def _init_from_Hrepresentation(self, ambient_dim, ieqs, eqns, minimize=True):
"""
Construct polyhedron from H-representation data.
INPUT:
- ``ambient_dim`` -- integer. The dimension of the ambient space.
- ``ieqs`` -- list of inequalities. Each line can be specified
as any iterable container of
:meth:`~sage.geometry.polyhedron.base.base_ring` elements.
- ``eqns`` -- list of equalities. Each line can be specified
as any iterable container of
:meth:`~sage.geometry.polyhedron.base.base_ring` elements.
EXAMPLES::
sage: p = Polyhedron(backend='ppl')
sage: from sage.geometry.polyhedron.backend_ppl import Polyhedron_QQ_ppl
sage: Polyhedron_QQ_ppl._init_from_Hrepresentation(p, 2, [], [])
"""
cs = Constraint_System()
if ieqs is None: ieqs = []
for ieq in ieqs:
d = lcm([denominator(ieq_i) for ieq_i in ieq])
dieq = [ ZZ(d*ieq_i) for ieq_i in ieq ]
b = dieq[0]
A = dieq[1:]
cs.insert(Linear_Expression(A, b) >= 0)
if eqns is None: eqns = []
for eqn in eqns:
d = lcm([denominator(eqn_i) for eqn_i in eqn])
deqn = [ ZZ(d*eqn_i) for eqn_i in eqn ]
b = deqn[0]
A = deqn[1:]
cs.insert(Linear_Expression(A, b) == 0)
self._ppl_polyhedron = C_Polyhedron(cs)
self._init_Vrepresentation_from_ppl(minimize)
self._init_Hrepresentation_from_ppl(minimize)
def _init_from_Hrepresentation(self,
ieqs,
eqns,
minimize=True,
verbose=False):
"""
Construct polyhedron from H-representation data.
INPUT:
- ``ieqs`` -- list of inequalities. Each line can be specified
as any iterable container of
:meth:`~sage.geometry.polyhedron.base.base_ring` elements.
- ``eqns`` -- list of equalities. Each line can be specified
as any iterable container of
:meth:`~sage.geometry.polyhedron.base.base_ring` elements.
- ``verbose`` -- boolean (default: ``False``). Whether to print
verbose output for debugging purposes.
EXAMPLES::
sage: p = Polyhedron(backend='ppl')
sage: from sage.geometry.polyhedron.backend_ppl import Polyhedron_ppl
sage: Polyhedron_ppl._init_from_Hrepresentation(p, [], [])
"""
cs = Constraint_System()
if ieqs is None: ieqs = []
for ieq in ieqs:
d = LCM_list([denominator(ieq_i) for ieq_i in ieq])
dieq = [ZZ(d * ieq_i) for ieq_i in ieq]
b = dieq[0]
A = dieq[1:]
cs.insert(Linear_Expression(A, b) >= 0)
if eqns is None: eqns = []
for eqn in eqns:
d = LCM_list([denominator(eqn_i) for eqn_i in eqn])
deqn = [ZZ(d * eqn_i) for eqn_i in eqn]
b = deqn[0]
A = deqn[1:]
cs.insert(Linear_Expression(A, b) == 0)
if cs.empty():
self._ppl_polyhedron = C_Polyhedron(self.ambient_dim(), 'universe')
else:
self._ppl_polyhedron = C_Polyhedron(cs)
self._init_Vrepresentation_from_ppl(minimize)
self._init_Hrepresentation_from_ppl(minimize)
def _init_from_Hrepresentation(self, ieqs, eqns, minimize=True, verbose=False):
"""
Construct polyhedron from H-representation data.
INPUT:
- ``ieqs`` -- list of inequalities. Each line can be specified
as any iterable container of
:meth:`~sage.geometry.polyhedron.base.base_ring` elements.
- ``eqns`` -- list of equalities. Each line can be specified
as any iterable container of
:meth:`~sage.geometry.polyhedron.base.base_ring` elements.
- ``verbose`` -- boolean (default: ``False``). Whether to print
verbose output for debugging purposes.
EXAMPLES::
sage: p = Polyhedron(backend='ppl')
sage: from sage.geometry.polyhedron.backend_ppl import Polyhedron_ppl
sage: Polyhedron_ppl._init_from_Hrepresentation(p, [], [])
"""
cs = Constraint_System()
if ieqs is None: ieqs = []
for ieq in ieqs:
d = LCM_list([denominator(ieq_i) for ieq_i in ieq])
dieq = [ ZZ(d*ieq_i) for ieq_i in ieq ]
b = dieq[0]
A = dieq[1:]
cs.insert(Linear_Expression(A, b) >= 0)
if eqns is None: eqns = []
for eqn in eqns:
d = LCM_list([denominator(eqn_i) for eqn_i in eqn])
deqn = [ ZZ(d*eqn_i) for eqn_i in eqn ]
b = deqn[0]
A = deqn[1:]
cs.insert(Linear_Expression(A, b) == 0)
if cs.empty():
self._ppl_polyhedron = C_Polyhedron(self.ambient_dim(), 'universe')
else:
self._ppl_polyhedron = C_Polyhedron(cs)
self._init_Vrepresentation_from_ppl(minimize)
self._init_Hrepresentation_from_ppl(minimize)
def _init_from_Hrepresentation(self,
ambient_dim,
ieqs,
eqns,
minimize=True):
"""
Construct polyhedron from H-representation data.
INPUT:
- ``ambient_dim`` -- integer. The dimension of the ambient space.
- ``ieqs`` -- list of inequalities. Each line can be specified
as any iterable container of
:meth:`~sage.geometry.polyhedron.base.base_ring` elements.
- ``eqns`` -- list of equalities. Each line can be specified
as any iterable container of
:meth:`~sage.geometry.polyhedron.base.base_ring` elements.
EXAMPLES::
sage: p = Polyhedron(backend='ppl')
sage: from sage.geometry.polyhedron.backend_ppl import Polyhedron_QQ_ppl
sage: Polyhedron_QQ_ppl._init_from_Hrepresentation(p, 2, [], [])
"""
cs = Constraint_System()
if ieqs is None: ieqs = []
for ieq in ieqs:
d = lcm([denominator(ieq_i) for ieq_i in ieq])
dieq = [ZZ(d * ieq_i) for ieq_i in ieq]
b = dieq[0]
A = dieq[1:]
cs.insert(Linear_Expression(A, b) >= 0)
if eqns is None: eqns = []
for eqn in eqns:
d = lcm([denominator(eqn_i) for eqn_i in eqn])
deqn = [ZZ(d * eqn_i) for eqn_i in eqn]
b = deqn[0]
A = deqn[1:]
cs.insert(Linear_Expression(A, b) == 0)
self._ppl_polyhedron = C_Polyhedron(cs)
self._init_Vrepresentation_from_ppl(minimize)
self._init_Hrepresentation_from_ppl(minimize)
def normal_cone(self):
r"""
Return the (closure of the) normal cone of the triangulation.
Recall that a regular triangulation is one that equals the
"crease lines" of a convex piecewise-linear function. This
support function is not unique, for example, you can scale it
by a positive constant. The set of all piecewise-linear
functions with fixed creases forms an open cone. This cone can
be interpreted as the cone of normal vectors at a point of the
secondary polytope, which is why we call it normal cone. See
[GKZ]_ Section 7.1 for details.
OUTPUT:
The closure of the normal cone. The `i`-th entry equals the
value of the piecewise-linear function at the `i`-th point of
the configuration.
For an irregular triangulation, the normal cone is empty. In
this case, a single point (the origin) is returned.
EXAMPLES::
sage: triangulation = polytopes.n_cube(2).triangulate(engine='internal')
sage: triangulation
(<0,1,3>, <0,2,3>)
sage: N = triangulation.normal_cone(); N
4-d cone in 4-d lattice
sage: N.rays()
(-1, 0, 0, 0),
( 1, 0, 1, 0),
(-1, 0, -1, 0),
( 1, 0, 0, -1),
(-1, 0, 0, 1),
( 1, 1, 0, 0),
(-1, -1, 0, 0)
in Ambient free module of rank 4
over the principal ideal domain Integer Ring
sage: N.dual().rays()
(-1, 1, 1, -1)
in Ambient free module of rank 4
over the principal ideal domain Integer Ring
TESTS::
sage: polytopes.n_simplex(2).triangulate().normal_cone()
3-d cone in 3-d lattice
sage: _.dual().is_trivial()
True
"""
if not self.point_configuration().base_ring().is_subring(QQ):
raise NotImplementedError("Only base rings ZZ and QQ are supported")
from sage.libs.ppl import Variable, Constraint, Constraint_System, Linear_Expression, C_Polyhedron
from sage.matrix.constructor import matrix
from sage.misc.misc import uniq
from sage.rings.arith import lcm
pc = self.point_configuration()
cs = Constraint_System()
for facet in self.interior_facets():
s0, s1 = self._boundary_simplex_dictionary()[facet]
p = set(s0).difference(facet).pop()
q = set(s1).difference(facet).pop()
origin = pc.point(p).reduced_affine_vector()
base_indices = [i for i in s0 if i != p]
base = matrix([pc.point(i).reduced_affine_vector() - origin for i in base_indices])
sol = base.solve_left(pc.point(q).reduced_affine_vector() - origin)
relation = [0] * pc.n_points()
relation[p] = sum(sol) - 1
relation[q] = 1
for i, base_i in enumerate(base_indices):
relation[base_i] = -sol[i]
rel_denom = lcm([QQ(r).denominator() for r in relation])
relation = [ZZ(r * rel_denom) for r in relation]
ex = Linear_Expression(relation, 0)
cs.insert(ex >= 0)
from sage.modules.free_module import FreeModule
ambient = FreeModule(ZZ, self.point_configuration().n_points())
if cs.empty():
cone = C_Polyhedron(ambient.dimension(), "universe")
else:
cone = C_Polyhedron(cs)
from sage.geometry.cone import _Cone_from_PPL
return _Cone_from_PPL(cone, lattice=ambient)
def normal_cone(self):
r"""
Return the (closure of the) normal cone of the triangulation.
Recall that a regular triangulation is one that equals the
"crease lines" of a convex piecewise-linear function. This
support function is not unique, for example, you can scale it
by a positive constant. The set of all piecewise-linear
functions with fixed creases forms an open cone. This cone can
be interpreted as the cone of normal vectors at a point of the
secondary polytope, which is why we call it normal cone. See
[GKZ]_ Section 7.1 for details.
OUTPUT:
The closure of the normal cone. The `i`-th entry equals the
value of the piecewise-linear function at the `i`-th point of
the configuration.
For an irregular triangulation, the normal cone is empty. In
this case, a single point (the origin) is returned.
EXAMPLES::
sage: triangulation = polytopes.hypercube(2).triangulate(engine='internal')
sage: triangulation
(<0,1,3>, <0,2,3>)
sage: N = triangulation.normal_cone(); N
4-d cone in 4-d lattice
sage: N.rays()
(-1, 0, 0, 0),
( 1, 0, 1, 0),
(-1, 0, -1, 0),
( 1, 0, 0, -1),
(-1, 0, 0, 1),
( 1, 1, 0, 0),
(-1, -1, 0, 0)
in Ambient free module of rank 4
over the principal ideal domain Integer Ring
sage: N.dual().rays()
(-1, 1, 1, -1)
in Ambient free module of rank 4
over the principal ideal domain Integer Ring
TESTS::
sage: polytopes.simplex(2).triangulate().normal_cone()
3-d cone in 3-d lattice
sage: _.dual().is_trivial()
True
"""
if not self.point_configuration().base_ring().is_subring(QQ):
raise NotImplementedError(
'Only base rings ZZ and QQ are supported')
from sage.libs.ppl import Variable, Constraint, Constraint_System, Linear_Expression, C_Polyhedron
from sage.matrix.constructor import matrix
from sage.misc.misc import uniq
from sage.rings.arith import lcm
pc = self.point_configuration()
cs = Constraint_System()
for facet in self.interior_facets():
s0, s1 = self._boundary_simplex_dictionary()[facet]
p = set(s0).difference(facet).pop()
q = set(s1).difference(facet).pop()
origin = pc.point(p).reduced_affine_vector()
base_indices = [i for i in s0 if i != p]
base = matrix([
pc.point(i).reduced_affine_vector() - origin
for i in base_indices
])
sol = base.solve_left(pc.point(q).reduced_affine_vector() - origin)
relation = [0] * pc.n_points()
relation[p] = sum(sol) - 1
relation[q] = 1
for i, base_i in enumerate(base_indices):
relation[base_i] = -sol[i]
rel_denom = lcm([QQ(r).denominator() for r in relation])
relation = [ZZ(r * rel_denom) for r in relation]
ex = Linear_Expression(relation, 0)
cs.insert(ex >= 0)
from sage.modules.free_module import FreeModule
ambient = FreeModule(ZZ, self.point_configuration().n_points())
if cs.empty():
cone = C_Polyhedron(ambient.dimension(), 'universe')
else:
cone = C_Polyhedron(cs)
from sage.geometry.cone import _Cone_from_PPL
return _Cone_from_PPL(cone, lattice=ambient)