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 generateRay_cons(self, generators, vs, lvs, all_vars):
from ppl import Variable, Linear_Expression, ray
from termination.algorithm.utils import get_free_name
g_points = [g for g in generators if g.is_point()]
g_lines = [g for g in generators if g.is_line()]
g_rays = [g for g in generators if g.is_ray()]
for l in g_lines:
exp = Linear_Expression(0)
for i in range(len(vs)):
ci = int(l.coefficient(Variable(i)))
exp += ci * Variable(i)
g_rays.append(ray(exp))
g_rays.append(ray(-exp))
p_vars = get_free_name(all_vars, name="_a", num=len(g_points))
r_vars = get_free_name(all_vars, name="_br", num=len(g_rays))
ray_cons = []
relation = {}
for i in range(len(vs)):
exp = Term(0)
for pi in range(len(g_points)):
if i == 0:
relation[p_vars[pi]] = g_points[pi]
exp += Term(p_vars[pi], g_points[pi].divisor()) * int(g_points[pi].coefficient(Variable(i)))
for ri in range(len(g_rays)):
if i == 0:
relation[r_vars[ri]] = g_rays[ri]
ci = int(g_rays[ri].coefficient(Variable(i)))
if ci == 0:
continue
exp += Term(r_vars[ri]) * ci
ray_cons.append(exp == Term(vs[i]))
exp = Term(0)
for a in p_vars:
ai = Term(a)
exp += ai
ray_cons.append(ai >= 0)
ray_cons.append(exp == 1)
exp = Term(0)
for b in r_vars:
bi = Term(b)
exp += bi
ray_cons.append(bi >= 0)
ray_cons.append(exp >= 1)
ors = []
for b in r_vars:
bi = Term(b)
ors.append(exp == bi)
ray_cons.append(Or(ors))
return ray_cons, relation
def ppl_positive_cone(n):
r"""
Return the positive cone in R^n
EXAMPLES::
sage: from surface_dynamics.misc.ppl_utils import ppl_positive_cone # optional - pplpy
sage: C = ppl_positive_cone(3) # optional - pplpy
sage: C.minimized_generators() # optional - pplpy
Generator_System {point(0/1, 0/1, 0/1), ray(0, 0, 1), ray(0, 1, 0), ray(1, 0, 0)}
"""
gs = ppl.Generator_System(ppl_zero_point(n))
for i in range(n):
gs.insert(ppl.ray(ppl.Variable(i)))
return ppl.C_Polyhedron(gs)
def ppl_cone(rays):
r"""
Convert the list ``rays`` into a ppl cone
EXAMPLES::
sage: from surface_dynamics.misc.ppl_utils import ppl_cone # optional - pplpy
sage: C = ppl_cone([(0,1,2),(1,1,1),(1,0,1)]) # optional - pplpy
sage: C.minimized_generators() # optional - pplpy
Generator_System {point(0/1, 0/1, 0/1), ray(0, 1, 2), ray(1, 0, 1), ray(1, 1, 1)}
"""
n = len(rays[0])
gs = ppl.Generator_System(ppl_zero_point(n))
for r in rays:
gs.insert(ppl.ray(sum(j * ppl.Variable(i) for i, j in enumerate(r))))
return ppl.C_Polyhedron(gs)
def ppl_cone(rays):
r"""
Convert the list ``rays`` into a ppl cone
EXAMPLES::
sage: from surface_dynamics.misc.ppl_utils import ppl_cone # optional - pplpy
sage: C = ppl_cone([(0,1,2),(1,1,1),(1,0,1)]) # optional - pplpy
sage: C.minimized_generators() # optional - pplpy
Generator_System {point(0/1, 0/1, 0/1), ray(0, 1, 2), ray(1, 0, 1), ray(1, 1, 1)}
"""
n = len(rays[0])
gs = ppl.Generator_System(ppl_zero_point(n))
for r in rays:
gs.insert(ppl.ray(sum(int(j) * ppl.Variable(i) for i,j in enumerate(r))))
return ppl.C_Polyhedron(gs)
def ppl_positive_cone(n):
r"""
Return the positive cone in R^n
EXAMPLES::
sage: from surface_dynamics.misc.ppl_utils import ppl_positive_cone # optional - pplpy
sage: C = ppl_positive_cone(3) # optional - pplpy
sage: C.minimized_generators() # optional - pplpy
Generator_System {point(0/1, 0/1, 0/1), ray(0, 0, 1), ray(0, 1, 0), ray(1, 0, 0)}
"""
gs = ppl.Generator_System(ppl_zero_point(n))
l = [0]*n
for i in range(n):
gs.insert(ppl.ray(ppl.Variable(i)))
return ppl.C_Polyhedron(gs)
def hyperplanes(self):
"""
**Description:**
Returns the inward-pointing normals to the hyperplanes that define the
cone.
**Arguments:**
None.
**Returns:**
*(numpy.ndarray)* The list of inward-pointing normals to the
hyperplanes that define the cone.
**Example:**
We construct two cones and find their hyperplane normals.
```python {3,6}
c1 = Cone([[0,1],[1,1]])
c2 = Cone(hyperplanes=[[0,1],[1,1]])
c1.hyperplanes()
# array([[ 1, 0],
# [-1, 1]])
c2.hyperplanes()
# array([[0, 1],
# [1, 1]])
```
"""
if self._hyperplanes is not None:
return np.array(self._hyperplanes)
if self._ambient_dim >= 12 and len(self.rays()) != self._ambient_dim:
print("Warning: This operation might take a while for d > ~12 "
"and is likely impossible for d > ~18.")
gs = ppl.Generator_System()
vrs = [ppl.Variable(i) for i in range(self._ambient_dim)]
gs.insert(ppl.point(0))
for r in self.extremal_rays():
gs.insert(
ppl.ray(sum(r[i] * vrs[i] for i in range(self._ambient_dim))))
cone = ppl.C_Polyhedron(gs)
hyperplanes = []
for cstr in cone.minimized_constraints():
hyperplanes.append(tuple(int(c) for c in cstr.coefficients()))
if cstr.is_equality():
hyperplanes.append(tuple(-int(c) for c in cstr.coefficients()))
self._hyperplanes = np.array(hyperplanes, dtype=int)
return np.array(self._hyperplanes)
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 _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)