Coverage for local/lib/python2.7/site-packages/sage/matroids/matroids_plot_helpers.py : 76%

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. 

# http://www.gnu.org/licenses/ 

# ***************************************************************************** 

from __future__ import print_function 

import scipy 

import scipy.interpolate 

import numpy as np 

from sage.plot.all import Graphics, line, text, polygon2d, point, points 

from sage.plot.colors import Color 

from sage.sets.set import Set 

from sage.matroids.advanced import newlabel 

def it(M, B1, nB1, lps): 

""" 

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) co-ordinates 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: print(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: print(trilines) 

[[3, 4, 5], [2, 5, 6], [1, 4, 6]] 

sage: print(nontripts) 

[0] 

sage: print(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 co-ordinates 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: print(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. 

""" 

n = 0 

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 allong 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 

j = 0 

for p in nontripts_labels: 

ptsdict[p] = tuple(gridpts[j]) 

j = j + 1 

return ptsdict 

def createline(ptsdict, ll, lineorders2=None): 

""" 

Return ordered lists of co-ordinates 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 x is False: 

# convert dictionary to list of lists 

linepts = [list(ptsdict[i]) for i in ll] 

xpts = [x[0] for x in linepts] 

ypts = [y[1] for y 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 is True: 

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]) 

[{7}, {8, 9, 10}] 

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]) 

[{7}, {0, 9, 10}] 

.. NOTE:: 

This method does NOT do any checks. 

""" 

L = set(M1.loops()) 

sg = sorted(M1.simplify().groundset()) 

nP = L | set(M1.simplify().groundset()) 

P = set(M1.groundset())-nP 

if len(P) > 0: 

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 len(pcl_in_basis) > 0: 

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 len(L) > 0: 

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 len(P) > 0: 

# create list of lists where inner lists are parallel classes 

pcls = [] 

gnd = sorted(list(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: 

if len(lodrs) > 0: 

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 x is False: 

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()) 

sg = sorted(M1.simplify().groundset()) 

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.keys() for x in pcl_list]): 

return False 

allP = [] 

for pcl in pcls: 

allP.extend(list(pcl)) 

if not all([x in pos_dict.keys() 

for x in list(set(M1.groundset()) - (L | set(allP)))]): 

return False 

return True 

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.grondset()`` 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 len(P) == 0: 

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], [pt[0] for pt in pl], 

[pt[1] for pt 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 is True: 

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.keys() 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 \