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(0 <= x, x <= 2, 0 <= y, y <= 2)
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(1 <= x, x <= 3, 1 <= y, y <= 3)
36 The usual polyhedral operations are available, including intersection:
38 >>> inter = square1.intersection(square2) # or square1 & square2
39 >>> inter
40 And(1 <= x, x <= 2, 1 <= y, y <= 2)
42 convex union:
44 >>> hull = square1.convex_union(square2)
45 >>> hull
46 And(0 <= x, 0 <= y, x <= y + 2, y <= x + 2, x <= 3, y <= 3)
48 and projection:
50 >>> proj = square1.project([y])
51 >>> proj
52 And(0 <= x, x <= 2)
54 Equality and inclusion tests are also provided.
55 Special values :data:`Empty` and :data:`Universe` represent the empty and universe polyhedra.
57 >>> inter <= square1
58 True
59 >>> inter == Empty
60 False
63 .. _tutorial_domains:
65 Domains
66 -------
68 LinPy is also able to manipulate polyhedral *domains*, that is, unions of polyhedra.
69 An example of domain is the set union (as opposed to convex union) of polyhedra ``square1`` and ``square2``.
70 The result is a :class:`Domain` object.
72 >>> union = square1.union(square2) # or square1 | square2
73 >>> union
74 Or(And(x <= 2, 0 <= x, y <= 2, 0 <= y), And(x <= 3, 1 <= x, y <= 3, 1 <= y))
75 >>> union <= hull
76 True
78 Unlike polyhedra, domains allow exact computation of union, subtraction and complementary operations.
80 >>> diff = square1.difference(square2) # or square1 - square2
81 >>> diff
82 Or(And(x == 0, 0 <= y, y <= 2), And(y == 0, 1 <= x, x <= 2))
83 >>> ~square1
84 Or(x + 1 <= 0, 3 <= x, And(0 <= x, x <= 2, y + 1 <= 0), And(0 <= x, x <= 2, 3 <= y))
87 .. _tutorial_plot:
89 Plotting
90 --------
92 LinPy can use the :mod:`matplotlib` plotting library, if available, to plot bounded polyhedra and domains.
94 >>> import matplotlib.pyplot as plt
95 >>> from matplotlib import pylab
96 >>> fig = plt.figure()
97 >>> plot = fig.add_subplot(1, 1, 1, aspect='equal')
98 >>> square1.plot(plot, facecolor='red', alpha=0.3)
99 >>> square2.plot(plot, facecolor='blue', alpha=0.3)
100 >>> hull.plot(plot, facecolor='blue', alpha=0.3)
101 >>> pylab.show()
103 Note that you can pass a plot object to the :meth:`Domain.plot` method, which provides great flexibility.
104 Also, keyword arguments can be passed such as color and the degree of transparency of a polygon.
106 .. figure:: images/union.jpg
107 :align: center
109 3D plots are also supported:
111 >>> import matplotlib.pyplot as plt
112 >>> from matplotlib import pylab
113 >>> from mpl_toolkits.mplot3d import Axes3D
114 >>> from linpy import *
115 >>> x, y, z = symbols('x y z')
116 >>> fig = plt.figure()
117 >>> plot = fig.add_subplot(1, 1, 1, projection='3d', aspect='equal')
118 >>> plot.set_title('Chamfered cube')
119 >>> poly = Le(0, x, 3) & Le(0, y, 3) & Le(0, z, 3) & \
120 Le(z - 2, x) & Le(x, z + 2) & Le(1 - z, x) & Le(x, 5 - z) & \
121 Le(z - 2, y) & Le(y, z + 2) & Le(1 - z, y) & Le(y, 5 - z) & \
122 Le(y - 2, x) & Le(x, y + 2) & Le(1 - y, x) & Le(x, 5 - y)
123 >>> poly.plot(plot, facecolor='red', alpha=0.75)
124 >>> pylab.show()
126 .. figure:: images/cham_cube.jpg
127 :align: center