_helpers
          sage: M=matroids.named_matroids.P7()
          sage: G=matroids_plot_helpers.geomrep(M)
          sage: G.show(xmin=-2, xmax=3, ymin=-2, ymax=3)
          sage: M=matroids.named_matroids.P7()
          sage: G=matroids_plot_helpers.geomrep(M,lineorders1=[['f','e','d']])
          sage: G.show(xmin=-2, xmax=3, ymin=-2, ymax=3)
  
      .. NOTE::
  
              This method does NOT do any checks.
      """
      G = Graphics()
      # create lists of loops and parallel elements and simplify given matroid
      [M, L, P] = slp(M1, pos_dict=pd, B=B1)
      if B1 is None:
          B1 = list(M.basis())
      M._cached_info = M1._cached_info
  
      if M.rank() == 0:
          limits = None
          loops = L
          looptext = ", ".join(str(l) for l in loops)
          rectx = -1
          recty = -1
          rectw = 0.5 + 0.4 * len(loops) + 0.5  # controlled based on len(loops)
          recth = 0.6
          G += polygon2d([[rectx, recty], [rectx, recty+recth],
                          [rectx+rectw, recty+recth], [rectx+rectw, recty]],
                         color='black', fill=False, thickness=4)
          G += text(looptext, (rectx+0.5, recty+0.3), color='black',
                    fontsize=13)
          G += point((rectx+0.2, recty+0.3), color=Color('#BDBDBD'), size=300,
                     zorder=2)
          G += text('Loop(s)', (rectx+0.5+0.4*len(loops)+0.1, recty+0.3),
                    fontsize=13, color='black')
          limits = tracklims(limits, [rectx, rectx+rectw], [recty, recty+recth])
          G.axes(False)
          G.axes_range(xmin=limits[0]-0.5, xmax=limits[1]+0.5,
                       ymin=limits[2]-0.5, ymax=limits[3]+0.5)
          return G
      elif M.rank() == 1:
          if M._cached_info is not None and \
             'plot_positions' in M._cached_info.keys() and \
             M._cached_info['plot_positions'] is not None:
              pts = M._cached_info['plot_positions']
          else:
              pts = {}
              gnd = sorted(M.groundset())
          pts[gnd[0]] = (1, float(2)/3)
          G += point((1, float(2)/3), size=300, color=Color('#BDBDBD'), zorder=2)
          pt = [1, float(2)/3]
          if not P:
              G += text(gnd[0], (float(pt[0]), float(pt[1])), color='black',
                        fontsize=13)
          pts2 = pts
          # track limits [xmin,xmax,ymin,ymax]
          pl = [list(x) for x in pts2.values()]
          lims = tracklims([None, None, None, None], [pnt[0] for pnt in pl],
                           [pnt[1] for pnt in pl])
      elif M.rank() == 2:
          nB1 = list(set(list(M.groundset())) - set(B1))
          bline = []
          for j in nB1:
              if M.is_dependent([j, B1[0], B1[1]]):
                  bline.append(j)
          interval = len(bline)+1
          if M._cached_info is not None and \
             'plot_positions' in M._cached_info.keys() and \
             M._cached_info['plot_positions'] is not None:
              pts2 = M._cached_info['plot_positions']
          else:
              pts2 = {}
              pts2[B1[0]] = (0, 0)
              pts2[B1[1]] = (2, 0)
              lpt = list(pts2[B1[0]])
              rpt = list(pts2[B1[1]])
              for k in range(len(bline)):
                  cc = (float(1)/interval)*(k+1)
                  pts2[bline[k]] = (cc*lpt[0]+(1-cc)*rpt[0],
                                    cc*lpt[1]+(1-cc)*rpt[1])
              if sp:
                  M._cached_info['plot_positions'] = pts2
          # track limits [xmin,xmax,ymin,ymax]
          pl = [list(x) for x in pts2.values()]
          lims = tracklims([None, None, None, None], [pt[0] for pt in pl],
                           [pt[1] for pt in pl])
          bline.extend(B1)
          ptsx, ptsy, x_i, y_i = createline(pts2, bline, lineorders1)
          lims = tracklims(lims, x_i, y_i)
          G += line(zip(x_i, y_i), color='black', thickness=3, zorder=1)
          pels = [p for p in pts2
                  if any(M1.rank([p, q]) == 1 for q in P)]
          allpts = [list(pts2[i]) for i in M.groundset()]
          xpts = [float(k[0]) for k in allpts]
          ypts = [float(k[1]) for k in allpts]
          G += points(zip(xpts, ypts), color=Color('#BDBDBD'), size=300,
                      zorder=2)
          for i in pts2:
              if i not in pels:
                  pt = list(pts2[i])
                  G += text(i, (float(pt[0]), float(pt[1])), color='black',
                            fontsize=13)
      else:
          if M._cached_info is None or \
             'plot_positions' not in M._cached_info.keys() or \
             M._cached_info['plot_positions'] is None:
              (pts, trilines,
               nontripts, curvedlines) = it(M1, B1,
                                            list(set(M.groundset())-set(B1)),
                                            list(set(L) | set(P)))
              pts2 = addnontripts([B1[0], B1[1], B1[2]], nontripts, pts)
              trilines.extend(curvedlines)
          else:
              pts2 = M._cached_info['plot_positions']
              trilines = [list(set(list(x)).difference(L | P))
                          for x in M1.flats(2)
                          if len(list(x)) >= 3]
          pl = [list(x) for x in pts2.values()]
          lims = tracklims([None, None, None, None], [pt[0] for pt in pl],
                           [pt[1] for pt in pl])
          j = 0
          for ll in trilines:
              if len(ll) >= 3:
                  ptsx, ptsy, x_i, y_i = createline(pts2, ll, lineorders1)
                  lims = tracklims(lims, x_i, y_i)
                  G += line(zip(x_i, y_i), color='black', thickness=3, zorder=1)
          pels = [p for p in pts2
                  if any(M1.rank([p, q]) == 1 for q in P)]
          allpts = [list(pts2[i]) for i in M.groundset()]
          xpts = [float(k[0]) for k in allpts]
          ypts = [float(k[1]) for k in allpts]
          G += points(zip(xpts, ypts), color=Color('#BDBDBD'), size=300,
                      zorder=2)
          for i in pts2:
              if i not in pels:
                  pt = list(pts2[i])
                  G += text(i, (float(pt[0]), float(pt[1])), color='black',
                            fontsize=13)
          if sp:
              M1._cached_info['plot_positions'] = pts2
              M1._cached_info['plot_lineorders'] = lineorders1
      # deal with loops and parallel elements
      G, lims = addlp(M1, M, L, P, pts2, G, lims)
      G.axes(False)
      G.axes_range(xmin=lims[0]-0.5, xmax=lims[1]+0.5, ymin=lims[2]-0.5,
                   ymax=lims[3]+0.5)
      return G
  