1 LinPy Examples
2 ==============
4 Basic Examples
5 -------------
6 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.
8 >>> from linpy import *
9 >>> x, y = symbols('x y')
10 >>> # define the constraints of the polyhedron
11 >>> square1 = Le(0, x) & Le(x, 2) & Le(0, y) & Le(y, 2)
12 >>> print(square1)
13 And(Ge(x, 0), Ge(-x + 2, 0), Ge(y, 0), Ge(-y + 2, 0))
15 Binary operations and properties examples:
17 >>> square2 = Le(2, x) & Le(x, 4) & Le(2, y) & Le(y, 4)
18 >>> #test equality
19 >>> square1 == square2
20 False
21 >>> # compute the union of two polygons
22 >>> square1 | square2
23 Or(And(Ge(x, 0), Ge(-x + 2, 0), Ge(y, 0), Ge(-y + 2, 0)), And(Ge(x - 2, 0), Ge(-x + 4, 0), Ge(y - 2, 0), Ge(-y + 4, 0)))
24 >>> # check if square1 and square2 are disjoint
25 >>> square1.disjoint(square2)
26 False
27 >>> # compute the intersection of two polygons
28 >>> square1 & square2
29 And(Eq(y - 2, 0), Eq(x - 2, 0))
30 >>> # compute the convex union of two polygons
31 >>> Polyhedron(square1 | sqaure2)
32 And(Ge(x, 0), Ge(-x + 4, 0), Ge(y, 0), Ge(-y + 4, 0), Ge(x - y + 2, 0), Ge(-x + y + 2, 0))
34 Unary operation and properties examples:
36 >>> square1.isempty()
37 False
38 >>> square1.symbols()
39 (x, y)
40 >>> square1.inequalities
41 (x, -x + 2, y, -y + 2)
42 >>> # project out the variable x
43 >>> square1.project([x])
44 And(Ge(-y + 2, 0), Ge(y, 0))
46 Plot Examples
47 -------------
49 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.
51 >>> import matplotlib.pyplot as plt
52 >>> from matplotlib import pylab
53 >>> from mpl_toolkits.mplot3d import Axes3D
54 >>> from linpy import *
55 >>> # define the symbols
56 >>> x, y, z = symbols('x y z')
57 >>> fig = plt.figure()
58 >>> cham_plot = fig.add_subplot(2, 2, 3, projection='3d')
59 >>> cham_plot.set_title('Chamfered cube')
60 >>> 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)
61 >>> cham.plot(cham_plot, facecolors=(1, 0, 0, 0.75))
62 >>> pylab.show()
64 .. figure:: images/cube.jpg
65 :align: center
67 LinPy can also inspect a polygon's vertices and the integer points included in the polygon.
69 >>> diamond = Ge(y, x - 1) & Le(y, x + 1) & Ge(y, -x - 1) & Le(y, -x + 1)
70 >>> diamond.vertices()
71 [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)})]
72 >>> diamond.points()
73 [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})]
75 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.
77 >>> from linpy import *
78 >>> import matplotlib.pyplot as plt
79 >>> from matplotlib import pylab
80 >>> x, y = symbols('x y')
81 >>> square1 = Le(0, x) & Le(x, 2) & Le(0, y) & Le(y, 2)
82 >>> square2 = Le(1, x) & Le(x, 3) & Le(1, y) & Le(y, 3)
83 >>> fig = plt.figure()
84 >>> plot = fig.add_subplot(1, 1, 1, aspect='equal')
85 >>> square1.plot(plot, facecolor='red', alpha=0.3)
86 >>> square2.plot(plot, facecolor='blue', alpha=0.3)
87 >>> squares = Polyhedron(square1 + square2)
88 >>> squares.plot(plot, facecolor='blue', alpha=0.3)
89 >>> pylab.show()
91 .. figure:: images/union.jpg
92 :align: center