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[linpy.git] / doc / examples.rst

diff --git a/doc/examples.rst b/doc/examples.rst
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+++ b/doc/examples.rst
@@ -1,67 +1,106 @@
-Linpy Examples
+LinPy Examples

 ==============
 
 ==============
 

-Creating a Polyhedron
------------------
-    To create any polyhedron, first define the symbols used. Then use the polyhedron functions to define the constraints for the polyhedron. This example creates a square.
-    
-    >>> from pypol import *
+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)
     >>> x, y = symbols('x y')
     >>> # define the constraints of the polyhedron
     >>> square1 = Le(0, x) & Le(x, 2) & Le(0, y) & Le(y, 2)

-    >>> print(square1)
+    >>> square1

     And(Ge(x, 0), Ge(-x + 2, 0), Ge(y, 0), Ge(-y + 2, 0))
 
     And(Ge(x, 0), Ge(-x + 2, 0), Ge(y, 0), Ge(-y + 2, 0))
 

-Urnary Operations
------------------    
-    
+    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.isempty()
     False

-    >>> square1.isbounded()
-    True
-    
-Binary Operations
------------------
-     
-     >>> square2 = Le(2, x) & Le(x, 4) & Le(2, y) & Le(y, 4)
-     >>> square1 + square2
-     Or(And(Ge(x, 0), Ge(-x + 2, 0), Ge(y, 0), Ge(-y + 2, 0)), And(Ge(x - 2, 0), Ge(-x + 4, 0), Ge(y - 2, 0), Ge(-y + 4, 0)))
-     >>> # check if square1 and square2 are disjoint    
-     >>> square1.disjoint(square2) 
-     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
 
 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. 
-         
+-------------
+
+     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
      >>> import matplotlib.pyplot as plt
      >>> from matplotlib import pylab
      >>> from mpl_toolkits.mplot3d import Axes3D

-     >>> from pypol import *
+     >>> from linpy import *

      >>> # define the symbols
      >>> x, y, z = symbols('x y z')
      >>> fig = plt.figure()
      >>> # define the symbols
      >>> x, y, z = symbols('x y z')
      >>> fig = plt.figure()

-     >>> cham_plot = fig.add_subplot(2, 2, 3, projection='3d')
+     >>> cham_plot = fig.add_subplot(1, 1, 1, projection='3d', aspect='equal')

      >>> cham_plot.set_title('Chamfered cube')
      >>> 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, facecolors=(1, 0, 0, 0.75))
+     >>> 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()
      >>> pylab.show()

-     
-     .. figure:: images/cube.jpg
+
+     .. figure:: images/cham_cube.jpg

         :align:  center
         :align:  center

-     
-     The user can also inspect a polygon's vertices and the integer points included in the polygon.   
-     
+
+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()
      >>> 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)})]
+     [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()
      >>> 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})]
-     
-     
-     
-     
-     
-     
-  
+     [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
+
+
+
+