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+.. _tutorial:
+
+Tutorial
+========
+
+Polyhedra
+---------
+
+The following example shows how we can manipulate polyhedra using LinPy.
+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}``.
+First, we need define the symbols used, for instance with the :func:`symbols` function.
+
+>>> from linpy import *
+>>> x, y = symbols('x y')
+
+Then, we can build the :class:`Polyhedron` object ``square1`` from its constraints:
+
+>>> square1 = Le(0, x, 2) & Le(0, y, 2)
+>>> square1
+And(Ge(x, 0), Ge(-x + 2, 0), Ge(y, 0), Ge(-y + 2, 0))
+
+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.
+Alternatively, a polyhedron can be built from a string:
+
+>>> square2 = Polyhedron('1 <= x <= 3, 1 <= y <= 3')
+>>> square2
+And(Ge(x - 1, 0), Ge(-x + 3, 0), Ge(y - 1, 0), Ge(-y + 3, 0))
+
+The usual polyhedral operations are available, including intersection:
+
+>>> inter = square1.intersection(square2)
+>>> inter
+And(Ge(x - 1, 0), Ge(-x + 2, 0), Ge(y - 1, 0), Ge(-y + 2, 0))
+
+convex union:
+
+>>> hull = square1.convex_union(square2)
+>>> hull
+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))
+
+and projection:
+
+>>> square1.project([y])
+And(Ge(x, 0), Ge(-x + 2, 0))
+
+Equality and inclusion tests are also provided.
+Special values :data:`Empty` and :data:`Universe` represent the empty and universe polyhedra.
+
+>>> inter <= square1
+True
+>>> inter == Empty
+False
+
+
+Domains
+-------
+
+LinPy is also able to manipulate polyhedral *domains*, that is, unions of polyhedra.
+An example of domain is the set union (as opposed to convex union) of polyhedra ``square1`` and ``square2``.
+The result is a :class:`Domain` object.
+
+>>> union = square1 | square2
+>>> union
+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)))
+>>> union <= hull
+True
+
+Unlike polyhedra, domains allow exact computation of union, subtraction and complementary operations.
+
+>>> diff = square1 - square2
+>>> diff
+Or(And(Eq(x, 0), Ge(y, 0), Ge(-y + 2, 0)), And(Eq(y, 0), Ge(x - 1, 0), Ge(-x + 2, 0)))
+>>> ~square1
+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)))
+
+
+Plotting
+--------
+
+LinPy can use the :mod:`matplotlib` plotting library, if available, to plot bounded polyhedra and domains.
+
+>>> import matplotlib.pyplot as plt
+>>> from matplotlib import pylab
+>>> fig = plt.figure()
+>>> plot = fig.add_subplot(1, 1, 1, aspect='equal')
+>>> square1.plot(plot, facecolor='red', alpha=0.3)
+>>> square2.plot(plot, facecolor='blue', alpha=0.3)
+>>> hull.plot(plot, facecolor='blue', alpha=0.3)
+>>> pylab.show()
+
+Note that you can pass a plot object to the :meth:`Domain.plot` method, which provides great flexibility.
+Also, keyword arguments can be passed such as color and the degree of transparency of a polygon.
+
+.. figure:: images/union.jpg
+    :align:  center
+
+3D plots are also supported.
+
+>>> import matplotlib.pyplot as plt
+>>> from matplotlib import pylab
+>>> from mpl_toolkits.mplot3d import Axes3D
+>>> from linpy import *
+>>> x, y, z = symbols('x y z')
+>>> fig = plt.figure()
+>>> plot = fig.add_subplot(1, 1, 1, projection='3d', aspect='equal')
+>>> plot.set_title('Chamfered cube')
+>>> poly = Le(0, x, 3) & Le(0, y, 3) & Le(0, z, 3) & \
+           Le(z - 2, x) & Le(x, z + 2) & Le(1 - z, x) & Le(x, 5 - z) & \
+           Le(z - 2, y) & Le(y, z + 2) & Le(1 - z, y) & Le(y, 5 - z) & \
+           Le(y - 2, x) & Le(x, y + 2) & Le(1 - y, x) & Le(x, 5 - y)
+>>> poly.plot(plot, facecolor='red', alpha=0.75)
+>>> pylab.show()
+
+.. figure:: images/cham_cube.jpg
+    :align:  center