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

Urnary Operations
-----------------    
    
    >>> 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  

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 pypol import *
     >>> # 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.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))
     >>> pylab.show()
     
     .. figure:: images/cube.jpg
        :align:  center
     
     The user 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})]