.. _examples:

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 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
-------------

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.

>>> 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