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