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     >>> square2 = Le(1, x) & Le(x, 3) & Le(1, y) & Le(y, 3)
  19     >>> #test equality
  20     >>> square1 == square2
  21     False
  22     >>> # compute the union of two polyhedrons
  23     >>> square1 | square2
  24     Or(And(Ge(x, 0), Ge(-x + 2, 0), Ge(y, 0), Ge(-y + 2, 0)), \
  25     And(Ge(x - 1, 0), Ge(-x + 3, 0), Ge(y - 1, 0), Ge(-y + 3, 0)))
  26     >>> # check if square1 and square2 are disjoint
  27     >>> square1.disjoint(square2)
  28     False
  29     >>> # compute the intersection of two polyhedrons
  30     >>> square1 & square2
  31     And(Ge(x - 1, 0), Ge(-x + 2, 0), Ge(y - 1, 0), Ge(-y + 2, 0))
  32     >>> # compute the convex union of two polyhedrons
  33     >>> Polyhedron(square1 | sqaure2)
  34     And(Ge(x, 0), Ge(y, 0), Ge(-y + 3, 0), Ge(-x + 3, 0), \
  35     Ge(x - y + 2, 0), Ge(-x + y + 2, 0))
  36
  37     Unary operation and properties examples:
  38
  39     >>> square1.isempty()
  40     False
  41     >>> square1.symbols()
  42     (x, y)
  43     >>> square1.inequalities
  44     (x, -x + 2, y, -y + 2)
  45     >>> # project out the variable x
  46     >>> square1.project([x])
  47     And(Ge(-y + 2, 0), Ge(y, 0))
  48
  49 Plot Examples
  50 -------------
  51
  52      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.
  53
  54      >>> import matplotlib.pyplot as plt
  55      >>> from matplotlib import pylab
  56      >>> from mpl_toolkits.mplot3d import Axes3D
  57      >>> from linpy import *
  58      >>> # define the symbols
  59      >>> x, y, z = symbols('x y z')
  60      >>> fig = plt.figure()
  61      >>> cham_plot = fig.add_subplot(1, 1, 1, projection='3d', aspect='equal')
  62      >>> cham_plot.set_title('Chamfered cube')
  63      >>> cham = Le(0, x) & Le(x, 3) & Le(0, y) & Le(y, 3) & Le(0, z) & \
  64      Le(z, 3) & Le(z - 2, x) & Le(x, z + 2) & Le(1 - z, x) & \
  65      Le(x, 5 - z) & Le(z - 2, y) & Le(y, z + 2) & Le(1 - z, y) & \
  66      Le(y, 5 - z) & Le(y - 2, x) & Le(x, y + 2) & Le(1 - y, x) & Le(x, 5 - y)
  67      >>> cham.plot(cham_plot, facecolor='red', alpha=0.75)
  68      >>> pylab.show()
  69
  70      .. figure:: images/cham_cube.jpg
  71         :align:  center
  72
  73 LinPy can also inspect a polygon's vertices and the integer points included in the polygon.
  74
  75      >>> diamond = Ge(y, x - 1) & Le(y, x + 1) & Ge(y, -x - 1) & Le(y, -x + 1)
  76      >>> diamond.vertices()
  77      [Point({x: Fraction(0, 1), y: Fraction(1, 1)}), \
  78      Point({x: Fraction(-1, 1), y: Fraction(0, 1)}), \
  79      Point({x: Fraction(1, 1), y: Fraction(0, 1)}), \
  80      Point({x: Fraction(0, 1), y: Fraction(-1, 1)})]
  81      >>> diamond.points()
  82      [Point({x: -1, y: 0}), Point({x: 0, y: -1}), Point({x: 0, y: 0}), \
  83      Point({x: 0, y: 1}), Point({x: 1, y: 0})]
  84
  85 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.
  86
  87     >>> from linpy import *
  88     >>> import matplotlib.pyplot as plt
  89     >>> from matplotlib import pylab
  90     >>> x, y = symbols('x y')
  91     >>> square1 = Le(0, x) & Le(x, 2) & Le(0, y) & Le(y, 2)
  92     >>> square2 = Le(1, x) & Le(x, 3) & Le(1, y) & Le(y, 3)
  93     >>> fig = plt.figure()
  94     >>> plot = fig.add_subplot(1, 1, 1, aspect='equal')
  95     >>> square1.plot(plot, facecolor='red', alpha=0.3)
  96     >>> square2.plot(plot, facecolor='blue', alpha=0.3)
  97     >>> squares = Polyhedron(square1 + square2)
  98     >>> squares.plot(plot, facecolor='blue', alpha=0.3)
  99     >>> pylab.show()
 100
 101     .. figure:: images/union.jpg
 102        :align:  center
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