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 - 0to- 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 ``Nonein which case
- alignis- Trueif- separatoris 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.constructorfor a definition of vertex/ray/line.- INPUT: - index– either an integer or- None
 - OUTPUT: - The optional argument is an index running from - 0to- 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: - pplthe Parma Polyhedra Library
- cddCDD
- normaliznormaliz
- polymakepolymake
- fielda 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 - RDFshould 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- RDFmay 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- Equationobjects.- 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- Inequalityobjects.- 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 - Trueif 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 - Trueif 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- Lineobjects.- 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_verticescorresponds 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- Rayobjects.- 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- Vertexobjects.- 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_ringwhose columns are the coordinates of the vertices. A- TypeErroris 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)