.. _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