doc/examples.rst

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