.. _tutorial:

Tutorial
========

This section a short introduction to some of LinPy's features.
For a comprehensive description of its functionalities, please consult the :ref:`reference`.


.. _tutorial_polyhedra:

Polyhedra
---------

The following example shows how we can manipulate polyhedra using LinPy.
Let us define two square polyhedra, corresponding to the sets ``square1 = {(x, y) | 0 <= x <= 2, 0 <= y <= 2}`` and ``square2 = {(x, y) | 2 <= x <= 4, 2 <= y <= 4}``.
First, we need define the symbols used, for instance with the :func:`symbols` function.

>>> from linpy import *
>>> x, y = symbols('x y')

Then, we can build the :class:`Polyhedron` object ``square1`` from its constraints:

>>> square1 = Le(0, x, 2) & Le(0, y, 2)
>>> square1
And(Ge(x, 0), Ge(-x + 2, 0), Ge(y, 0), Ge(-y + 2, 0))

LinPy provides comparison functions :func:`Lt`, :func:`Le`, :func:`Eq`, :func:`Ne`, :func:`Ge` and :func:`Gt` to build constraints, and logical operators :func:`And`, :func:`Or`, :func:`Not` to combine them.
Alternatively, a polyhedron can be built from a string:

>>> square2 = Polyhedron('1 <= x <= 3, 1 <= y <= 3')
>>> square2
And(Ge(x - 1, 0), Ge(-x + 3, 0), Ge(y - 1, 0), Ge(-y + 3, 0))

The usual polyhedral operations are available, including intersection:

>>> inter = square1.intersection(square2)
>>> inter
And(Ge(x - 1, 0), Ge(-x + 2, 0), Ge(y - 1, 0), Ge(-y + 2, 0))

convex union:

>>> hull = square1.convex_union(square2)
>>> hull
And(Ge(x, 0), Ge(y, 0), Ge(-x + y + 2, 0), Ge(x - y + 2, 0), Ge(-x + 3, 0), Ge(-y + 3, 0))

and projection:

>>> square1.project([y])
And(Ge(x, 0), Ge(-x + 2, 0))

Equality and inclusion tests are also provided.
Special values :data:`Empty` and :data:`Universe` represent the empty and universe polyhedra.

>>> inter <= square1
True
>>> inter == Empty
False


.. _tutorial_domains:

Domains
-------

LinPy is also able to manipulate polyhedral *domains*, that is, unions of polyhedra.
An example of domain is the set union (as opposed to convex union) of polyhedra ``square1`` and ``square2``.
The result is a :class:`Domain` object.

>>> union = square1 | square2
>>> union
Or(And(Ge(-x + 2, 0), Ge(x, 0), Ge(-y + 2, 0), Ge(y, 0)), And(Ge(-x + 3, 0), Ge(x - 1, 0), Ge(-y + 3, 0), Ge(y - 1, 0)))
>>> union <= hull
True

Unlike polyhedra, domains allow exact computation of union, subtraction and complementary operations.

>>> diff = square1 - square2
>>> diff
Or(And(Eq(x, 0), Ge(y, 0), Ge(-y + 2, 0)), And(Eq(y, 0), Ge(x - 1, 0), Ge(-x + 2, 0)))
>>> ~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)))


.. _tutorial_plot:

Plotting
--------

LinPy can use the :mod:`matplotlib` plotting library, if available, to plot bounded polyhedra and domains.

>>> import matplotlib.pyplot as plt
>>> from matplotlib import pylab
>>> 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)
>>> hull.plot(plot, facecolor='blue', alpha=0.3)
>>> pylab.show()

Note that you can pass a plot object 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.

.. figure:: images/union.jpg
    :align:  center

3D plots are also supported:

>>> import matplotlib.pyplot as plt
>>> from matplotlib import pylab
>>> from mpl_toolkits.mplot3d import Axes3D
>>> from linpy import *
>>> x, y, z = symbols('x y z')
>>> fig = plt.figure()
>>> plot = fig.add_subplot(1, 1, 1, projection='3d', aspect='equal')
>>> plot.set_title('Chamfered cube')
>>> poly = Le(0, x, 3) & Le(0, y, 3) & Le(0, 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)
>>> poly.plot(plot, facecolor='red', alpha=0.75)
>>> pylab.show()

.. figure:: images/cham_cube.jpg
    :align:  center