doc/tutorial.rst

   1
   2 .. _tutorial:
   3
   4 Tutorial
   5 ========
   6
   7 This section a short introduction to some of LinPy's features.
   8 For a comprehensive description of its functionalities, please consult the :ref:`reference` section.
   9
  10
  11 Polyhedra
  12 ---------
  13
  14 The following example shows how we can manipulate polyhedra using LinPy.
  15 Let us define two square polyhedra, corresponding to the sets ``square1 = {(x, y) | 0 <= x <= 2, 0 <= y <= 2}`` and ``square2 = {(x, y) | 2 <= x <= 4, 2 <= y <= 4}``.
  16 First, we need define the symbols used, for instance with the :func:`symbols` function.
  17
  18 >>> from linpy import *
  19 >>> x, y = symbols('x y')
  20
  21 Then, we can build the :class:`Polyhedron` object ``square1`` from its constraints:
  22
  23 >>> square1 = Le(0, x, 2) & Le(0, y, 2)
  24 >>> square1
  25 And(Ge(x, 0), Ge(-x + 2, 0), Ge(y, 0), Ge(-y + 2, 0))
  26
  27 LinPy provides comparison functions :func:`Lt`, :func:`Le`, :func:`Eq`, :func:`Ne`, :func:`Ge` and :func:`Gt` to build constraints, and logical operators :func:`And`, :func:`Or`, :func:`Not` to combine them.
  28 Alternatively, a polyhedron can be built from a string:
  29
  30 >>> square2 = Polyhedron('1 <= x <= 3, 1 <= y <= 3')
  31 >>> square2
  32 And(Ge(x - 1, 0), Ge(-x + 3, 0), Ge(y - 1, 0), Ge(-y + 3, 0))
  33
  34 The usual polyhedral operations are available, including intersection:
  35
  36 >>> inter = square1.intersection(square2)
  37 >>> inter
  38 And(Ge(x - 1, 0), Ge(-x + 2, 0), Ge(y - 1, 0), Ge(-y + 2, 0))
  39
  40 convex union:
  41
  42 >>> hull = square1.convex_union(square2)
  43 >>> hull
  44 And(Ge(x, 0), Ge(y, 0), Ge(-x + y + 2, 0), Ge(x - y + 2, 0), Ge(-x + 3, 0), Ge(-y + 3, 0))
  45
  46 and projection:
  47
  48 >>> square1.project([y])
  49 And(Ge(x, 0), Ge(-x + 2, 0))
  50
  51 Equality and inclusion tests are also provided.
  52 Special values :data:`Empty` and :data:`Universe` represent the empty and universe polyhedra.
  53
  54 >>> inter <= square1
  55 True
  56 >>> inter == Empty
  57 False
  58
  59
  60 Domains
  61 -------
  62
  63 LinPy is also able to manipulate polyhedral *domains*, that is, unions of polyhedra.
  64 An example of domain is the set union (as opposed to convex union) of polyhedra ``square1`` and ``square2``.
  65 The result is a :class:`Domain` object.
  66
  67 >>> union = square1 | square2
  68 >>> union
  69 Or(And(Ge(-x + 2, 0), Ge(x, 0), Ge(-y + 2, 0), Ge(y, 0)), And(Ge(-x + 3, 0), Ge(x - 1, 0), Ge(-y + 3, 0), Ge(y - 1, 0)))
  70 >>> union <= hull
  71 True
  72
  73 Unlike polyhedra, domains allow exact computation of union, subtraction and complementary operations.
  74
  75 >>> diff = square1 - square2
  76 >>> diff
  77 Or(And(Eq(x, 0), Ge(y, 0), Ge(-y + 2, 0)), And(Eq(y, 0), Ge(x - 1, 0), Ge(-x + 2, 0)))
  78 >>> ~square1
  79 Or(Ge(-x - 1, 0), Ge(x - 3, 0), And(Ge(x, 0), Ge(-x + 2, 0), Ge(-y - 1, 0)), And(Ge(x, 0), Ge(-x + 2, 0), Ge(y - 3, 0)))
  80
  81
  82 Plotting
  83 --------
  84
  85 LinPy can use the :mod:`matplotlib` plotting library, if available, to plot bounded polyhedra and domains.
  86
  87 >>> import matplotlib.pyplot as plt
  88 >>> from matplotlib import pylab
  89 >>> fig = plt.figure()
  90 >>> plot = fig.add_subplot(1, 1, 1, aspect='equal')
  91 >>> square1.plot(plot, facecolor='red', alpha=0.3)
  92 >>> square2.plot(plot, facecolor='blue', alpha=0.3)
  93 >>> hull.plot(plot, facecolor='blue', alpha=0.3)
  94 >>> pylab.show()
  95
  96 Note that you can pass a plot object to the :meth:`Domain.plot` method, which provides great flexibility.
  97 Also, keyword arguments can be passed such as color and the degree of transparency of a polygon.
  98
  99 .. figure:: images/union.jpg
 100     :align:  center
 101
 102 3D plots are also supported:
 103
 104 >>> import matplotlib.pyplot as plt
 105 >>> from matplotlib import pylab
 106 >>> from mpl_toolkits.mplot3d import Axes3D
 107 >>> from linpy import *
 108 >>> x, y, z = symbols('x y z')
 109 >>> fig = plt.figure()
 110 >>> plot = fig.add_subplot(1, 1, 1, projection='3d', aspect='equal')
 111 >>> plot.set_title('Chamfered cube')
 112 >>> poly = Le(0, x, 3) & Le(0, y, 3) & Le(0, z, 3) & \
 113            Le(z - 2, x) & Le(x, z + 2) & Le(1 - z, x) & Le(x, 5 - z) & \
 114            Le(z - 2, y) & Le(y, z + 2) & Le(1 - z, y) & Le(y, 5 - z) & \
 115            Le(y - 2, x) & Le(x, y + 2) & Le(1 - y, x) & Le(x, 5 - y)
 116 >>> poly.plot(plot, facecolor='red', alpha=0.75)
 117 >>> pylab.show()
 118
 119 .. figure:: images/cham_cube.jpg
 120     :align:  center