==============
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)
- >>> print(square1)
+ >>> square1
And(Ge(x, 0), Ge(-x + 2, 0), Ge(y, 0), Ge(-y + 2, 0))
-
+
Binary operations and properties examples:
-
- >>> square2 = Le(2, x) & Le(x, 4) & Le(2, y) & Le(y, 4)
- >>> #test equality
+
+ >>> # create a polyhedron from a string
+ >>> square2 = Polyhedron('1 <= x') & Polyhedron('x <= 3') & \
+ Polyhedron('1 <= y') & Polyhedron('y <= 3')
+ >>> #test equality
>>> square1 == square2
False
- >>> # find the union of two polygons
- >>> square1 + square2
- 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)))
+ >>> # 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
- >>> # find the intersection of two polygons
+ >>> # compute the intersection of two polyhedrons
>>> square1 & square2
- And(Eq(y - 2, 0), Eq(x - 2, 0))
- >>> # find the convex union of two polygons
+ 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(-x + 4, 0), Ge(y, 0), Ge(-y + 4, 0), Ge(x - y + 2, 0), Ge(-x + y + 2, 0))
-
+ 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
-------------
>>> # define the symbols
>>> x, y, z = symbols('x y z')
>>> fig = plt.figure()
- >>> cham_plot = fig.add_subplot(2, 2, 3, projection='3d')
+ >>> 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, facecolors=(1, 0, 0, 0.75))
+ >>> 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/cube.jpg
+ .. figure:: images/cham_cube.jpg
:align: center
- LinPy can also inspect a polygon's vertices and the integer points included in the polygon.
+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)})]
+ [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})]
+ [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
+
+
+
projects / linpy.git / blobdiff