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 ppl_convert(P):
r"""
Convert a Sage polyhedron to a ppl polyhedron
EXAMPLES::
sage: from surface_dynamics.misc.ppl_utils import ppl_convert # optional - pplpy
sage: P = ppl_convert(Polyhedron(vertices=[(0,1,0),(1,0,1)], rays=[(0,0,1),[3,2,1]])) # optional - pplpy
sage: P.minimized_generators() # optional - pplpy
Generator_System {ray(0, 0, 1), point(0/1, 1/1, 0/1), point(1/1, 0/1, 1/1), ray(3, 2, 1)}
"""
if isinstance(P, ppl.C_Polyhedron):
return P
gs = ppl.Generator_System()
for v in P.vertices_list():
gs.insert(ppl.point(sum(j * ppl.Variable(i) for i, j in enumerate(v))))
for r in P.rays_list():
gs.insert(ppl.ray(sum(j * ppl.Variable(i) for i, j in enumerate(r))))
for l in P.lines_list():
gs.insert(ppl.line(sum(j * ppl.Variable(i) for i, j in enumerate(l))))
return ppl.C_Polyhedron(gs)
def ppl_convert(P):
r"""
Convert a Sage polyhedron to a ppl polyhedron
EXAMPLES::
sage: from surface_dynamics.misc.ppl_utils import ppl_convert # optional - pplpy
sage: P = ppl_convert(Polyhedron(vertices=[(0,1,0),(1,0,1)], rays=[(0,0,1),[3,2,1]])) # optional - pplpy
sage: P.minimized_generators() # optional - pplpy
Generator_System {ray(0, 0, 1), point(0/1, 1/1, 0/1), point(1/1, 0/1, 1/1), ray(3, 2, 1)}
"""
if isinstance(P, ppl.C_Polyhedron):
return P
gs = ppl.Generator_System()
for v in P.vertices_list():
gs.insert(ppl.point(sum(int(j) * ppl.Variable(i) for i,j in enumerate(v))))
for r in P.rays_list():
gs.insert(ppl.ray(sum(int(j) * ppl.Variable(i) for i,j in enumerate(r))))
for l in P.lines_list():
gs.insert(ppl.line(sum(int(j) * ppl.Variable(i) for i,j in enumerate(l))))
return ppl.C_Polyhedron(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()
# "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 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()
# "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 _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)