def _init_from_Vrepresentation(self, vertices, rays, lines, minimize=True, verbose=False):
"""
Construct polyhedron from V-representation data.
INPUT:
- ``vertices`` -- list of point. Each point can be specified
as any iterable container of
:meth:`~sage.geometry.polyhedron.base.base_ring` elements.
- ``rays`` -- list of rays. Each ray can be specified as any
iterable container of
:meth:`~sage.geometry.polyhedron.base.base_ring` elements.
- ``lines`` -- list of lines. 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_Vrepresentation(p, [], [], [])
"""
gs = Generator_System()
if vertices is None: vertices = []
for v in vertices:
d = LCM_list([denominator(v_i) for v_i in v])
if d.is_one():
gs.insert(point(Linear_Expression(v, 0)))
else:
dv = [ d*v_i for v_i in v ]
gs.insert(point(Linear_Expression(dv, 0), d))
if rays is None: rays = []
for r in rays:
d = LCM_list([denominator(r_i) for r_i in r])
if d.is_one():
gs.insert(ray(Linear_Expression(r, 0)))
else:
dr = [ d*r_i for r_i in r ]
gs.insert(ray(Linear_Expression(dr, 0)))
if lines is None: lines = []
for l in lines:
d = LCM_list([denominator(l_i) for l_i in l])
if d.is_one():
gs.insert(line(Linear_Expression(l, 0)))
else:
dl = [ d*l_i for l_i in l ]
gs.insert(line(Linear_Expression(dl, 0)))
if gs.empty():
self._ppl_polyhedron = C_Polyhedron(self.ambient_dim(), 'empty')
else:
self._ppl_polyhedron = C_Polyhedron(gs)
self._init_Vrepresentation_from_ppl(minimize)
self._init_Hrepresentation_from_ppl(minimize)
def _init_from_Vrepresentation(self, vertices, rays, lines, minimize=True):
"""
Construct polyhedron from V-representation data.
INPUT:
- ``vertices`` -- list of point. Each point can be specified
as any iterable container of
:meth:`~sage.geometry.polyhedron.base.base_ring` elements.
- ``rays`` -- list of rays. Each ray can be specified as any
iterable container of
:meth:`~sage.geometry.polyhedron.base.base_ring` elements.
- ``lines`` -- list of lines. 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_ppl
sage: Polyhedron_ppl._init_from_Vrepresentation(p, [], [], [])
"""
gs = Generator_System()
if vertices is None: vertices = []
for v in vertices:
d = LCM_list([denominator(v_i) for v_i in v])
if d.is_one():
gs.insert(point(Linear_Expression(v, 0)))
else:
dv = [d * v_i for v_i in v]
gs.insert(point(Linear_Expression(dv, 0), d))
if rays is None: rays = []
for r in rays:
d = LCM_list([denominator(r_i) for r_i in r])
if d.is_one():
gs.insert(ray(Linear_Expression(r, 0)))
else:
dr = [d * r_i for r_i in r]
gs.insert(ray(Linear_Expression(dr, 0)))
if lines is None: lines = []
for l in lines:
d = LCM_list([denominator(l_i) for l_i in l])
if d.is_one():
gs.insert(line(Linear_Expression(l, 0)))
else:
dl = [d * l_i for l_i in l]
gs.insert(line(Linear_Expression(dl, 0)))
if gs.empty():
self._ppl_polyhedron = C_Polyhedron(self.ambient_dim(), 'empty')
else:
self._ppl_polyhedron = C_Polyhedron(gs)
self._init_Vrepresentation_from_ppl(minimize)
self._init_Hrepresentation_from_ppl(minimize)
def has_IP_property(self):
"""
Whether the lattice polytope has the IP property.
That is, the polytope is full-dimensional and the origin is a
interior point not on the boundary.
OUTPUT:
Boolean.
EXAMPLES::
sage: from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
sage: LatticePolytope_PPL((-1,-1),(0,1),(1,0)).has_IP_property()
True
sage: LatticePolytope_PPL((-1,-1),(1,1)).has_IP_property()
False
"""
origin = C_Polyhedron(point(0*Variable(self.space_dimension())))
is_included = Poly_Con_Relation.is_included()
saturates = Poly_Con_Relation.saturates()
for c in self.constraints():
rel = origin.relation_with(c)
if (not rel.implies(is_included)) or rel.implies(saturates):
return False
return True
def contains(self, point_coordinates):
r"""
Test whether point is contained in the polytope.
INPUT:
- ``point_coordinates`` -- a list/tuple/iterable of rational
numbers. The coordinates of the point.
OUTPUT:
Boolean.
EXAMPLES::
sage: from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
sage: line = LatticePolytope_PPL((1,2,3), (-1,-2,-3))
sage: line.contains([0,0,0])
True
sage: line.contains([1,0,0])
False
"""
p = C_Polyhedron(point(Linear_Expression(list(point_coordinates), 1)))
is_included = Poly_Con_Relation.is_included()
for c in self.constraints():
if not p.relation_with(c).implies(is_included):
return False
return True
def has_IP_property(self):
"""
Whether the lattice polytope has the IP property.
That is, the polytope is full-dimensional and the origin is a
interior point not on the boundary.
OUTPUT:
Boolean.
EXAMPLES::
sage: from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
sage: LatticePolytope_PPL((-1,-1),(0,1),(1,0)).has_IP_property()
True
sage: LatticePolytope_PPL((-1,-1),(1,1)).has_IP_property()
False
"""
origin = C_Polyhedron(point(0*Variable(self.space_dimension())))
is_included = Poly_Con_Relation.is_included()
saturates = Poly_Con_Relation.saturates()
for c in self.constraints():
rel = origin.relation_with(c)
if (not rel.implies(is_included)) or rel.implies(saturates):
return False
return True
def contains(self, point_coordinates):
r"""
Test whether point is contained in the polytope.
INPUT:
- ``point_coordinates`` -- a list/tuple/iterable of rational
numbers. The coordinates of the point.
OUTPUT:
Boolean.
EXAMPLES::
sage: from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
sage: line = LatticePolytope_PPL((1,2,3), (-1,-2,-3))
sage: line.contains([0,0,0])
True
sage: line.contains([1,0,0])
False
"""
p = C_Polyhedron(point(Linear_Expression(list(point_coordinates), 1)))
is_included = Poly_Con_Relation.is_included()
for c in self.constraints():
if not p.relation_with(c).implies(is_included):
return False
return True
def _init_from_Vrepresentation(self, ambient_dim, vertices, rays, lines, minimize=True):
"""
Construct polyhedron from V-representation data.
INPUT:
- ``ambient_dim`` -- integer. The dimension of the ambient space.
- ``vertices`` -- list of point. Each point can be specified
as any iterable container of
:meth:`~sage.geometry.polyhedron.base.base_ring` elements.
- ``rays`` -- list of rays. Each ray can be specified as any
iterable container of
:meth:`~sage.geometry.polyhedron.base.base_ring` elements.
- ``lines`` -- list of lines. 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_Vrepresentation(p, 2, [], [], [])
"""
gs = Generator_System()
if vertices is None: vertices = []
for v in vertices:
d = lcm([denominator(v_i) for v_i in v])
dv = [ ZZ(d*v_i) for v_i in v ]
gs.insert(point(Linear_Expression(dv, 0), d))
if rays is None: rays = []
for r in rays:
d = lcm([denominator(r_i) for r_i in r])
dr = [ ZZ(d*r_i) for r_i in r ]
gs.insert(ray(Linear_Expression(dr, 0)))
if lines is None: lines = []
for l in lines:
d = lcm([denominator(l_i) for l_i in l])
dl = [ ZZ(d*l_i) for l_i in l ]
gs.insert(line(Linear_Expression(dl, 0)))
self._ppl_polyhedron = C_Polyhedron(gs)
self._init_Vrepresentation_from_ppl(minimize)
self._init_Hrepresentation_from_ppl(minimize)
def fibration_generator(self, dim):
"""
Generate the lattice polytope fibrations.
For the purposes of this function, a lattice polytope fiber is
a sub-lattice polytope. Projecting the plane spanned by the
subpolytope to a point yields another lattice polytope, the
base of the fibration.
INPUT:
- ``dim`` -- integer. The dimension of the lattice polytope
fiber.
OUTPUT:
A generator yielding the distinct lattice polytope fibers of
given dimension.
EXAMPLES::
sage: from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
sage: p = LatticePolytope_PPL((-9,-6,-1,-1),(0,0,0,1),(0,0,1,0),(0,1,0,0),(1,0,0,0))
sage: list( p.fibration_generator(2) )
[A 2-dimensional lattice polytope in ZZ^4 with 3 vertices]
"""
assert self.is_full_dimensional()
codim = self.space_dimension() - dim
# "points" are the potential vertices of the fiber. They are
# in the $codim$-skeleton of the polytope, which is contained
# in the points that saturate at least $dim$ equations.
points = [ p for p in self._integral_points_saturating() if len(p[1])>=dim ]
points = sorted(points, key=lambda x:len(x[1]))
# iterate over point combinations subject to all points being on one facet.
def point_combinations_iterator(n, i0=0, saturated=None):
for i in range(i0, len(points)):
p, ieqs = points[i]
if saturated is None:
saturated_ieqs = ieqs
else:
saturated_ieqs = saturated.intersection(ieqs)
if len(saturated_ieqs)==0:
continue
if n == 1:
yield [i]
else:
for c in point_combinations_iterator(n-1, i+1, saturated_ieqs):
yield [i] + c
point_lines = [ line(Linear_Expression(p[0].list(),0)) for p in points ]
origin = point()
fibers = set()
gs = Generator_System()
for indices in point_combinations_iterator(dim):
gs.clear()
gs.insert(origin)
for i in indices:
gs.insert(point_lines[i])
plane = C_Polyhedron(gs)
if plane.affine_dimension() != dim:
continue
plane.intersection_assign(self)
if (not self.is_full_dimensional()) and (plane.affine_dimension() != dim):
continue
try:
fiber = LatticePolytope_PPL(plane)
except TypeError: # not a lattice polytope
continue
fiber_vertices = tuple(sorted(fiber.vertices()))
if fiber_vertices not in fibers:
yield fiber
fibers.update([fiber_vertices])
def LatticePolytope_PPL(*args):
"""
Construct a new instance of the PPL-based lattice polytope class.
EXAMPLES::
sage: from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
sage: LatticePolytope_PPL((0,0),(1,0),(0,1))
A 2-dimensional lattice polytope in ZZ^2 with 3 vertices
sage: from sage.libs.ppl import point, Generator_System, C_Polyhedron, Linear_Expression, Variable
sage: p = point(Linear_Expression([2,3],0)); p
point(2/1, 3/1)
sage: LatticePolytope_PPL(p)
A 0-dimensional lattice polytope in ZZ^2 with 1 vertex
sage: P = C_Polyhedron(Generator_System(p)); P
A 0-dimensional polyhedron in QQ^2 defined as the convex hull of 1 point
sage: LatticePolytope_PPL(P)
A 0-dimensional lattice polytope in ZZ^2 with 1 vertex
A ``TypeError`` is raised if the arguments do not specify a lattice polytope::
sage: from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
sage: LatticePolytope_PPL((0,0),(1/2,1))
Traceback (most recent call last):
...
TypeError: no conversion of this rational to integer
sage: from sage.libs.ppl import point, Generator_System, C_Polyhedron, Linear_Expression, Variable
sage: p = point(Linear_Expression([2,3],0), 5); p
point(2/5, 3/5)
sage: LatticePolytope_PPL(p)
Traceback (most recent call last):
...
TypeError: generator is not a lattice polytope generator
sage: P = C_Polyhedron(Generator_System(p)); P
A 0-dimensional polyhedron in QQ^2 defined as the convex hull of 1 point
sage: LatticePolytope_PPL(P)
Traceback (most recent call last):
...
TypeError: polyhedron has non-integral generators
"""
polytope_class = LatticePolytope_PPL_class
if len(args)==1 and isinstance(args[0], C_Polyhedron):
polyhedron = args[0]
polytope_class = _class_for_LatticePolytope(polyhedron.space_dimension())
if not all(p.is_point() and p.divisor().is_one() for p in polyhedron.generators()):
raise TypeError('polyhedron has non-integral generators')
return polytope_class(polyhedron)
if len(args)==1 \
and isinstance(args[0], (list, tuple)) \
and isinstance(args[0][0], (list,tuple)):
vertices = args[0]
else:
vertices = args
gs = Generator_System()
for v in vertices:
if isinstance(v, Generator):
if (not v.is_point()) or (not v.divisor().is_one()):
raise TypeError('generator is not a lattice polytope generator')
gs.insert(v)
else:
gs.insert(point(Linear_Expression(v, 0)))
if not gs.empty():
dim = next(Generator_System_iterator(gs)).space_dimension()
polytope_class = _class_for_LatticePolytope(dim)
return polytope_class(gs)
def fibration_generator(self, dim):
"""
Generate the lattice polytope fibrations.
For the purposes of this function, a lattice polytope fiber is
a sub-lattice polytope. Projecting the plane spanned by the
subpolytope to a point yields another lattice polytope, the
base of the fibration.
INPUT:
- ``dim`` -- integer. The dimension of the lattice polytope
fiber.
OUTPUT:
A generator yielding the distinct lattice polytope fibers of
given dimension.
EXAMPLES::
sage: from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
sage: p = LatticePolytope_PPL((-9,-6,-1,-1),(0,0,0,1),(0,0,1,0),(0,1,0,0),(1,0,0,0))
sage: list( p.fibration_generator(2) )
[A 2-dimensional lattice polytope in ZZ^4 with 3 vertices]
"""
assert self.is_full_dimensional()
codim = self.space_dimension() - dim
# "points" are the potential vertices of the fiber. They are
# in the $codim$-skeleton of the polytope, which is contained
# in the points that saturate at least $dim$ equations.
points = [ p for p in self._integral_points_saturating() if len(p[1])>=dim ]
points = sorted(points, key=lambda x:len(x[1]))
# iterate over point combinations subject to all points being on one facet.
def point_combinations_iterator(n, i0=0, saturated=None):
for i in range(i0, len(points)):
p, ieqs = points[i]
if saturated is None:
saturated_ieqs = ieqs
else:
saturated_ieqs = saturated.intersection(ieqs)
if len(saturated_ieqs)==0:
continue
if n == 1:
yield [i]
else:
for c in point_combinations_iterator(n-1, i+1, saturated_ieqs):
yield [i] + c
point_lines = [ line(Linear_Expression(p[0].list(),0)) for p in points ]
origin = point()
fibers = set()
gs = Generator_System()
for indices in point_combinations_iterator(dim):
gs.clear()
gs.insert(origin)
for i in indices:
gs.insert(point_lines[i])
plane = C_Polyhedron(gs)
if plane.affine_dimension() != dim:
continue
plane.intersection_assign(self)
if (not self.is_full_dimensional()) and (plane.affine_dimension() != dim):
continue
try:
fiber = LatticePolytope_PPL(plane)
except TypeError: # not a lattice polytope
continue
fiber_vertices = tuple(sorted(fiber.vertices()))
if fiber_vertices not in fibers:
yield fiber
fibers.update([fiber_vertices])
def LatticePolytope_PPL(*args):
"""
Construct a new instance of the PPL-based lattice polytope class.
EXAMPLES::
sage: from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
sage: LatticePolytope_PPL((0,0),(1,0),(0,1))
A 2-dimensional lattice polytope in ZZ^2 with 3 vertices
sage: from sage.libs.ppl import point, Generator_System, C_Polyhedron, Linear_Expression, Variable
sage: p = point(Linear_Expression([2,3],0)); p
point(2/1, 3/1)
sage: LatticePolytope_PPL(p)
A 0-dimensional lattice polytope in ZZ^2 with 1 vertex
sage: P = C_Polyhedron(Generator_System(p)); P
A 0-dimensional polyhedron in QQ^2 defined as the convex hull of 1 point
sage: LatticePolytope_PPL(P)
A 0-dimensional lattice polytope in ZZ^2 with 1 vertex
A ``TypeError`` is raised if the arguments do not specify a lattice polytope::
sage: from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
sage: LatticePolytope_PPL((0,0),(1/2,1))
Traceback (most recent call last):
...
TypeError: no conversion of this rational to integer
sage: from sage.libs.ppl import point, Generator_System, C_Polyhedron, Linear_Expression, Variable
sage: p = point(Linear_Expression([2,3],0), 5); p
point(2/5, 3/5)
sage: LatticePolytope_PPL(p)
Traceback (most recent call last):
...
TypeError: generator is not a lattice polytope generator
sage: P = C_Polyhedron(Generator_System(p)); P
A 0-dimensional polyhedron in QQ^2 defined as the convex hull of 1 point
sage: LatticePolytope_PPL(P)
Traceback (most recent call last):
...
TypeError: polyhedron has non-integral generators
"""
polytope_class = LatticePolytope_PPL_class
if len(args)==1 and isinstance(args[0], C_Polyhedron):
polyhedron = args[0]
polytope_class = _class_for_LatticePolytope(polyhedron.space_dimension())
if not all(p.is_point() and p.divisor().is_one() for p in polyhedron.generators()):
raise TypeError('polyhedron has non-integral generators')
return polytope_class(polyhedron)
if len(args)==1 \
and isinstance(args[0], (list, tuple)) \
and isinstance(args[0][0], (list,tuple)):
vertices = args[0]
else:
vertices = args
gs = Generator_System()
for v in vertices:
if isinstance(v, Generator):
if (not v.is_point()) or (not v.divisor().is_one()):
raise TypeError('generator is not a lattice polytope generator')
gs.insert(v)
else:
gs.insert(point(Linear_Expression(v, 0)))
if not gs.empty():
dim = Generator_System_iterator(gs).next().space_dimension()
polytope_class = _class_for_LatticePolytope(dim)
return polytope_class(gs)
def LatticePolytope_PPL(*args):
"""
Construct a new instance of the PPL-based lattice polytope class.
EXAMPLES::
sage: from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
sage: LatticePolytope_PPL((0,0),(1,0),(0,1))
A 2-dimensional lattice polytope in ZZ^2 with 3 vertices
sage: from sage.libs.ppl import point, Generator_System, C_Polyhedron, Linear_Expression, Variable
sage: p = point(Linear_Expression([2,3],0)); p
point(2/1, 3/1)
sage: LatticePolytope_PPL(p)
A 0-dimensional lattice polytope in ZZ^2 with 1 vertex
sage: P = C_Polyhedron(Generator_System(p)); P
A 0-dimensional polyhedron in QQ^2 defined as the convex hull of 1 point
sage: LatticePolytope_PPL(P)
A 0-dimensional lattice polytope in ZZ^2 with 1 vertex
A ``TypeError`` is raised if the arguments do not specify a lattice polytope::
sage: from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
sage: LatticePolytope_PPL((0,0),(1/2,1))
Traceback (most recent call last):
...
TypeError: no conversion of this rational to integer
sage: from sage.libs.ppl import point, Generator_System, C_Polyhedron, Linear_Expression, Variable
sage: p = point(Linear_Expression([2,3],0), 5); p
point(2/5, 3/5)
sage: LatticePolytope_PPL(p)
Traceback (most recent call last):
...
TypeError: The generator is not a lattice polytope generator.
sage: P = C_Polyhedron(Generator_System(p)); P
A 0-dimensional polyhedron in QQ^2 defined as the convex hull of 1 point
sage: LatticePolytope_PPL(P)
Traceback (most recent call last):
...
TypeError: The polyhedron has non-integral generators.
"""
if len(args)==1 and isinstance(args[0], C_Polyhedron):
polyhedron = args[0]
#if not polyhedron.is_bounded():
# raise TypeError('The polyhedron is unbounded.')
if not all(p.is_point() and p.divisor().is_one() for p in polyhedron.generators()):
raise TypeError('The polyhedron has non-integral generators.')
return LatticePolytope_PPL_class(polyhedron)
if len(args)==1:
vertices = args[0]
try:
return LatticePolytope_PPL(*vertices)
except TypeError:
pass
vertices = args
gs = Generator_System()
for v in vertices:
if isinstance(v, Generator):
if (not v.is_point()) or (not v.divisor().is_one()):
raise TypeError('The generator is not a lattice polytope generator.')
gs.insert(v)
else:
gs.insert(point(Linear_Expression(v, 0)))
return LatticePolytope_PPL_class(gs)
