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:

    >>> # 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 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
    >>> # 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 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