doc/examples.rst

   1 LinPy Examples
   2 ==============
   3
   4 Basic Examples
   5 -------------
   6     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.
   7
   8     >>> from linpy import *
   9     >>> x, y = symbols('x y')
  10     >>> # define the constraints of the polyhedron
  11     >>> square1 = Le(0, x) & Le(x, 2) & Le(0, y) & Le(y, 2)
  12     >>> square1
  13     And(Ge(x, 0), Ge(-x + 2, 0), Ge(y, 0), Ge(-y + 2, 0))
  14
  15     Binary operations and properties examples:
  16
  17     >>> square2 = Le(1, x) & Le(x, 3) & Le(1, y) & Le(y, 3)
  18     >>> #test equality
  19     >>> square1 == square2
  20     False
  21     >>> # compute the union of two polyhedrons
  22     >>> square1 | square2
  23     Or(And(Ge(x, 0), Ge(-x + 2, 0), Ge(y, 0), Ge(-y + 2, 0)), And(Ge(x - 1, 0), Ge(-x + 3, 0), Ge(y - 1, 0), Ge(-y + 3, 0)))
  24     >>> # check if square1 and square2 are disjoint
  25     >>> square1.disjoint(square2)
  26     False
  27     >>> # compute the intersection of two polyhedrons
  28     >>> square1 & square2
  29     And(Ge(x - 1, 0), Ge(-x + 2, 0), Ge(y - 1, 0), Ge(-y + 2, 0))
  30     >>> # compute the convex union of two polyhedrons
  31     >>> Polyhedron(square1 | sqaure2)
  32     And(Ge(x, 0), Ge(y, 0), Ge(-y + 3, 0), Ge(-x + 3, 0), Ge(x - y + 2, 0), Ge(-x + y + 2, 0))
  33
  34     Unary operation and properties examples:
  35
  36     >>> square1.isempty()
  37     False
  38     >>> square1.symbols()
  39     (x, y)
  40     >>> square1.inequalities
  41     (x, -x + 2, y, -y + 2)
  42     >>> # project out the variable x
  43     >>> square1.project([x])
  44     And(Ge(-y + 2, 0), Ge(y, 0))
  45
  46 Plot Examples
  47 -------------
  48
  49      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.
  50
  51      >>> import matplotlib.pyplot as plt
  52      >>> from matplotlib import pylab
  53      >>> from mpl_toolkits.mplot3d import Axes3D
  54      >>> from linpy import *
  55      >>> # define the symbols
  56      >>> x, y, z = symbols('x y z')
  57      >>> fig = plt.figure()
  58      >>> cham_plot = fig.add_subplot(1, 1, 1, projection='3d', aspect='equal')
  59      >>> cham_plot.set_title('Chamfered cube')
  60      >>> cham = Le(0, x) & Le(x, 3) & Le(0, y) & Le(y, 3) & Le(0, z) & \
  61      Le(z, 3) & Le(z - 2, x) & Le(x, z + 2) & Le(1 - z, x) & \
  62      Le(x, 5 - z) & Le(z - 2, y) & Le(y, z + 2) & Le(1 - z, y) & \
  63      Le(y, 5 - z) & Le(y - 2, x) & Le(x, y + 2) & Le(1 - y, x) & Le(x, 5 - y)
  64      >>> cham.plot(cham_plot, facecolor='red', alpha=0.75)
  65      >>> pylab.show()
  66
  67      .. figure:: images/cham_cube.jpg
  68         :align:  center
  69
  70 LinPy can also inspect a polygon's vertices and the integer points included in the polygon.
  71
  72      >>> diamond = Ge(y, x - 1) & Le(y, x + 1) & Ge(y, -x - 1) & Le(y, -x + 1)
  73      >>> diamond.vertices()
  74      [Point({x: Fraction(0, 1), y: Fraction(1, 1)}), \
  75      Point({x: Fraction(-1, 1), y: Fraction(0, 1)}), \
  76      Point({x: Fraction(1, 1), y: Fraction(0, 1)}), \
  77      Point({x: Fraction(0, 1), y: Fraction(-1, 1)})]
  78      >>> diamond.points()
  79      [Point({x: -1, y: 0}), Point({x: 0, y: -1}), Point({x: 0, y: 0}), \
  80      Point({x: 0, y: 1}), Point({x: 1, y: 0})]
  81
  82 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.
  83
  84     >>> from linpy import *
  85     >>> import matplotlib.pyplot as plt
  86     >>> from matplotlib import pylab
  87     >>> x, y = symbols('x y')
  88     >>> square1 = Le(0, x) & Le(x, 2) & Le(0, y) & Le(y, 2)
  89     >>> square2 = Le(1, x) & Le(x, 3) & Le(1, y) & Le(y, 3)
  90     >>> fig = plt.figure()
  91     >>> plot = fig.add_subplot(1, 1, 1, aspect='equal')
  92     >>> square1.plot(plot, facecolor='red', alpha=0.3)
  93     >>> square2.plot(plot, facecolor='blue', alpha=0.3)
  94     >>> squares = Polyhedron(square1 + square2)
  95     >>> squares.plot(plot, facecolor='blue', alpha=0.3)
  96     >>> pylab.show()
  97
  98     .. figure:: images/union.jpg
  99        :align:  center
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