Drop useless slot inheritance in EmptyType and UniverseType

[linpy.git] / doc / examples.rst

diff --git a/doc/examples.rst b/doc/examples.rst
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+++ b/doc/examples.rst
@@ -1,103 +1,118 @@
-LinPy Examples
-==============
+
+.. _examples:
+
+Examples
+========

 
 Basic 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))
-    
+--------------
+
+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
 -------------
 
 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 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.
 
 
 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 
-     
-     
-     
+>>> 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