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-
-.. _examples:
-
-Examples
-========
-
-Basic Examples
---------------
-
-To create any polyhedron, first define the symbols used.
-Then use the polyhedron functions to define the constraints.
-The following is a simple running example illustrating some different operations and properties that can be performed by LinPy with two squares.
-
->>> from linpy import *
->>> x, y = symbols('x y')
->>> # define the constraints of the polyhedron
->>> square1 = Le(0, x) & Le(x, 2) & Le(0, y) & Le(y, 2)
->>> square1
-And(Ge(x, 0), Ge(-x + 2, 0), Ge(y, 0), Ge(-y + 2, 0))
-
-Binary operations and properties examples:
-
->>> # create a polyhedron from a string
->>> square2 = Polyhedron('1 <= x') & Polyhedron('x <= 3') & \
-        Polyhedron('1 <= y') & Polyhedron('y <= 3')
->>> #test equality
->>> square1 == square2
-False
->>> # compute the union of two polyhedra
->>> square1 | square2
-Or(And(Ge(x, 0), Ge(-x + 2, 0), Ge(y, 0), Ge(-y + 2, 0)), \
-    And(Ge(x - 1, 0), Ge(-x + 3, 0), Ge(y - 1, 0), Ge(-y + 3, 0)))
->>> # check if square1 and square2 are disjoint
->>> square1.disjoint(square2)
-False
->>> # compute the intersection of two polyhedra
->>> square1 & square2
-And(Ge(x - 1, 0), Ge(-x + 2, 0), Ge(y - 1, 0), Ge(-y + 2, 0))
->>> # compute the convex union of two polyhedra
->>> Polyhedron(square1 | sqaure2)
-And(Ge(x, 0), Ge(y, 0), Ge(-y + 3, 0), Ge(-x + 3, 0), \
-    Ge(x - y + 2, 0), Ge(-x + y + 2, 0))
-
-Unary operation and properties examples:
-
->>> square1.isempty()
-False
->>> # compute the complement of square1
->>> ~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)))
->>> square1.symbols()
-(x, y)
->>> square1.inequalities
-(x, -x + 2, y, -y + 2)
->>> # project out the variable x
->>> square1.project([x])
-And(Ge(-y + 2, 0), Ge(y, 0))
-
-Plot Examples
--------------
-
-LinPy can use the matplotlib plotting library to plot 2D and 3D polygons.
-This can be a useful tool to visualize and compare polygons.
-The user has the option to pass plot objects 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.
-
->>> import matplotlib.pyplot as plt
->>> from matplotlib import pylab
->>> from mpl_toolkits.mplot3d import Axes3D
->>> from linpy import *
->>> # define the symbols
->>> x, y, z = symbols('x y z')
->>> fig = plt.figure()
->>> cham_plot = fig.add_subplot(1, 1, 1, projection='3d', aspect='equal')
->>> cham_plot.set_title('Chamfered cube')
->>> cham = Le(0, x) & Le(x, 3) & Le(0, y) & Le(y, 3) & Le(0, z) & \
-        Le(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)
->>> cham.plot(cham_plot, facecolor='red', alpha=0.75)
->>> pylab.show()
-
-.. figure:: images/cham_cube.jpg
-    :align:  center
-
-LinPy can also inspect a polygon's vertices and the integer points included in the polygon.
-
->>> diamond = Ge(y, x - 1) & Le(y, x + 1) & Ge(y, -x - 1) & Le(y, -x + 1)
->>> diamond.vertices()
-[Point({x: Fraction(0, 1), y: Fraction(1, 1)}), \
-    Point({x: Fraction(-1, 1), y: Fraction(0, 1)}), \
-    Point({x: Fraction(1, 1), y: Fraction(0, 1)}), \
-    Point({x: Fraction(0, 1), y: Fraction(-1, 1)})]
->>> diamond.points()
-[Point({x: -1, y: 0}), Point({x: 0, y: -1}), Point({x: 0, y: 0}), \
-    Point({x: 0, y: 1}), Point({x: 1, y: 0})]
-
-The user also can pass another plot to the :meth:`Domain.plot` method.
-This can be useful to compare two polyhedra on the same axis.
-This example illustrates the union of two squares.
-
->>> from linpy import *
->>> import matplotlib.pyplot as plt
->>> from matplotlib import pylab
->>> x, y = symbols('x y')
->>> square1 = Le(0, x) & Le(x, 2) & Le(0, y) & Le(y, 2)
->>> square2 = Le(1, x) & Le(x, 3) & Le(1, y) & Le(y, 3)
->>> 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)
->>> squares = Polyhedron(square1 + square2)
->>> squares.plot(plot, facecolor='blue', alpha=0.3)
->>> pylab.show()
-
-.. figure:: images/union.jpg
-    :align:  center