def vertices(self):
r"""
Return the vertices as a tuple of `\ZZ`-vectors.
OUTPUT:
A tuple of `\ZZ`-vectors. Each entry is the coordinate vector
of an integral points of the lattice polytope.
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: p.vertices()
((-9, -6, -1, -1), (0, 0, 0, 1), (0, 0, 1, 0), (0, 1, 0, 0), (1, 0, 0, 0))
sage: p.minimized_generators()
Generator_System {point(-9/1, -6/1, -1/1, -1/1), point(0/1, 0/1, 0/1, 1/1),
point(0/1, 0/1, 1/1, 0/1), point(0/1, 1/1, 0/1, 0/1), point(1/1, 0/1, 0/1, 0/1)}
"""
d = self.space_dimension()
v = vector(ZZ, d)
points = []
for g in self.minimized_generators():
for i in range(0,d):
v[i] = g.coefficient(Variable(i))
v_copy = copy.copy(v)
v_copy.set_immutable()
points.append(v_copy)
return tuple(points)
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 bounding_box(self):
r"""
Return the coordinates of a rectangular box containing the non-empty polytope.
OUTPUT:
A pair of tuples ``(box_min, box_max)`` where ``box_min`` are
the coordinates of a point bounding the coordinates of the
polytope from below and ``box_max`` bounds the coordinates
from above.
EXAMPLES::
sage: from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL
sage: LatticePolytope_PPL((0,0),(1,0),(0,1)).bounding_box()
((0, 0), (1, 1))
"""
box_min = []
box_max = []
if self.is_empty():
raise ValueError('empty polytope is not allowed')
for i in range(0, self.space_dimension()):
x = Variable(i)
coords = [ v.coefficient(x) for v in self.generators() ]
max_coord = max(coords)
min_coord = min(coords)
box_max.append(max_coord)
box_min.append(min_coord)
return (tuple(box_min), tuple(box_max))
