X-Git-Url: https://scm.cri.ensmp.fr/git/linpy.git/blobdiff_plain/0f40fdb54d0fc6505342d008deaa1e9c4c9dbcee..a08ebc700e22f6aee8147cb5b5323a6c040b12db:/doc/examples.rst diff --git a/doc/examples.rst b/doc/examples.rst index ea044b8..b552b7f 100644 --- a/doc/examples.rst +++ b/doc/examples.rst @@ -1,104 +1,118 @@ -LinPy Examples -============== + +.. _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: - - >>> square2 = Le(1, x) & Le(x, 3) & Le(1, y) & Le(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 - >>> 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)) +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 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 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:`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 - - - - +>>> 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