Base class for polyhedra: Initialization and access to Vrepresentation and Hrepresentation

class sage.geometry.polyhedron.base0.Polyhedron_base0(parent, Vrep, Hrep, Vrep_minimal=None, Hrep_minimal=None, pref_rep=None, mutable=False, **kwds)[source]

Bases: Element, Polyhedron

Initialization and basic access for polyhedra.

See sage.geometry.polyhedron.base.Polyhedron_base.

Hrep_generator()[source]

Return an iterator over the objects of the H-representation (inequalities or equations).

EXAMPLES:

sage: p = polytopes.hypercube(3)
sage: next(p.Hrep_generator())
An inequality (-1, 0, 0) x + 1 >= 0

>>> from sage.all import *
>>> p = polytopes.hypercube(Integer(3))
>>> next(p.Hrep_generator())
An inequality (-1, 0, 0) x + 1 >= 0
Hrepresentation(index=None)[source]

Return the objects of the H-representation. Each entry is either an inequality or a equation.

INPUT:

  • index – either an integer or None

OUTPUT:

The optional argument is an index running from 0 to self.n_Hrepresentation()-1. If present, the H-representation object at the given index will be returned. Without an argument, returns the list of all H-representation objects.

EXAMPLES:

sage: p = polytopes.hypercube(3, backend='field')
sage: p.Hrepresentation(0)
An inequality (-1, 0, 0) x + 1 >= 0
sage: p.Hrepresentation(0) == p.Hrepresentation()[0]
True

>>> from sage.all import *
>>> p = polytopes.hypercube(Integer(3), backend='field')
>>> p.Hrepresentation(Integer(0))
An inequality (-1, 0, 0) x + 1 >= 0
>>> p.Hrepresentation(Integer(0)) == p.Hrepresentation()[Integer(0)]
True
Hrepresentation_str(separator='\n', latex=False, style='>=', align=None, **kwds)[source]

Return a human-readable string representation of the Hrepresentation of this polyhedron.

INPUT:

  • separator – string (default: '\n')

  • latex – boolean (default: False)

  • style – either 'positive' (making all coefficients positive)

    or '<=', or '>='; default is '>='

  • align – boolean or None''; default is ``None in which case

    align is True if separator is the newline character. If set, then the lines of the output string are aligned by the comparison symbol by padding blanks.

Keyword parameters of repr_pretty() are passed on:

  • prefix – string

  • indices – tuple or other iterable

OUTPUT: string

EXAMPLES:

sage: P = polytopes.permutahedron(3)
sage: print(P.Hrepresentation_str())
x0 + x1 + x2 ==  6
     x0 + x1 >=  3
    -x0 - x1 >= -5
          x1 >=  1
         -x0 >= -3
          x0 >=  1
         -x1 >= -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

sage: print(P.Hrepresentation_str(latex=True))
\begin{array}{rcl}
x_{0} + x_{1} + x_{2} & =    &  6 \\
        x_{0} + x_{1} & \geq &  3 \\
       -x_{0} - x_{1} & \geq & -5 \\
                x_{1} & \geq &  1 \\
               -x_{0} & \geq & -3 \\
                x_{0} & \geq &  1 \\
               -x_{1} & \geq & -3
\end{array}

sage: print(P.Hrepresentation_str(align=False))
x0 + x1 + x2 == 6
x0 + x1 >= 3
-x0 - x1 >= -5
x1 >= 1
-x0 >= -3
x0 >= 1
-x1 >= -3

sage: c = polytopes.cube()
sage: c.Hrepresentation_str(separator=', ', style='positive')
'1 >= x0, 1 >= x1, 1 >= x2, 1 + x0 >= 0, 1 + x2 >= 0, 1 + x1 >= 0'

>>> from sage.all import *
>>> P = polytopes.permutahedron(Integer(3))
>>> print(P.Hrepresentation_str())
x0 + x1 + x2 ==  6
     x0 + x1 >=  3
    -x0 - x1 >= -5
          x1 >=  1
         -x0 >= -3
          x0 >=  1
         -x1 >= -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

>>> print(P.Hrepresentation_str(latex=True))
\begin{array}{rcl}
x_{0} + x_{1} + x_{2} & =    &  6 \\
        x_{0} + x_{1} & \geq &  3 \\
       -x_{0} - x_{1} & \geq & -5 \\
                x_{1} & \geq &  1 \\
               -x_{0} & \geq & -3 \\
                x_{0} & \geq &  1 \\
               -x_{1} & \geq & -3
\end{array}

>>> print(P.Hrepresentation_str(align=False))
x0 + x1 + x2 == 6
x0 + x1 >= 3
-x0 - x1 >= -5
x1 >= 1
-x0 >= -3
x0 >= 1
-x1 >= -3

>>> c = polytopes.cube()
>>> c.Hrepresentation_str(separator=', ', style='positive')
'1 >= x0, 1 >= x1, 1 >= x2, 1 + x0 >= 0, 1 + x2 >= 0, 1 + x1 >= 0'
Vrep_generator()[source]

Return an iterator over the objects of the V-representation (vertices, rays, and lines).

EXAMPLES:

sage: p = polytopes.cyclic_polytope(3,4)
sage: vg = p.Vrep_generator()
sage: next(vg)
A vertex at (0, 0, 0)
sage: next(vg)
A vertex at (1, 1, 1)

>>> from sage.all import *
>>> p = polytopes.cyclic_polytope(Integer(3),Integer(4))
>>> vg = p.Vrep_generator()
>>> next(vg)
A vertex at (0, 0, 0)
>>> next(vg)
A vertex at (1, 1, 1)
Vrepresentation(index=None)[source]

Return the objects of the V-representation. Each entry is either a vertex, a ray, or a line.

See sage.geometry.polyhedron.constructor for a definition of vertex/ray/line.

INPUT:

  • index – either an integer or None

OUTPUT:

The optional argument is an index running from 0 to self.n_Vrepresentation()-1. If present, the V-representation object at the given index will be returned. Without an argument, returns the list of all V-representation objects.

EXAMPLES:

sage: p = polytopes.simplex(4, project=True)
sage: p.Vrepresentation(0)
A vertex at (0.7071067812, 0.4082482905, 0.2886751346, 0.2236067977)
sage: p.Vrepresentation(0) == p.Vrepresentation() [0]
True

>>> from sage.all import *
>>> p = polytopes.simplex(Integer(4), project=True)
>>> p.Vrepresentation(Integer(0))
A vertex at (0.7071067812, 0.4082482905, 0.2886751346, 0.2236067977)
>>> p.Vrepresentation(Integer(0)) == p.Vrepresentation() [Integer(0)]
True
backend()[source]

Return the backend used.

OUTPUT:

The name of the backend used for computations. It will be one of the following backends:

  • ppl the Parma Polyhedra Library

  • cdd CDD

  • normaliz normaliz

  • polymake polymake

  • field a generic Sage implementation

EXAMPLES:

sage: triangle = Polyhedron(vertices=[[1, 0], [0, 1], [1, 1]])
sage: triangle.backend()
'ppl'
sage: D = polytopes.dodecahedron()                                          # needs sage.groups sage.rings.number_field
sage: D.backend()                                                           # needs sage.groups sage.rings.number_field
'field'
sage: P = Polyhedron([[1.23]])
sage: P.backend()
'cdd'

>>> from sage.all import *
>>> triangle = Polyhedron(vertices=[[Integer(1), Integer(0)], [Integer(0), Integer(1)], [Integer(1), Integer(1)]])
>>> triangle.backend()
'ppl'
>>> D = polytopes.dodecahedron()                                          # needs sage.groups sage.rings.number_field
>>> D.backend()                                                           # needs sage.groups sage.rings.number_field
'field'
>>> P = Polyhedron([[RealNumber('1.23')]])
>>> P.backend()
'cdd'
base_extend(base_ring, backend=None)[source]

Return a new polyhedron over a larger base ring.

This method can also be used to change the backend.

INPUT:

  • base_ring – the new base ring

  • backend – the new backend, see Polyhedron() If None (the default), attempt to keep the same backend. Otherwise, use the same defaulting behavior as described there.

OUTPUT: the same polyhedron, but over a larger base ring and possibly with a changed backend

EXAMPLES:

sage: P = Polyhedron(vertices=[(1,0), (0,1)], rays=[(1,1)], base_ring=ZZ);  P
A 2-dimensional polyhedron in ZZ^2 defined as the convex hull of 2 vertices and 1 ray
sage: P.base_extend(QQ)
A 2-dimensional polyhedron in QQ^2 defined as the convex hull of 2 vertices and 1 ray
sage: P.base_extend(QQ) == P
True

>>> from sage.all import *
>>> P = Polyhedron(vertices=[(Integer(1),Integer(0)), (Integer(0),Integer(1))], rays=[(Integer(1),Integer(1))], base_ring=ZZ);  P
A 2-dimensional polyhedron in ZZ^2 defined as the convex hull of 2 vertices and 1 ray
>>> P.base_extend(QQ)
A 2-dimensional polyhedron in QQ^2 defined as the convex hull of 2 vertices and 1 ray
>>> P.base_extend(QQ) == P
True
base_ring()[source]

Return the base ring.

OUTPUT:

The ring over which the polyhedron is defined. Must be a sub-ring of the reals to define a polyhedron, in particular comparison must be defined. Popular choices are

  • ZZ (the ring of integers, lattice polytope),

  • QQ (exact arithmetic using gmp),

  • RDF (double precision floating-point arithmetic), or

  • AA (real algebraic field).

EXAMPLES:

sage: triangle = Polyhedron(vertices = [[1,0],[0,1],[1,1]])
sage: triangle.base_ring() == ZZ
True

>>> from sage.all import *
>>> triangle = Polyhedron(vertices = [[Integer(1),Integer(0)],[Integer(0),Integer(1)],[Integer(1),Integer(1)]])
>>> triangle.base_ring() == ZZ
True
cdd_Hrepresentation()[source]

Write the inequalities/equations data of the polyhedron in cdd’s H-representation format.

See also

write_cdd_Hrepresentation() – export the polyhedron as a H-representation to a file.

OUTPUT: string

EXAMPLES:

sage: p = polytopes.hypercube(2)
sage: print(p.cdd_Hrepresentation())
H-representation
begin
 4 3 rational
 1 -1 0
 1 0 -1
 1 1 0
 1 0 1
end


sage: triangle = Polyhedron(vertices=[[1,0], [0,1], [1,1]], base_ring=AA)   # needs sage.rings.number_field
sage: triangle.base_ring()                                                  # needs sage.rings.number_field
Algebraic Real Field
sage: triangle.cdd_Hrepresentation()                                        # needs sage.rings.number_field
Traceback (most recent call last):
...
TypeError: the base ring must be ZZ, QQ, or RDF

>>> from sage.all import *
>>> p = polytopes.hypercube(Integer(2))
>>> print(p.cdd_Hrepresentation())
H-representation
begin
 4 3 rational
 1 -1 0
 1 0 -1
 1 1 0
 1 0 1
end
<BLANKLINE>

>>> triangle = Polyhedron(vertices=[[Integer(1),Integer(0)], [Integer(0),Integer(1)], [Integer(1),Integer(1)]], base_ring=AA)   # needs sage.rings.number_field
>>> triangle.base_ring()                                                  # needs sage.rings.number_field
Algebraic Real Field
>>> triangle.cdd_Hrepresentation()                                        # needs sage.rings.number_field
Traceback (most recent call last):
...
TypeError: the base ring must be ZZ, QQ, or RDF
cdd_Vrepresentation()[source]

Write the vertices/rays/lines data of the polyhedron in cdd’s V-representation format.

See also

write_cdd_Vrepresentation() – export the polyhedron as a V-representation to a file.

OUTPUT: string

EXAMPLES:

sage: q = Polyhedron(vertices = [[1,1],[0,0],[1,0],[0,1]])
sage: print(q.cdd_Vrepresentation())
V-representation
begin
 4 3 rational
 1 0 0
 1 0 1
 1 1 0
 1 1 1
end

>>> from sage.all import *
>>> q = Polyhedron(vertices = [[Integer(1),Integer(1)],[Integer(0),Integer(0)],[Integer(1),Integer(0)],[Integer(0),Integer(1)]])
>>> print(q.cdd_Vrepresentation())
V-representation
begin
 4 3 rational
 1 0 0
 1 0 1
 1 1 0
 1 1 1
end
change_ring(base_ring, backend=None)[source]

Return the polyhedron obtained by coercing the entries of the vertices/lines/rays of this polyhedron into the given ring.

This method can also be used to change the backend.

INPUT:

  • base_ring – the new base ring

  • backend – the new backend or None (default), see Polyhedron(). If None (the default), attempt to keep the same backend. Otherwise, use the same defaulting behavior as described there.

EXAMPLES:

sage: P = Polyhedron(vertices=[(1,0), (0,1)], rays=[(1,1)], base_ring=QQ); P
A 2-dimensional polyhedron in QQ^2 defined as the convex hull of 2 vertices and 1 ray
sage: P.change_ring(ZZ)
A 2-dimensional polyhedron in ZZ^2 defined as the convex hull of 2 vertices and 1 ray
sage: P.change_ring(ZZ) == P
True

sage: P = Polyhedron(vertices=[(-1.3,0), (0,2.3)], base_ring=RDF); P.vertices()
(A vertex at (-1.3, 0.0), A vertex at (0.0, 2.3))
sage: P.change_ring(QQ).vertices()
(A vertex at (-13/10, 0), A vertex at (0, 23/10))
sage: P == P.change_ring(QQ)
True
sage: P.change_ring(ZZ)
Traceback (most recent call last):
...
TypeError: cannot change the base ring to the Integer Ring

sage: P = polytopes.regular_polygon(3); P                                   # needs sage.rings.number_field
A 2-dimensional polyhedron in AA^2 defined as the convex hull of 3 vertices
sage: P.vertices()                                                          # needs sage.rings.number_field
(A vertex at (0.?e-16, 1.000000000000000?),
 A vertex at (0.866025403784439?, -0.500000000000000?),
 A vertex at (-0.866025403784439?, -0.500000000000000?))
sage: P.change_ring(QQ)                                                     # needs sage.rings.number_field
Traceback (most recent call last):
...
TypeError: cannot change the base ring to the Rational Field

>>> from sage.all import *
>>> P = Polyhedron(vertices=[(Integer(1),Integer(0)), (Integer(0),Integer(1))], rays=[(Integer(1),Integer(1))], base_ring=QQ); P
A 2-dimensional polyhedron in QQ^2 defined as the convex hull of 2 vertices and 1 ray
>>> P.change_ring(ZZ)
A 2-dimensional polyhedron in ZZ^2 defined as the convex hull of 2 vertices and 1 ray
>>> P.change_ring(ZZ) == P
True

>>> P = Polyhedron(vertices=[(-RealNumber('1.3'),Integer(0)), (Integer(0),RealNumber('2.3'))], base_ring=RDF); P.vertices()
(A vertex at (-1.3, 0.0), A vertex at (0.0, 2.3))
>>> P.change_ring(QQ).vertices()
(A vertex at (-13/10, 0), A vertex at (0, 23/10))
>>> P == P.change_ring(QQ)
True
>>> P.change_ring(ZZ)
Traceback (most recent call last):
...
TypeError: cannot change the base ring to the Integer Ring

>>> P = polytopes.regular_polygon(Integer(3)); P                                   # needs sage.rings.number_field
A 2-dimensional polyhedron in AA^2 defined as the convex hull of 3 vertices
>>> P.vertices()                                                          # needs sage.rings.number_field
(A vertex at (0.?e-16, 1.000000000000000?),
 A vertex at (0.866025403784439?, -0.500000000000000?),
 A vertex at (-0.866025403784439?, -0.500000000000000?))
>>> P.change_ring(QQ)                                                     # needs sage.rings.number_field
Traceback (most recent call last):
...
TypeError: cannot change the base ring to the Rational Field

Warning

The base ring RDF should be used with care. As it is not an exact ring, certain computations may break or silently produce wrong results, for example changing the base ring from an exact ring into RDF may cause a loss of data:

sage: P = Polyhedron([[2/3,0],[6666666666666667/10^16,0]], base_ring=AA); P         # needs sage.rings.number_field
A 1-dimensional polyhedron in AA^2 defined as the convex hull of 2 vertices
sage: Q = P.change_ring(RDF); Q                                         # needs sage.rings.number_field
A 0-dimensional polyhedron in RDF^2 defined as the convex hull of 1 vertex
sage: P.n_vertices() == Q.n_vertices()                                  # needs sage.rings.number_field
False

>>> from sage.all import *
>>> P = Polyhedron([[Integer(2)/Integer(3),Integer(0)],[Integer(6666666666666667)/Integer(10)**Integer(16),Integer(0)]], base_ring=AA); P         # needs sage.rings.number_field
A 1-dimensional polyhedron in AA^2 defined as the convex hull of 2 vertices
>>> Q = P.change_ring(RDF); Q                                         # needs sage.rings.number_field
A 0-dimensional polyhedron in RDF^2 defined as the convex hull of 1 vertex
>>> P.n_vertices() == Q.n_vertices()                                  # needs sage.rings.number_field
False
equation_generator()[source]

Return a generator for the linear equations satisfied by the polyhedron.

EXAMPLES:

sage: p = polytopes.regular_polygon(8,base_ring=RDF)
sage: p3 = Polyhedron(vertices = [x+[0] for x in p.vertices()], base_ring=RDF)
sage: next(p3.equation_generator())
An equation (0.0, 0.0, 1.0) x + 0.0 == 0

>>> from sage.all import *
>>> p = polytopes.regular_polygon(Integer(8),base_ring=RDF)
>>> p3 = Polyhedron(vertices = [x+[Integer(0)] for x in p.vertices()], base_ring=RDF)
>>> next(p3.equation_generator())
An equation (0.0, 0.0, 1.0) x + 0.0 == 0
equations()[source]

Return all linear constraints of the polyhedron.

OUTPUT: a tuple of equations

EXAMPLES:

sage: test_p = Polyhedron(vertices = [[1,2,3,4],[2,1,3,4],[4,3,2,1],[3,4,1,2]])
sage: test_p.equations()
(An equation (1, 1, 1, 1) x - 10 == 0,)

>>> from sage.all import *
>>> test_p = Polyhedron(vertices = [[Integer(1),Integer(2),Integer(3),Integer(4)],[Integer(2),Integer(1),Integer(3),Integer(4)],[Integer(4),Integer(3),Integer(2),Integer(1)],[Integer(3),Integer(4),Integer(1),Integer(2)]])
>>> test_p.equations()
(An equation (1, 1, 1, 1) x - 10 == 0,)
equations_list()[source]

Return the linear constraints of the polyhedron. As with inequalities, each constraint is given as [b -a1 -a2 … an] where for variables x1, x2,…, xn, the polyhedron satisfies the equation b = a1*x1 + a2*x2 + … + an*xn.

Note

It is recommended to use equations() or equation_generator() instead to iterate over the list of Equation objects.

EXAMPLES:

sage: test_p = Polyhedron(vertices = [[1,2,3,4],[2,1,3,4],[4,3,2,1],[3,4,1,2]])
sage: test_p.equations_list()
[[-10, 1, 1, 1, 1]]

>>> from sage.all import *
>>> test_p = Polyhedron(vertices = [[Integer(1),Integer(2),Integer(3),Integer(4)],[Integer(2),Integer(1),Integer(3),Integer(4)],[Integer(4),Integer(3),Integer(2),Integer(1)],[Integer(3),Integer(4),Integer(1),Integer(2)]])
>>> test_p.equations_list()
[[-10, 1, 1, 1, 1]]
inequalities()[source]

Return all inequalities.

OUTPUT: a tuple of inequalities

EXAMPLES:

sage: p = Polyhedron(vertices = [[0,0,0],[0,0,1],[0,1,0],[1,0,0],[2,2,2]])
sage: p.inequalities()[0:3]
(An inequality (1, 0, 0) x + 0 >= 0,
 An inequality (0, 1, 0) x + 0 >= 0,
 An inequality (0, 0, 1) x + 0 >= 0)

sage: # needs sage.combinat
sage: p3 = Polyhedron(vertices=Permutations([1, 2, 3, 4]))
sage: ieqs = p3.inequalities()
sage: ieqs[0]
An inequality (0, 1, 1, 1) x - 6 >= 0
sage: list(_)
[-6, 0, 1, 1, 1]

>>> from sage.all import *
>>> p = Polyhedron(vertices = [[Integer(0),Integer(0),Integer(0)],[Integer(0),Integer(0),Integer(1)],[Integer(0),Integer(1),Integer(0)],[Integer(1),Integer(0),Integer(0)],[Integer(2),Integer(2),Integer(2)]])
>>> p.inequalities()[Integer(0):Integer(3)]
(An inequality (1, 0, 0) x + 0 >= 0,
 An inequality (0, 1, 0) x + 0 >= 0,
 An inequality (0, 0, 1) x + 0 >= 0)

>>> # needs sage.combinat
>>> p3 = Polyhedron(vertices=Permutations([Integer(1), Integer(2), Integer(3), Integer(4)]))
>>> ieqs = p3.inequalities()
>>> ieqs[Integer(0)]
An inequality (0, 1, 1, 1) x - 6 >= 0
>>> list(_)
[-6, 0, 1, 1, 1]
inequalities_list()[source]

Return a list of inequalities as coefficient lists.

Note

It is recommended to use inequalities() or inequality_generator() instead to iterate over the list of Inequality objects.

EXAMPLES:

sage: p = Polyhedron(vertices = [[0,0,0],[0,0,1],[0,1,0],[1,0,0],[2,2,2]])
sage: p.inequalities_list()[0:3]
[[0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]]

sage: # needs sage.combinat
sage: p3 = Polyhedron(vertices=Permutations([1, 2, 3, 4]))
sage: ieqs = p3.inequalities_list()
sage: ieqs[0]
[-6, 0, 1, 1, 1]
sage: ieqs[-1]
[-3, 0, 1, 0, 1]
sage: ieqs == [list(x) for x in p3.inequality_generator()]
True

>>> from sage.all import *
>>> p = Polyhedron(vertices = [[Integer(0),Integer(0),Integer(0)],[Integer(0),Integer(0),Integer(1)],[Integer(0),Integer(1),Integer(0)],[Integer(1),Integer(0),Integer(0)],[Integer(2),Integer(2),Integer(2)]])
>>> p.inequalities_list()[Integer(0):Integer(3)]
[[0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]]

>>> # needs sage.combinat
>>> p3 = Polyhedron(vertices=Permutations([Integer(1), Integer(2), Integer(3), Integer(4)]))
>>> ieqs = p3.inequalities_list()
>>> ieqs[Integer(0)]
[-6, 0, 1, 1, 1]
>>> ieqs[-Integer(1)]
[-3, 0, 1, 0, 1]
>>> ieqs == [list(x) for x in p3.inequality_generator()]
True
inequality_generator()[source]

Return a generator for the defining inequalities of the polyhedron.

OUTPUT: a generator of the inequality Hrepresentation objects

EXAMPLES:

sage: triangle = Polyhedron(vertices=[[1,0],[0,1],[1,1]])
sage: for v in triangle.inequality_generator(): print(v)
An inequality (1, 1) x - 1 >= 0
An inequality (0, -1) x + 1 >= 0
An inequality (-1, 0) x + 1 >= 0
sage: [ v for v in triangle.inequality_generator() ]
[An inequality (1, 1) x - 1 >= 0,
 An inequality (0, -1) x + 1 >= 0,
 An inequality (-1, 0) x + 1 >= 0]
sage: [ [v.A(), v.b()] for v in triangle.inequality_generator() ]
[[(1, 1), -1], [(0, -1), 1], [(-1, 0), 1]]

>>> from sage.all import *
>>> triangle = Polyhedron(vertices=[[Integer(1),Integer(0)],[Integer(0),Integer(1)],[Integer(1),Integer(1)]])
>>> for v in triangle.inequality_generator(): print(v)
An inequality (1, 1) x - 1 >= 0
An inequality (0, -1) x + 1 >= 0
An inequality (-1, 0) x + 1 >= 0
>>> [ v for v in triangle.inequality_generator() ]
[An inequality (1, 1) x - 1 >= 0,
 An inequality (0, -1) x + 1 >= 0,
 An inequality (-1, 0) x + 1 >= 0]
>>> [ [v.A(), v.b()] for v in triangle.inequality_generator() ]
[[(1, 1), -1], [(0, -1), 1], [(-1, 0), 1]]
is_compact()[source]

Test for boundedness of the polytope.

EXAMPLES:

sage: p = polytopes.icosahedron()                                           # needs sage.groups sage.rings.number_field
sage: p.is_compact()                                                        # needs sage.groups sage.rings.number_field
True
sage: p = Polyhedron(ieqs=[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,-1,0,0]])
sage: p.is_compact()
False

>>> from sage.all import *
>>> p = polytopes.icosahedron()                                           # needs sage.groups sage.rings.number_field
>>> p.is_compact()                                                        # needs sage.groups sage.rings.number_field
True
>>> p = Polyhedron(ieqs=[[Integer(0),Integer(1),Integer(0),Integer(0)],[Integer(0),Integer(0),Integer(1),Integer(0)],[Integer(0),Integer(0),Integer(0),Integer(1)],[Integer(1),-Integer(1),Integer(0),Integer(0)]])
>>> p.is_compact()
False
is_immutable()[source]

Return True if the polyhedron is immutable, i.e. it cannot be modified in place.

EXAMPLES:

sage: p = polytopes.cube(backend='field')
sage: p.is_immutable()
True

>>> from sage.all import *
>>> p = polytopes.cube(backend='field')
>>> p.is_immutable()
True
is_mutable()[source]

Return True if the polyhedron is mutable, i.e. it can be modified in place.

EXAMPLES:

sage: p = polytopes.cube(backend='field')
sage: p.is_mutable()
False

>>> from sage.all import *
>>> p = polytopes.cube(backend='field')
>>> p.is_mutable()
False
line_generator()[source]

Return a generator for the lines of the polyhedron.

EXAMPLES:

sage: pr = Polyhedron(rays = [[1,0],[-1,0],[0,1]], vertices = [[-1,-1]])
sage: next(pr.line_generator()).vector()
(1, 0)

>>> from sage.all import *
>>> pr = Polyhedron(rays = [[Integer(1),Integer(0)],[-Integer(1),Integer(0)],[Integer(0),Integer(1)]], vertices = [[-Integer(1),-Integer(1)]])
>>> next(pr.line_generator()).vector()
(1, 0)
lines()[source]

Return all lines of the polyhedron.

OUTPUT: a tuple of lines

EXAMPLES:

sage: p = Polyhedron(rays = [[1,0],[-1,0],[0,1],[1,1]], vertices = [[-2,-2],[2,3]])
sage: p.lines()
(A line in the direction (1, 0),)

>>> from sage.all import *
>>> p = Polyhedron(rays = [[Integer(1),Integer(0)],[-Integer(1),Integer(0)],[Integer(0),Integer(1)],[Integer(1),Integer(1)]], vertices = [[-Integer(2),-Integer(2)],[Integer(2),Integer(3)]])
>>> p.lines()
(A line in the direction (1, 0),)
lines_list()[source]

Return a list of lines of the polyhedron. The line data is given as a list of coordinates rather than as a Hrepresentation object.

Note

It is recommended to use line_generator() instead to iterate over the list of Line objects.

EXAMPLES:

sage: p = Polyhedron(rays = [[1,0],[-1,0],[0,1],[1,1]], vertices = [[-2,-2],[2,3]])
sage: p.lines_list()
[[1, 0]]
sage: p.lines_list() == [list(x) for x in p.line_generator()]
True

>>> from sage.all import *
>>> p = Polyhedron(rays = [[Integer(1),Integer(0)],[-Integer(1),Integer(0)],[Integer(0),Integer(1)],[Integer(1),Integer(1)]], vertices = [[-Integer(2),-Integer(2)],[Integer(2),Integer(3)]])
>>> p.lines_list()
[[1, 0]]
>>> p.lines_list() == [list(x) for x in p.line_generator()]
True
n_Hrepresentation()[source]

Return the number of objects that make up the H-representation of the polyhedron.

OUTPUT: integer

EXAMPLES:

sage: p = polytopes.cross_polytope(4)
sage: p.n_Hrepresentation()
16
sage: p.n_Hrepresentation() == p.n_inequalities() + p.n_equations()
True

>>> from sage.all import *
>>> p = polytopes.cross_polytope(Integer(4))
>>> p.n_Hrepresentation()
16
>>> p.n_Hrepresentation() == p.n_inequalities() + p.n_equations()
True
n_Vrepresentation()[source]

Return the number of objects that make up the V-representation of the polyhedron.

OUTPUT: integer

EXAMPLES:

sage: p = polytopes.simplex(4)
sage: p.n_Vrepresentation()
5
sage: p.n_Vrepresentation() == p.n_vertices() + p.n_rays() + p.n_lines()
True

>>> from sage.all import *
>>> p = polytopes.simplex(Integer(4))
>>> p.n_Vrepresentation()
5
>>> p.n_Vrepresentation() == p.n_vertices() + p.n_rays() + p.n_lines()
True
n_equations()[source]

Return the number of equations. The representation will always be minimal, so the number of equations is the codimension of the polyhedron in the ambient space.

EXAMPLES:

sage: p = Polyhedron(vertices = [[1,0,0],[0,1,0],[0,0,1]])
sage: p.n_equations()
1

>>> from sage.all import *
>>> p = Polyhedron(vertices = [[Integer(1),Integer(0),Integer(0)],[Integer(0),Integer(1),Integer(0)],[Integer(0),Integer(0),Integer(1)]])
>>> p.n_equations()
1
n_facets()[source]

Return the number of inequalities. The representation will always be minimal, so the number of inequalities is the number of facets of the polyhedron in the ambient space.

EXAMPLES:

sage: p = Polyhedron(vertices = [[1,0,0],[0,1,0],[0,0,1]])
sage: p.n_inequalities()
3

sage: p = Polyhedron(vertices = [[t,t^2,t^3] for t in range(6)])
sage: p.n_facets()
8

>>> from sage.all import *
>>> p = Polyhedron(vertices = [[Integer(1),Integer(0),Integer(0)],[Integer(0),Integer(1),Integer(0)],[Integer(0),Integer(0),Integer(1)]])
>>> p.n_inequalities()
3

>>> p = Polyhedron(vertices = [[t,t**Integer(2),t**Integer(3)] for t in range(Integer(6))])
>>> p.n_facets()
8
n_inequalities()[source]

Return the number of inequalities. The representation will always be minimal, so the number of inequalities is the number of facets of the polyhedron in the ambient space.

EXAMPLES:

sage: p = Polyhedron(vertices = [[1,0,0],[0,1,0],[0,0,1]])
sage: p.n_inequalities()
3

sage: p = Polyhedron(vertices = [[t,t^2,t^3] for t in range(6)])
sage: p.n_facets()
8

>>> from sage.all import *
>>> p = Polyhedron(vertices = [[Integer(1),Integer(0),Integer(0)],[Integer(0),Integer(1),Integer(0)],[Integer(0),Integer(0),Integer(1)]])
>>> p.n_inequalities()
3

>>> p = Polyhedron(vertices = [[t,t**Integer(2),t**Integer(3)] for t in range(Integer(6))])
>>> p.n_facets()
8
n_lines()[source]

Return the number of lines. The representation will always be minimal.

EXAMPLES:

sage: p = Polyhedron(vertices = [[0,0]], rays=[[0,1],[0,-1]])
sage: p.n_lines()
1

>>> from sage.all import *
>>> p = Polyhedron(vertices = [[Integer(0),Integer(0)]], rays=[[Integer(0),Integer(1)],[Integer(0),-Integer(1)]])
>>> p.n_lines()
1
n_rays()[source]

Return the number of rays. The representation will always be minimal.

EXAMPLES:

sage: p = Polyhedron(vertices = [[1,0],[0,1]], rays=[[1,1]])
sage: p.n_rays()
1

>>> from sage.all import *
>>> p = Polyhedron(vertices = [[Integer(1),Integer(0)],[Integer(0),Integer(1)]], rays=[[Integer(1),Integer(1)]])
>>> p.n_rays()
1
n_vertices()[source]

Return the number of vertices. The representation will always be minimal.

Warning

If the polyhedron has lines, return the number of vertices in the Vrepresentation. As the represented polyhedron has no 0-dimensional faces (i.e. vertices), n_vertices corresponds to the number of \(k\)-faces, where \(k\) is the number of lines:

sage: P = Polyhedron(rays=[[1,0,0]],lines=[[0,1,0]])
sage: P.n_vertices()
1
sage: P.faces(0)
()
sage: P.f_vector()
(1, 0, 1, 1)

sage: P = Polyhedron(rays=[[1,0,0]],lines=[[0,1,0],[0,1,1]])
sage: P.n_vertices()
1
sage: P.f_vector()
(1, 0, 0, 1, 1)

>>> from sage.all import *
>>> P = Polyhedron(rays=[[Integer(1),Integer(0),Integer(0)]],lines=[[Integer(0),Integer(1),Integer(0)]])
>>> P.n_vertices()
1
>>> P.faces(Integer(0))
()
>>> P.f_vector()
(1, 0, 1, 1)

>>> P = Polyhedron(rays=[[Integer(1),Integer(0),Integer(0)]],lines=[[Integer(0),Integer(1),Integer(0)],[Integer(0),Integer(1),Integer(1)]])
>>> P.n_vertices()
1
>>> P.f_vector()
(1, 0, 0, 1, 1)

EXAMPLES:

sage: p = Polyhedron(vertices = [[1,0],[0,1],[1,1]], rays=[[1,1]])
sage: p.n_vertices()
2

>>> from sage.all import *
>>> p = Polyhedron(vertices = [[Integer(1),Integer(0)],[Integer(0),Integer(1)],[Integer(1),Integer(1)]], rays=[[Integer(1),Integer(1)]])
>>> p.n_vertices()
2
ray_generator()[source]

Return a generator for the rays of the polyhedron.

EXAMPLES:

sage: pi = Polyhedron(ieqs = [[1,1,0],[1,0,1]])
sage: pir = pi.ray_generator()
sage: [x.vector() for x in pir]
[(1, 0), (0, 1)]

>>> from sage.all import *
>>> pi = Polyhedron(ieqs = [[Integer(1),Integer(1),Integer(0)],[Integer(1),Integer(0),Integer(1)]])
>>> pir = pi.ray_generator()
>>> [x.vector() for x in pir]
[(1, 0), (0, 1)]
rays()[source]

Return a list of rays of the polyhedron.

OUTPUT: a tuple of rays

EXAMPLES:

sage: p = Polyhedron(ieqs = [[0,0,0,1],[0,0,1,0],[1,1,0,0]])
sage: p.rays()
(A ray in the direction (1, 0, 0),
 A ray in the direction (0, 1, 0),
 A ray in the direction (0, 0, 1))

>>> from sage.all import *
>>> p = Polyhedron(ieqs = [[Integer(0),Integer(0),Integer(0),Integer(1)],[Integer(0),Integer(0),Integer(1),Integer(0)],[Integer(1),Integer(1),Integer(0),Integer(0)]])
>>> p.rays()
(A ray in the direction (1, 0, 0),
 A ray in the direction (0, 1, 0),
 A ray in the direction (0, 0, 1))
rays_list()[source]

Return a list of rays as coefficient lists.

Note

It is recommended to use rays() or ray_generator() instead to iterate over the list of Ray objects.

OUTPUT: list of rays as lists of coordinates

EXAMPLES:

sage: p = Polyhedron(ieqs = [[0,0,0,1],[0,0,1,0],[1,1,0,0]])
sage: p.rays_list()
[[1, 0, 0], [0, 1, 0], [0, 0, 1]]
sage: p.rays_list() == [list(r) for r in p.ray_generator()]
True

>>> from sage.all import *
>>> p = Polyhedron(ieqs = [[Integer(0),Integer(0),Integer(0),Integer(1)],[Integer(0),Integer(0),Integer(1),Integer(0)],[Integer(1),Integer(1),Integer(0),Integer(0)]])
>>> p.rays_list()
[[1, 0, 0], [0, 1, 0], [0, 0, 1]]
>>> p.rays_list() == [list(r) for r in p.ray_generator()]
True
vertex_generator()[source]

Return a generator for the vertices of the polyhedron.

Warning

If the polyhedron has lines, return a generator for the vertices of the Vrepresentation. However, the represented polyhedron has no 0-dimensional faces (i.e. vertices):

sage: P = Polyhedron(rays=[[1,0,0]],lines=[[0,1,0]])
sage: list(P.vertex_generator())
[A vertex at (0, 0, 0)]
sage: P.faces(0)
()

>>> from sage.all import *
>>> P = Polyhedron(rays=[[Integer(1),Integer(0),Integer(0)]],lines=[[Integer(0),Integer(1),Integer(0)]])
>>> list(P.vertex_generator())
[A vertex at (0, 0, 0)]
>>> P.faces(Integer(0))
()

EXAMPLES:

sage: triangle = Polyhedron(vertices=[[1,0],[0,1],[1,1]])
sage: for v in triangle.vertex_generator(): print(v)
A vertex at (0, 1)
A vertex at (1, 0)
A vertex at (1, 1)
sage: v_gen = triangle.vertex_generator()
sage: next(v_gen)   # the first vertex
A vertex at (0, 1)
sage: next(v_gen)   # the second vertex
A vertex at (1, 0)
sage: next(v_gen)   # the third vertex
A vertex at (1, 1)
sage: try: next(v_gen)   # there are only three vertices
....: except StopIteration: print("STOP")
STOP
sage: type(v_gen)
<... 'generator'>
sage: [ v for v in triangle.vertex_generator() ]
[A vertex at (0, 1), A vertex at (1, 0), A vertex at (1, 1)]

>>> from sage.all import *
>>> triangle = Polyhedron(vertices=[[Integer(1),Integer(0)],[Integer(0),Integer(1)],[Integer(1),Integer(1)]])
>>> for v in triangle.vertex_generator(): print(v)
A vertex at (0, 1)
A vertex at (1, 0)
A vertex at (1, 1)
>>> v_gen = triangle.vertex_generator()
>>> next(v_gen)   # the first vertex
A vertex at (0, 1)
>>> next(v_gen)   # the second vertex
A vertex at (1, 0)
>>> next(v_gen)   # the third vertex
A vertex at (1, 1)
>>> try: next(v_gen)   # there are only three vertices
... except StopIteration: print("STOP")
STOP
>>> type(v_gen)
<... 'generator'>
>>> [ v for v in triangle.vertex_generator() ]
[A vertex at (0, 1), A vertex at (1, 0), A vertex at (1, 1)]
vertices()[source]

Return all vertices of the polyhedron.

OUTPUT: a tuple of vertices

Warning

If the polyhedron has lines, return the vertices of the Vrepresentation. However, the represented polyhedron has no 0-dimensional faces (i.e. vertices):

sage: P = Polyhedron(rays=[[1,0,0]],lines=[[0,1,0]])
sage: P.vertices()
(A vertex at (0, 0, 0),)
sage: P.faces(0)
()

>>> from sage.all import *
>>> P = Polyhedron(rays=[[Integer(1),Integer(0),Integer(0)]],lines=[[Integer(0),Integer(1),Integer(0)]])
>>> P.vertices()
(A vertex at (0, 0, 0),)
>>> P.faces(Integer(0))
()

EXAMPLES:

sage: triangle = Polyhedron(vertices=[[1,0],[0,1],[1,1]])
sage: triangle.vertices()
(A vertex at (0, 1), A vertex at (1, 0), A vertex at (1, 1))
sage: a_simplex = Polyhedron(ieqs = [
....:          [0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]
....:      ], eqns = [[1,-1,-1,-1,-1]])
sage: a_simplex.vertices()
(A vertex at (1, 0, 0, 0), A vertex at (0, 1, 0, 0),
 A vertex at (0, 0, 1, 0), A vertex at (0, 0, 0, 1))

>>> from sage.all import *
>>> triangle = Polyhedron(vertices=[[Integer(1),Integer(0)],[Integer(0),Integer(1)],[Integer(1),Integer(1)]])
>>> triangle.vertices()
(A vertex at (0, 1), A vertex at (1, 0), A vertex at (1, 1))
>>> a_simplex = Polyhedron(ieqs = [
...          [Integer(0),Integer(1),Integer(0),Integer(0),Integer(0)],[Integer(0),Integer(0),Integer(1),Integer(0),Integer(0)],[Integer(0),Integer(0),Integer(0),Integer(1),Integer(0)],[Integer(0),Integer(0),Integer(0),Integer(0),Integer(1)]
...      ], eqns = [[Integer(1),-Integer(1),-Integer(1),-Integer(1),-Integer(1)]])
>>> a_simplex.vertices()
(A vertex at (1, 0, 0, 0), A vertex at (0, 1, 0, 0),
 A vertex at (0, 0, 1, 0), A vertex at (0, 0, 0, 1))
vertices_list()[source]

Return a list of vertices of the polyhedron.

Note

It is recommended to use vertex_generator() instead to iterate over the list of Vertex objects.

Warning

If the polyhedron has lines, return the vertices of the Vrepresentation. However, the represented polyhedron has no 0-dimensional faces (i.e. vertices):

sage: P = Polyhedron(rays=[[1,0,0]],lines=[[0,1,0]])
sage: P.vertices_list()
[[0, 0, 0]]
sage: P.faces(0)
()

>>> from sage.all import *
>>> P = Polyhedron(rays=[[Integer(1),Integer(0),Integer(0)]],lines=[[Integer(0),Integer(1),Integer(0)]])
>>> P.vertices_list()
[[0, 0, 0]]
>>> P.faces(Integer(0))
()

EXAMPLES:

sage: triangle = Polyhedron(vertices=[[1,0],[0,1],[1,1]])
sage: triangle.vertices_list()
[[0, 1], [1, 0], [1, 1]]
sage: a_simplex = Polyhedron(ieqs = [
....:          [0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]
....:      ], eqns = [[1,-1,-1,-1,-1]])
sage: a_simplex.vertices_list()
[[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]]
sage: a_simplex.vertices_list() == [list(v) for v in a_simplex.vertex_generator()]
True

>>> from sage.all import *
>>> triangle = Polyhedron(vertices=[[Integer(1),Integer(0)],[Integer(0),Integer(1)],[Integer(1),Integer(1)]])
>>> triangle.vertices_list()
[[0, 1], [1, 0], [1, 1]]
>>> a_simplex = Polyhedron(ieqs = [
...          [Integer(0),Integer(1),Integer(0),Integer(0),Integer(0)],[Integer(0),Integer(0),Integer(1),Integer(0),Integer(0)],[Integer(0),Integer(0),Integer(0),Integer(1),Integer(0)],[Integer(0),Integer(0),Integer(0),Integer(0),Integer(1)]
...      ], eqns = [[Integer(1),-Integer(1),-Integer(1),-Integer(1),-Integer(1)]])
>>> a_simplex.vertices_list()
[[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]]
>>> a_simplex.vertices_list() == [list(v) for v in a_simplex.vertex_generator()]
True
vertices_matrix(base_ring=None)[source]

Return the coordinates of the vertices as the columns of a matrix.

INPUT:

  • base_ring – a ring or None (default); the base ring of the returned matrix. If not specified, the base ring of the polyhedron is used.

OUTPUT:

A matrix over base_ring whose columns are the coordinates of the vertices. A TypeError is raised if the coordinates cannot be converted to base_ring.

Warning

If the polyhedron has lines, return the coordinates of the vertices of the Vrepresentation. However, the represented polyhedron has no 0-dimensional faces (i.e. vertices):

sage: P = Polyhedron(rays=[[1,0,0]],lines=[[0,1,0]])
sage: P.vertices_matrix()
[0]
[0]
[0]
sage: P.faces(0)
()

>>> from sage.all import *
>>> P = Polyhedron(rays=[[Integer(1),Integer(0),Integer(0)]],lines=[[Integer(0),Integer(1),Integer(0)]])
>>> P.vertices_matrix()
[0]
[0]
[0]
>>> P.faces(Integer(0))
()

EXAMPLES:

sage: triangle = Polyhedron(vertices=[[1,0],[0,1],[1,1]])
sage: triangle.vertices_matrix()
[0 1 1]
[1 0 1]
sage: (triangle/2).vertices_matrix()
[  0 1/2 1/2]
[1/2   0 1/2]
sage: (triangle/2).vertices_matrix(ZZ)
Traceback (most recent call last):
...
TypeError: no conversion of this rational to integer

>>> from sage.all import *
>>> triangle = Polyhedron(vertices=[[Integer(1),Integer(0)],[Integer(0),Integer(1)],[Integer(1),Integer(1)]])
>>> triangle.vertices_matrix()
[0 1 1]
[1 0 1]
>>> (triangle/Integer(2)).vertices_matrix()
[  0 1/2 1/2]
[1/2   0 1/2]
>>> (triangle/Integer(2)).vertices_matrix(ZZ)
Traceback (most recent call last):
...
TypeError: no conversion of this rational to integer
write_cdd_Hrepresentation(filename)[source]

Export the polyhedron as a H-representation to a file.

INPUT:

  • filename – the output file

See also

cdd_Hrepresentation() – return the H-representation of the polyhedron as a string.

EXAMPLES:

sage: from sage.misc.temporary_file import tmp_filename
sage: filename = tmp_filename(ext='.ext')
sage: polytopes.cube().write_cdd_Hrepresentation(filename)

>>> from sage.all import *
>>> from sage.misc.temporary_file import tmp_filename
>>> filename = tmp_filename(ext='.ext')
>>> polytopes.cube().write_cdd_Hrepresentation(filename)
write_cdd_Vrepresentation(filename)[source]

Export the polyhedron as a V-representation to a file.

INPUT:

  • filename – the output file

See also

cdd_Vrepresentation() – return the V-representation of the polyhedron as a string.

EXAMPLES:

sage: from sage.misc.temporary_file import tmp_filename
sage: filename = tmp_filename(ext='.ext')
sage: polytopes.cube().write_cdd_Vrepresentation(filename)

>>> from sage.all import *
>>> from sage.misc.temporary_file import tmp_filename
>>> filename = tmp_filename(ext='.ext')
>>> polytopes.cube().write_cdd_Vrepresentation(filename)