doc/tutorial.rst

   1
   2 .. _tutorial:
   3
   4 Tutorial
   5 ========
   6
   7 Polyhedra
   8 ---------
   9
  10 The following example shows how we can manipulate polyhedra using LinPy.
  11 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}``.
  12 First, we need define the symbols used, for instance with the :func:`symbols` function.
  13
  14 >>> from linpy import *
  15 >>> x, y = symbols('x y')
  16
  17 Then, we can build the :class:`Polyhedron` object ``square1`` from its constraints:
  18
  19 >>> square1 = Le(0, x, 2) & Le(0, y, 2)
  20 >>> square1
  21 And(Ge(x, 0), Ge(-x + 2, 0), Ge(y, 0), Ge(-y + 2, 0))
  22
  23 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.
  24 Alternatively, a polyhedron can be built from a string:
  25
  26 >>> square2 = Polyhedron('1 <= x <= 3, 1 <= y <= 3')
  27 >>> square2
  28 And(Ge(x - 1, 0), Ge(-x + 3, 0), Ge(y - 1, 0), Ge(-y + 3, 0))
  29
  30 The usual polyhedral operations are available, including intersection:
  31
  32 >>> inter = square1.intersection(square2)
  33 >>> inter
  34 And(Ge(x - 1, 0), Ge(-x + 2, 0), Ge(y - 1, 0), Ge(-y + 2, 0))
  35
  36 convex union:
  37
  38 >>> hull = square1.convex_union(square2)
  39 >>> hull
  40 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))
  41
  42 and projection:
  43
  44 >>> square1.project([y])
  45 And(Ge(x, 0), Ge(-x + 2, 0))
  46
  47 Equality and inclusion tests are also provided.
  48 Special values :data:`Empty` and :data:`Universe` represent the empty and universe polyhedra.
  49
  50 >>> inter <= square1
  51 True
  52 >>> inter == Empty
  53 False
  54
  55
  56 Domains
  57 -------
  58
  59 LinPy is also able to manipulate polyhedral *domains*, that is, unions of polyhedra.
  60 An example of domain is the set union (as opposed to convex union) of polyhedra ``square1`` and ``square2``.
  61 The result is a :class:`Domain` object.
  62
  63 >>> union = square1 | square2
  64 >>> union
  65 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)))
  66 >>> union <= hull
  67 True
  68
  69 Unlike polyhedra, domains allow exact computation of union, subtraction and complementary operations.
  70
  71 >>> diff = square1 - square2
  72 >>> diff
  73 Or(And(Eq(x, 0), Ge(y, 0), Ge(-y + 2, 0)), And(Eq(y, 0), Ge(x - 1, 0), Ge(-x + 2, 0)))
  74 >>> ~square1
  75 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)))
  76
  77
  78 Plotting
  79 --------
  80
  81 LinPy can use the :mod:`matplotlib` plotting library, if available, to plot bounded polyhedra and domains.
  82
  83 >>> import matplotlib.pyplot as plt
  84 >>> from matplotlib import pylab
  85 >>> fig = plt.figure()
  86 >>> plot = fig.add_subplot(1, 1, 1, aspect='equal')
  87 >>> square1.plot(plot, facecolor='red', alpha=0.3)
  88 >>> square2.plot(plot, facecolor='blue', alpha=0.3)
  89 >>> hull.plot(plot, facecolor='blue', alpha=0.3)
  90 >>> pylab.show()
  91
  92 Note that you can pass a plot object to the :meth:`Domain.plot` method, which provides great flexibility.
  93 Also, keyword arguments can be passed such as color and the degree of transparency of a polygon.
  94
  95 .. figure:: images/union.jpg
  96     :align:  center
  97
  98 3D plots are also supported.
  99
 100 >>> import matplotlib.pyplot as plt
 101 >>> from matplotlib import pylab
 102 >>> from mpl_toolkits.mplot3d import Axes3D
 103 >>> from linpy import *
 104 >>> x, y, z = symbols('x y z')
 105 >>> fig = plt.figure()
 106 >>> plot = fig.add_subplot(1, 1, 1, projection='3d', aspect='equal')
 107 >>> plot.set_title('Chamfered cube')
 108 >>> poly = Le(0, x, 3) & Le(0, y, 3) & Le(0, z, 3) & \
 109            Le(z - 2, x) & Le(x, z + 2) & Le(1 - z, x) & Le(x, 5 - z) & \
 110            Le(z - 2, y) & Le(y, z + 2) & Le(1 - z, y) & Le(y, 5 - z) & \
 111            Le(y - 2, x) & Le(x, y + 2) & Le(1 - y, x) & Le(x, 5 - y)
 112 >>> poly.plot(plot, facecolor='red', alpha=0.75)
 113 >>> pylab.show()
 114
 115 .. figure:: images/cham_cube.jpg
 116     :align:  center