Polyhedra tips and tricks

Author: Jean-Philippe Labbé <labbe@math.fu-berlin.de>

Operation shortcuts

You can obtain different operations using natural symbols:

Sage

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

Python

>>> 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 Live

Cube = polytopes.cube()
Octahedron = 3/2*Cube.polar()  # Dilation
Cube + Octahedron   # Minkowski sum
Cube & Octahedron   # Intersection
Cube * Octahedron   # Cartesian product
Cube - Polyhedron(vertices=[[-1,0,0],[1,0,0]])  # Minkowski difference

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

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)])

Python

>>> 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)])

Sage Live

Cube = polytopes.cube()
TCube = Cube.truncation().dilation(1/2)
sage_input(TCube)

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

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

Python

>>> 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

Sage Live

Nice_repr = TCube.Hrepresentation_str()
print(Nice_repr)
print(TCube.Hrepresentation_str(latex=True))
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

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

Python

>>> 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

Sage Live

P = polytopes.permutahedron(3)
print(P.Hrepresentation_str(style='<='))
print(P.Hrepresentation_str(style='positive'))

Make live