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+LinPy 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:
+    
+    >>> square2 = Le(1, x) & Le(x, 3) & Le(1, y) & Le(y, 3)
+    >>> #test equality 
+    >>> square1 == square2
+    False
+    >>> # compute the union of two polyhedrons
+    >>> 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 polyhedrons
+    >>> 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 polyhedrons
+    >>> 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
+    >>> 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 uses matplotlib plotting library to plot 2D and 3D polygons. The user has the option to pass subplots to the :meth:`plot` method. This can be a useful tool to compare polygons. Also, key word 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:`plot` method. This can be useful to compare two polyhedrons 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 
+     
+     
+     
+