doc/examples.rst

   1
   2 .. _examples:
   3
   4 Examples
   5 ========
   6
   7 Basic Examples
   8 --------------
   9
  10 To create any polyhedron, first define the symbols used.
  11 Then use the polyhedron functions to define the constraints.
  12 The following is a simple running example illustrating some different operations and properties that can be performed by LinPy with two squares.
  13
  14 >>> from linpy import *
  15 >>> x, y = symbols('x y')
  16 >>> # define the constraints of the polyhedron
  17 >>> square1 = Le(0, x) & Le(x, 2) & Le(0, y) & Le(y, 2)
  18 >>> square1
  19 And(Ge(x, 0), Ge(-x + 2, 0), Ge(y, 0), Ge(-y + 2, 0))
  20
  21 Binary operations and properties examples:
  22
  23 >>> # create a polyhedron from a string
  24 >>> square2 = Polyhedron('1 <= x') & Polyhedron('x <= 3') & \
  25         Polyhedron('1 <= y') & Polyhedron('y <= 3')
  26 >>> #test equality
  27 >>> square1 == square2
  28 False
  29 >>> # compute the union of two polyhedra
  30 >>> square1 | square2
  31 Or(And(Ge(x, 0), Ge(-x + 2, 0), Ge(y, 0), Ge(-y + 2, 0)), \
  32     And(Ge(x - 1, 0), Ge(-x + 3, 0), Ge(y - 1, 0), Ge(-y + 3, 0)))
  33 >>> # check if square1 and square2 are disjoint
  34 >>> square1.disjoint(square2)
  35 False
  36 >>> # compute the intersection of two polyhedra
  37 >>> square1 & square2
  38 And(Ge(x - 1, 0), Ge(-x + 2, 0), Ge(y - 1, 0), Ge(-y + 2, 0))
  39 >>> # compute the convex union of two polyhedra
  40 >>> Polyhedron(square1 | sqaure2)
  41 And(Ge(x, 0), Ge(y, 0), Ge(-y + 3, 0), Ge(-x + 3, 0), \
  42     Ge(x - y + 2, 0), Ge(-x + y + 2, 0))
  43
  44 Unary operation and properties examples:
  45
  46 >>> square1.isempty()
  47 False
  48 >>> # compute the complement of square1
  49 >>> ~square1
  50 Or(Ge(-x - 1, 0), Ge(x - 3, 0), And(Ge(x, 0), Ge(-x + 2, 0), \
  51     Ge(-y - 1, 0)), And(Ge(x, 0), Ge(-x + 2, 0), Ge(y - 3, 0)))
  52 >>> square1.symbols()
  53 (x, y)
  54 >>> square1.inequalities
  55 (x, -x + 2, y, -y + 2)
  56 >>> # project out the variable x
  57 >>> square1.project([x])
  58 And(Ge(-y + 2, 0), Ge(y, 0))
  59
  60 Plot Examples
  61 -------------
  62
  63 LinPy can use the matplotlib plotting library to plot 2D and 3D polygons.
  64 This can be a useful tool to visualize and compare polygons.
  65 The user has the option to pass plot objects to the :meth:`Domain.plot` method, which provides great flexibility.
  66 Also, keyword arguments can be passed such as color and the degree of transparency of a polygon.
  67
  68 >>> import matplotlib.pyplot as plt
  69 >>> from matplotlib import pylab
  70 >>> from mpl_toolkits.mplot3d import Axes3D
  71 >>> from linpy import *
  72 >>> # define the symbols
  73 >>> x, y, z = symbols('x y z')
  74 >>> fig = plt.figure()
  75 >>> cham_plot = fig.add_subplot(1, 1, 1, projection='3d', aspect='equal')
  76 >>> cham_plot.set_title('Chamfered cube')
  77 >>> cham = Le(0, x) & Le(x, 3) & Le(0, y) & Le(y, 3) & Le(0, z) & \
  78         Le(z, 3) & Le(z - 2, x) & Le(x, z + 2) & Le(1 - z, x) & \
  79         Le(x, 5 - z) & Le(z - 2, y) & Le(y, z + 2) & Le(1 - z, y) & \
  80         Le(y, 5 - z) & Le(y - 2, x) & Le(x, y + 2) & Le(1 - y, x) & Le(x, 5 - y)
  81 >>> cham.plot(cham_plot, facecolor='red', alpha=0.75)
  82 >>> pylab.show()
  83
  84 .. figure:: images/cham_cube.jpg
  85     :align:  center
  86
  87 LinPy can also inspect a polygon's vertices and the integer points included in the polygon.
  88
  89 >>> diamond = Ge(y, x - 1) & Le(y, x + 1) & Ge(y, -x - 1) & Le(y, -x + 1)
  90 >>> diamond.vertices()
  91 [Point({x: Fraction(0, 1), y: Fraction(1, 1)}), \
  92     Point({x: Fraction(-1, 1), y: Fraction(0, 1)}), \
  93     Point({x: Fraction(1, 1), y: Fraction(0, 1)}), \
  94     Point({x: Fraction(0, 1), y: Fraction(-1, 1)})]
  95 >>> diamond.points()
  96 [Point({x: -1, y: 0}), Point({x: 0, y: -1}), Point({x: 0, y: 0}), \
  97     Point({x: 0, y: 1}), Point({x: 1, y: 0})]
  98
  99 The user also can pass another plot to the :meth:`Domain.plot` method.
 100 This can be useful to compare two polyhedra on the same axis.
 101 This example illustrates the union of two squares.
 102
 103 >>> from linpy import *
 104 >>> import matplotlib.pyplot as plt
 105 >>> from matplotlib import pylab
 106 >>> x, y = symbols('x y')
 107 >>> square1 = Le(0, x) & Le(x, 2) & Le(0, y) & Le(y, 2)
 108 >>> square2 = Le(1, x) & Le(x, 3) & Le(1, y) & Le(y, 3)
 109 >>> fig = plt.figure()
 110 >>> plot = fig.add_subplot(1, 1, 1, aspect='equal')
 111 >>> square1.plot(plot, facecolor='red', alpha=0.3)
 112 >>> square2.plot(plot, facecolor='blue', alpha=0.3)
 113 >>> squares = Polyhedron(square1 + square2)
 114 >>> squares.plot(plot, facecolor='blue', alpha=0.3)
 115 >>> pylab.show()
 116
 117 .. figure:: images/union.jpg
 118     :align:  center