Polyhedra tips and tricks¶
Author: Jean-Philippe Labbé <labbe@math.fu-berlin.de>
Operation shortcuts¶
You can obtain different operations using natural symbols:
sage: Cube = polytopes.cube()
sage: Octahedron = 3/2*Cube.polar() # Dilation
sage: Cube + Octahedron # Minkowski sum
A 3-dimensional polyhedron in QQ^3 defined as the convex hull of 24 vertices
sage: Cube & Octahedron # Intersection
A 3-dimensional polyhedron in QQ^3 defined as the convex hull of 24 vertices
sage: Cube * Octahedron # Cartesian product
A 6-dimensional polyhedron in QQ^6 defined as the convex hull of 48 vertices
sage: Cube - Polyhedron(vertices=[[-1,0,0],[1,0,0]]) # Minkowski difference
A 2-dimensional polyhedron in QQ^3 defined as the convex hull of 4 vertices
>>> from sage.all import *
>>> Cube = polytopes.cube()
>>> Octahedron = Integer(3)/Integer(2)*Cube.polar() # Dilation
>>> Cube + Octahedron # Minkowski sum
A 3-dimensional polyhedron in QQ^3 defined as the convex hull of 24 vertices
>>> Cube & Octahedron # Intersection
A 3-dimensional polyhedron in QQ^3 defined as the convex hull of 24 vertices
>>> Cube * Octahedron # Cartesian product
A 6-dimensional polyhedron in QQ^6 defined as the convex hull of 48 vertices
>>> Cube - Polyhedron(vertices=[[-Integer(1),Integer(0),Integer(0)],[Integer(1),Integer(0),Integer(0)]]) # Minkowski difference
A 2-dimensional polyhedron in QQ^3 defined as the convex hull of 4 vertices
Sage input function¶
If you are working with a polyhedron that was difficult to construct and you would like to get back the proper Sage input code to reproduce this object, you can!
sage: Cube = polytopes.cube()
sage: TCube = Cube.truncation().dilation(1/2)
sage: sage_input(TCube)
Polyhedron(backend='ppl', base_ring=QQ, vertices=[(1/6, -1/2, -1/2),
(1/2, -1/6, -1/2), (1/2, 1/6, -1/2), (1/2, 1/2, -1/6), (1/2, 1/2, 1/6),
(1/2, 1/6, 1/2), (1/6, 1/2, 1/2), (1/2, -1/6, 1/2), (1/6, 1/2, -1/2),
(1/6, -1/2, 1/2), (1/2, -1/2, 1/6), (1/2, -1/2, -1/6), (-1/2, 1/6, -1/2),
(-1/2, -1/2, 1/6), (-1/2, 1/6, 1/2), (-1/2, 1/2, 1/6), (-1/6, 1/2, 1/2),
(-1/2, 1/2, -1/6), (-1/6, 1/2, -1/2), (-1/2, -1/6, 1/2), (-1/6, -1/2, 1/2),
(-1/2, -1/2, -1/6), (-1/6, -1/2, -1/2), (-1/2, -1/6, -1/2)])
>>> from sage.all import *
>>> Cube = polytopes.cube()
>>> TCube = Cube.truncation().dilation(Integer(1)/Integer(2))
>>> sage_input(TCube)
Polyhedron(backend='ppl', base_ring=QQ, vertices=[(1/6, -1/2, -1/2),
(1/2, -1/6, -1/2), (1/2, 1/6, -1/2), (1/2, 1/2, -1/6), (1/2, 1/2, 1/6),
(1/2, 1/6, 1/2), (1/6, 1/2, 1/2), (1/2, -1/6, 1/2), (1/6, 1/2, -1/2),
(1/6, -1/2, 1/2), (1/2, -1/2, 1/6), (1/2, -1/2, -1/6), (-1/2, 1/6, -1/2),
(-1/2, -1/2, 1/6), (-1/2, 1/6, 1/2), (-1/2, 1/2, 1/6), (-1/6, 1/2, 1/2),
(-1/2, 1/2, -1/6), (-1/6, 1/2, -1/2), (-1/2, -1/6, 1/2), (-1/6, -1/2, 1/2),
(-1/2, -1/2, -1/6), (-1/6, -1/2, -1/2), (-1/2, -1/6, -1/2)])
Hrepresentation_str¶
If you would like to visualize the \(H\)-representation nicely and even get the latex presentation, there is a method for that!
sage: Nice_repr = TCube.Hrepresentation_str()
sage: print(Nice_repr)
-6*x0 - 6*x1 - 6*x2 >= -7
-6*x0 - 6*x1 + 6*x2 >= -7
-6*x0 + 6*x1 - 6*x2 >= -7
-6*x0 + 6*x1 + 6*x2 >= -7
-2*x0 >= -1
-2*x1 >= -1
-2*x2 >= -1
6*x0 + 6*x1 + 6*x2 >= -7
2*x2 >= -1
2*x1 >= -1
2*x0 >= -1
6*x0 - 6*x1 - 6*x2 >= -7
6*x0 - 6*x1 + 6*x2 >= -7
6*x0 + 6*x1 - 6*x2 >= -7
sage: print(TCube.Hrepresentation_str(latex=True))
\begin{array}{rcl}
-6 x_{0} - 6 x_{1} - 6 x_{2} & \geq & -7 \\
-6 x_{0} - 6 x_{1} + 6 x_{2} & \geq & -7 \\
-6 x_{0} + 6 x_{1} - 6 x_{2} & \geq & -7 \\
-6 x_{0} + 6 x_{1} + 6 x_{2} & \geq & -7 \\
-2 x_{0} & \geq & -1 \\
-2 x_{1} & \geq & -1 \\
-2 x_{2} & \geq & -1 \\
6 x_{0} + 6 x_{1} + 6 x_{2} & \geq & -7 \\
2 x_{2} & \geq & -1 \\
2 x_{1} & \geq & -1 \\
2 x_{0} & \geq & -1 \\
6 x_{0} - 6 x_{1} - 6 x_{2} & \geq & -7 \\
6 x_{0} - 6 x_{1} + 6 x_{2} & \geq & -7 \\
6 x_{0} + 6 x_{1} - 6 x_{2} & \geq & -7
\end{array}
sage: Latex_repr = LatexExpr(TCube.Hrepresentation_str(latex=True))
sage: view(Latex_repr) # not tested
>>> from sage.all import *
>>> Nice_repr = TCube.Hrepresentation_str()
>>> print(Nice_repr)
-6*x0 - 6*x1 - 6*x2 >= -7
-6*x0 - 6*x1 + 6*x2 >= -7
-6*x0 + 6*x1 - 6*x2 >= -7
-6*x0 + 6*x1 + 6*x2 >= -7
-2*x0 >= -1
-2*x1 >= -1
-2*x2 >= -1
6*x0 + 6*x1 + 6*x2 >= -7
2*x2 >= -1
2*x1 >= -1
2*x0 >= -1
6*x0 - 6*x1 - 6*x2 >= -7
6*x0 - 6*x1 + 6*x2 >= -7
6*x0 + 6*x1 - 6*x2 >= -7
>>> print(TCube.Hrepresentation_str(latex=True))
\begin{array}{rcl}
-6 x_{0} - 6 x_{1} - 6 x_{2} & \geq & -7 \\
-6 x_{0} - 6 x_{1} + 6 x_{2} & \geq & -7 \\
-6 x_{0} + 6 x_{1} - 6 x_{2} & \geq & -7 \\
-6 x_{0} + 6 x_{1} + 6 x_{2} & \geq & -7 \\
-2 x_{0} & \geq & -1 \\
-2 x_{1} & \geq & -1 \\
-2 x_{2} & \geq & -1 \\
6 x_{0} + 6 x_{1} + 6 x_{2} & \geq & -7 \\
2 x_{2} & \geq & -1 \\
2 x_{1} & \geq & -1 \\
2 x_{0} & \geq & -1 \\
6 x_{0} - 6 x_{1} - 6 x_{2} & \geq & -7 \\
6 x_{0} - 6 x_{1} + 6 x_{2} & \geq & -7 \\
6 x_{0} + 6 x_{1} - 6 x_{2} & \geq & -7
\end{array}
>>> Latex_repr = LatexExpr(TCube.Hrepresentation_str(latex=True))
>>> view(Latex_repr) # not tested
The \(style\) parameter allows to change the way to print the \(H\)-relations:
sage: P = polytopes.permutahedron(3)
sage: print(P.Hrepresentation_str(style='<='))
-x0 - x1 - x2 == -6
-x0 - x1 <= -3
x0 + x1 <= 5
-x1 <= -1
x0 <= 3
-x0 <= -1
x1 <= 3
sage: print(P.Hrepresentation_str(style='positive'))
x0 + x1 + x2 == 6
x0 + x1 >= 3
5 >= x0 + x1
x1 >= 1
3 >= x0
x0 >= 1
3 >= x1
>>> from sage.all import *
>>> P = polytopes.permutahedron(Integer(3))
>>> print(P.Hrepresentation_str(style='<='))
-x0 - x1 - x2 == -6
-x0 - x1 <= -3
x0 + x1 <= 5
-x1 <= -1
x0 <= 3
-x0 <= -1
x1 <= 3
>>> print(P.Hrepresentation_str(style='positive'))
x0 + x1 + x2 == 6
x0 + x1 >= 3
5 >= x0 + x1
x1 >= 1
3 >= x0
x0 >= 1
3 >= x1