Conversions#
This module provides conversions to ListOfFaces
from
- an incidence matrix of a polyhedron or
- a tuple of facets (as tuple of vertices each).
Also this module provides a conversion from the data of ListOfFaces
,
which is a Bit-vector representing incidences of a face,
to a list of entries which are incident.
See also
EXAMPLES:
Obtain the facets of a polyhedron as ListOfFaces
:
sage: from sage.geometry.polyhedron.combinatorial_polyhedron.conversions \
....: import incidence_matrix_to_bit_rep_of_facets
sage: P = polytopes.simplex(4)
sage: inc = P.incidence_matrix()
sage: mod_inc = inc.delete_columns([i for i,V in enumerate(P.Hrepresentation()) if V.is_equation()])
sage: face_list = incidence_matrix_to_bit_rep_of_facets(mod_inc)
sage: face_list.compute_dimension()
4
Obtain the Vrepresentation of a polyhedron as facet-incidences stored in
ListOfFaces
:
sage: # needs sage.combinat
sage: from sage.geometry.polyhedron.combinatorial_polyhedron.conversions \
....: import incidence_matrix_to_bit_rep_of_Vrep
sage: P = polytopes.associahedron(['A',4])
sage: face_list = incidence_matrix_to_bit_rep_of_Vrep(P.incidence_matrix())
sage: face_list.compute_dimension()
4
Obtain the facets of a polyhedron as ListOfFaces
from a facet list:
sage: from sage.geometry.polyhedron.combinatorial_polyhedron.conversions \
....: import facets_tuple_to_bit_rep_of_facets
sage: facets = ((0,1,2), (0,1,3), (0,2,3), (1,2,3))
sage: face_list = facets_tuple_to_bit_rep_of_facets(facets, 4)
Likewise for the Vrep as facet-incidences:
sage: from sage.geometry.polyhedron.combinatorial_polyhedron.conversions \
....: import facets_tuple_to_bit_rep_of_Vrep
sage: facets = ((0,1,2), (0,1,3), (0,2,3), (1,2,3))
sage: face_list = facets_tuple_to_bit_rep_of_Vrep(facets, 4)
AUTHOR:
Jonathan Kliem (2019-04)
- sage.geometry.polyhedron.combinatorial_polyhedron.conversions.facets_tuple_to_bit_rep_of_Vrep(facets_input, n_Vrep)#
Initialize Vrepresentatives in Bit-representation as
ListOfFaces
.Each Vrepresentative is represented as the facets it is contained in. Those are the facets of the polar polyhedron, if it exists.
INPUT:
facets_input
– tuple of facets, each facet a tuple of Vrep, Vrep must be exactlyrange(n_Vrep)
n_Vrep
OUTPUT:
EXAMPLES:
sage: from sage.geometry.polyhedron.combinatorial_polyhedron.conversions \ ....: import facets_tuple_to_bit_rep_of_Vrep, \ ....: _bit_rep_to_Vrep_list_wrapper sage: bi_pyr = ((0,1,4), (1,2,4), (2,3,4), (3,0,4), ....: (0,1,5), (1,2,5), (2,3,5), (3,0,5)) sage: vertices = facets_tuple_to_bit_rep_of_Vrep(bi_pyr, 6) sage: for i in range(6): ....: print(_bit_rep_to_Vrep_list_wrapper(vertices, i)) (0, 3, 4, 7) (0, 1, 4, 5) (1, 2, 5, 6) (2, 3, 6, 7) (0, 1, 2, 3) (4, 5, 6, 7)
- sage.geometry.polyhedron.combinatorial_polyhedron.conversions.facets_tuple_to_bit_rep_of_facets(facets_input, n_Vrep)#
Initializes facets in Bit-representation as
ListOfFaces
.INPUT:
facets_input
– tuple of facets, each facet a tuple of Vrep, Vrep must be exactlyrange(n_Vrep)
n_Vrep
OUTPUT:
EXAMPLES:
sage: from sage.geometry.polyhedron.combinatorial_polyhedron.conversions \ ....: import facets_tuple_to_bit_rep_of_facets, \ ....: _bit_rep_to_Vrep_list_wrapper sage: bi_pyr = ((0,1,4), (1,2,4), (2,3,4), (3,0,4), ....: (0,1,5), (1,2,5), (2,3,5), (3,0,5)) sage: facets = facets_tuple_to_bit_rep_of_facets(bi_pyr, 6) sage: for i in range(8): ....: print(_bit_rep_to_Vrep_list_wrapper(facets, i)) (0, 1, 4) (1, 2, 4) (2, 3, 4) (0, 3, 4) (0, 1, 5) (1, 2, 5) (2, 3, 5) (0, 3, 5)
- sage.geometry.polyhedron.combinatorial_polyhedron.conversions.incidence_matrix_to_bit_rep_of_Vrep(matrix)#
Initialize Vrepresentatives in Bit-representation as
ListOfFaces
.Each Vrepresentative is represented as the facets it is contained in. Those are the facets of the polar polyhedron, if it exists.
INPUT:
matrix
– an incidence matrix as insage.geometry.polyhedron.base.Polyhedron_base.incidence_matrix()
with columns corresponding to equations deleted of typesage.matrix.matrix_dense.Matrix_dense
OUTPUT:
EXAMPLES:
sage: from sage.geometry.polyhedron.combinatorial_polyhedron.conversions \ ....: import incidence_matrix_to_bit_rep_of_Vrep, \ ....: _bit_rep_to_Vrep_list_wrapper sage: P = polytopes.permutahedron(4) sage: inc = P.incidence_matrix() sage: mod_inc = inc.delete_columns([i for i,V in enumerate(P.Hrepresentation()) if V.is_equation()]) sage: vertices = incidence_matrix_to_bit_rep_of_Vrep(mod_inc) sage: vertices.matrix().dimensions() (24, 14) sage: for row in vertices.matrix(): ....: row.nonzero_positions() [8, 9, 11] [8, 10, 11] [2, 3, 7] [1, 5, 7] [4, 5, 7] [1, 3, 7] [4, 6, 7] [2, 6, 7] [1, 5, 13] [8, 9, 13] [1, 9, 11] [2, 10, 11] [1, 3, 11] [2, 3, 11] [4, 5, 13] [4, 12, 13] [8, 12, 13] [1, 9, 13] [0, 8, 12] [0, 4, 12] [0, 2, 10] [0, 2, 6] [0, 8, 10] [0, 4, 6]
- sage.geometry.polyhedron.combinatorial_polyhedron.conversions.incidence_matrix_to_bit_rep_of_facets(matrix)#
Initialize facets in Bit-representation as
ListOfFaces
.INPUT:
matrix
– an incidence matrix as insage.geometry.polyhedron.base.Polyhedron_base.incidence_matrix()
with columns corresponding to equations deleted of typesage.matrix.matrix_dense.Matrix_dense
OUTPUT:
EXAMPLES:
sage: from sage.geometry.polyhedron.combinatorial_polyhedron.conversions \ ....: import incidence_matrix_to_bit_rep_of_facets, \ ....: _bit_rep_to_Vrep_list_wrapper sage: P = polytopes.permutahedron(4) sage: inc = P.incidence_matrix() sage: mod_inc = inc.delete_columns([i for i,V in enumerate(P.Hrepresentation()) if V.is_equation()]) sage: facets = incidence_matrix_to_bit_rep_of_facets(mod_inc) sage: facets.matrix().dimensions() (14, 24) sage: for row in facets.matrix(): ....: row.nonzero_positions() [18, 19, 20, 21, 22, 23] [3, 5, 8, 10, 12, 17] [2, 7, 11, 13, 20, 21] [2, 5, 12, 13] [4, 6, 14, 15, 19, 23] [3, 4, 8, 14] [6, 7, 21, 23] [2, 3, 4, 5, 6, 7] [0, 1, 9, 16, 18, 22] [0, 9, 10, 17] [1, 11, 20, 22] [0, 1, 10, 11, 12, 13] [15, 16, 18, 19] [8, 9, 14, 15, 16, 17]