+++ /dev/null
-
-.. _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
LinPy is a free software, licensed under the `GPLv3 license <http://www.gnu.org/licenses/gpl-3.0.txt>`_.
Its source code is available `here <https://scm.cri.ensmp.fr/git/linpy.git>`_.
-To have an overview of LinPy's functionalities, you may wish to consult the :ref:`examples` section.
+To have an overview of LinPy's functionalities, you may wish to consult the :ref:`tutorial` section.
.. only:: html
:maxdepth: 2
install.rst
- examples.rst
+ tutorial.rst
reference.rst
.. only:: html
+.. _install:
+
Installation
============
.. warning::
The project has not been published in PyPI yet, so this section is not relevant.
- Instead, see the :ref:`source` section to install LinPy.
+ Instead, see the :ref:`install_source` section to install LinPy.
LinPy can be installed using pip with the command::
sudo pip install linpy
-.. _source:
+.. _install_source:
Install From Source
-------------------
+.. _reference:
+
Module Reference
================
-------
A *domain* is a union of polyhedra.
-Unlike polyhedra, domains allow exact computation of union and complementary operations.
+Unlike polyhedra, domains allow exact computation of union, subtraction and complementary operations.
.. class:: Domain(*polyhedra)
Domain(string)
--- /dev/null
+
+.. _tutorial:
+
+Tutorial
+========
+
+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
+
+
+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)))
+
+
+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
class Domain(GeometricObject):
"""
A domain is a union of polyhedra. Unlike polyhedra, domains allow exact
- computation of union and complementary operations.
+ computation of union, subtraction and complementary operations.
A domain with a unique polyhedron is automatically subclassed as a
Polyhedron instance.
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