def cone(self):
r"""
Return the cone defined by `A`.
This method is for debugging only. Assumes that the base ring
is `\QQ`.
OUTPUT:
The cone defined by the inequalities as a
:func:`~sage.geometry.polyhedron.constructor.Polyhedron`,
using the PPL backend.
EXAMPLES::
sage: from sage.geometry.polyhedron.double_description import StandardAlgorithm
sage: A = matrix(QQ, [(1,0,1), (0,1,1), (-1,-1,1)])
sage: DD, _ = StandardAlgorithm(A).initial_pair()
sage: DD.cone().Hrepresentation()
(An inequality (-1, -1, 1) x + 0 >= 0,
An inequality (0, 1, 1) x + 0 >= 0,
An inequality (1, 0, 1) x + 0 >= 0)
"""
from sage.geometry.polyhedron.constructor import Polyhedron
assert self.problem.base_ring() == QQ # required for PPL backend
if not self.A:
return Polyhedron(vertices=[[0] * self.problem.dim()],
backend='ppl')
else:
ieqs = [[0] + list(a) for a in self.A]
return Polyhedron(ieqs=ieqs,
base_ring=self.problem.base_ring(),
backend='ppl')
def verify(self):
r"""
Validate the double description pair.
This method used the PPL backend to check that the double
description pair is valid. An assertion is triggered if it is
not. Does nothing if the base ring is not `\QQ`.
EXAMPLES::
sage: from sage.geometry.polyhedron.double_description import \
....: DoubleDescriptionPair, Problem
sage: A = matrix(QQ, [(1,0,1), (0,1,1), (-1,-1,1)])
sage: alg = Problem(A)
sage: DD = DoubleDescriptionPair(alg,
....: [(1, 0, 3), (0, 1, 1), (-1, -1, 1)],
....: [(2/3, -1/3, 1/3), (-1/3, 2/3, 1/3), (-1/3, -1/3, 1/3)])
sage: DD.verify()
Traceback (most recent call last):
...
assert A_cone == R_cone
AssertionError
"""
from sage.geometry.polyhedron.constructor import Polyhedron
if self.problem.base_ring() is not QQ:
return
A_cone = self.cone()
R_cone = Polyhedron(vertices=[[self.zero] * self.problem.dim()],
rays=self.R,
base_ring=self.problem.base_ring(),
backend='ppl')
assert A_cone == R_cone
assert A_cone.n_inequalities() <= len(self.A)
assert R_cone.n_rays() == len(self.R)
def arp_polyhedron(d=3):
r"""
Return the d-dimensional 1-cylinders of the ARP algorithm.
EXAMPLES::
sage: from slabbe.matrix_cocycle import arp_polyhedron
sage: A,P,L = arp_polyhedron(3)
sage: A.vertices_list()
[[0, 0, 0], [1/2, 1/2, 0], [1/2, 1/4, 1/4], [1, 0, 0]]
sage: P.vertices_list()
[[0, 0, 0], [1/2, 1/2, 0], [1/2, 1/4, 1/4], [1/3, 1/3, 1/3]]
::
sage: A,P,L = arp_polyhedron(4)
sage: A.vertices_list()
[[0, 0, 0, 0],
[1/2, 1/2, 0, 0],
[1/2, 1/6, 1/6, 1/6],
[1/2, 1/4, 1/4, 0],
[1, 0, 0, 0]]
sage: P.vertices_list()
[[0, 0, 0, 0],
[1/2, 1/2, 0, 0],
[1/2, 1/4, 1/4, 0],
[1/2, 1/6, 1/6, 1/6],
[1/4, 1/4, 1/4, 1/4],
[1/3, 1/3, 1/3, 0]]
::
sage: A,P,L = arp_polyhedron(5)
sage: A.vertices_list()
[[0, 0, 0, 0, 0],
[1/2, 1/2, 0, 0, 0],
[1/2, 1/8, 1/8, 1/8, 1/8],
[1/2, 1/6, 1/6, 1/6, 0],
[1/2, 1/4, 1/4, 0, 0],
[1, 0, 0, 0, 0]]
sage: P.vertices_list()
[[0, 0, 0, 0, 0],
[1/2, 1/2, 0, 0, 0],
[1/2, 1/6, 1/6, 1/6, 0],
[1/2, 1/8, 1/8, 1/8, 1/8],
[1/2, 1/4, 1/4, 0, 0],
[1/3, 1/3, 1/3, 0, 0],
[1/5, 1/5, 1/5, 1/5, 1/5],
[1/4, 1/4, 1/4, 1/4, 0]]
"""
from sage.geometry.polyhedron.constructor import Polyhedron
positive = [ [0]*i + [1] + [0]*(d-i) for i in range(1, d+1)]
atmostone = [[1] + [-1]*d]
ieq_arnoux = [[0]+[1]+[-1]*(d-1)]
ieq_arnoux_not = [[0]+[-1]+[1]*(d-1)]
ieq_sorted = [ [0]*i + [1,-1] + [0]*(d-i-1) for i in range(1,d)]
A = Polyhedron(ieqs=positive + atmostone + ieq_sorted + ieq_arnoux)
P = Polyhedron(ieqs=positive + atmostone + ieq_sorted + ieq_arnoux_not)
L = Polyhedron(ieqs=positive + atmostone + ieq_sorted)
return A,P,L
def is_Z3_convergent(a, b, c, d, e, f, g):
r"""
TESTS::
sage: from surface_dynamics.misc.generalized_multiple_zeta_values import is_Z3_convergent
Convergent examples::
sage: assert is_Z3_convergent(2,0,2,2,0,0,0)
sage: assert is_Z3_convergent(0,0,0,0,0,0,4)
sage: assert is_Z3_convergent(1,0,0,1,0,0,2)
sage: assert is_Z3_convergent(0,1,0,1,0,0,2)
sage: assert is_Z3_convergent(0,1,0,0,0,1,2)
Divergent examples::
sage: assert not is_Z3_convergent(0,0,0,1,1,1,0)
sage: assert not is_Z3_convergent(0,0,0,0,0,0,3)
"""
from sage.geometry.polyhedron.constructor import Polyhedron
x, y, z = ZZ['x,y,z'].gens()
poly = x**a * y**b * z**c * (x + y)**d * (x + z)**e * (y + z)**f * (x + y +
z)**g
newton_polytope = Polyhedron(vertices=poly.exponents(),
rays=[(-1, 0, 0), (0, -1, 0), (0, 0, -1)])
V = newton_polytope.intersection(Polyhedron(rays=[(1, 1, 1)])).vertices()
r = max(max(v.vector()) for v in V)
return r > 1
def populate_polyhedra(self):
from sage.geometry.polyhedron.constructor import Polyhedron
from sage.symbolic.ring import SR
def get_vector(inequality, vars):
coefficients = list(inequality.coefficient(var) for var in vars)
constant = inequality - sum(c*v for c, v in zip(coefficients, vars))
return [constant] + coefficients
d = self.dimension()
prefix = self.trees[0].PREFIX
vars = list(SR(prefix + "{}".format(j)) for j in range(d+1))
for tree in self.trees:
others = list(self.trees)
others.remove(tree)
ineqs = [other.partition_cost() - tree.partition_cost()
for other in others] + vars
ineq_matrix = [get_vector(ineq, vars) for ineq in ineqs]
P = Polyhedron(ieqs=ineq_matrix)
if self.is_disjoint():
P = polyhedron_break_tie(P)
tree.polyhedron = P
nonnegative_orthant = Polyhedron(ieqs=[dd*(0,) + (1,) + (d+1-dd)*(0,)
for dd in range(1, d+1+1)])
assert all(A.polyhedron & nonnegative_orthant == A.polyhedron
for A in self.trees)
if self.is_disjoint():
assert all((A.polyhedron & B.polyhedron).is_empty()
for A in self.trees for B in self.trees if A != B)
def verify(self):
r"""
Validate the double description pair.
This method used the PPL backend to check that the double
description pair is valid. An assertion is triggered if it is
not. Does nothing if the base ring is not `\QQ`.
EXAMPLES::
sage: from sage.geometry.polyhedron.double_description import \
....: DoubleDescriptionPair, Problem
sage: A = matrix(QQ, [(1,0,1), (0,1,1), (-1,-1,1)])
sage: alg = Problem(A)
sage: DD = DoubleDescriptionPair(alg,
....: [(1, 0, 3), (0, 1, 1), (-1, -1, 1)],
....: [(2/3, -1/3, 1/3), (-1/3, 2/3, 1/3), (-1/3, -1/3, 1/3)])
sage: DD.verify()
Traceback (most recent call last):
...
assert A_cone == R_cone
AssertionError
"""
from sage.geometry.polyhedron.constructor import Polyhedron
if self.problem.base_ring() is not QQ:
return
A_cone = self.cone()
R_cone = Polyhedron(vertices=[[self.zero] * self.problem.dim()], rays=self.R,
base_ring=self.problem.base_ring(), backend='ppl')
assert A_cone == R_cone
assert A_cone.n_inequalities() <= len(self.A)
assert R_cone.n_rays() == len(self.R)
def verify(self, inequalities, equations):
"""
Compare result to PPL if the base ring is QQ.
This method is for debugging purposes and compares the
computation with another backend if available.
INPUT:
- ``inequalities``, ``equations`` -- see :class:`Hrep2Vrep`.
EXAMPLES::
sage: from sage.geometry.polyhedron.double_description_inhomogeneous import Hrep2Vrep
sage: H = Hrep2Vrep(QQ, 1, [(1,2)], [])
sage: H.verify([(1,2)], [])
"""
from sage.rings.all import QQ
from sage.geometry.polyhedron.constructor import Polyhedron
if self.base_ring is not QQ:
return
P = Polyhedron(vertices=self.vertices, rays=self.rays, lines=self.lines,
base_ring=QQ, ambient_dim=self.dim, backend='ppl')
Q = Polyhedron(ieqs=inequalities, eqns=equations,
base_ring=QQ, ambient_dim=self.dim, backend='ppl')
if (P != Q) or \
(len(self.vertices) != P.n_vertices()) or \
(len(self.rays) != P.n_rays()) or \
(len(self.lines) != P.n_lines()):
print('incorrect!', end="")
print(Q.Vrepresentation())
print(P.Hrepresentation())
def compute_monomials(R, nverts, dresverts):
"""Eq. A.8 from arXiv:1411.1418"""
X = np.array(R.gens())
Delta = Polyhedron(vertices=nverts)
npoints = np.array(Delta.integral_points())
P_expon = (npoints @ dresverts.T) + 1
P_monoms = np.power(X, P_expon).prod(axis=1)
return P_monoms
def cassaigne_polyhedron(d=3):
r"""
Return the d-dimensional 1-cylinders of the Cassaigne algorithm.
(of the dual!)
EXAMPLES::
sage: from slabbe.matrix_cocycle import cassaigne_polyhedron
sage: L,La,Lb = cassaigne_polyhedron(3)
sage: L.vertices_list()
[[0, 0, 0], [0, 1/2, 1/2], [1/3, 1/3, 1/3], [1/2, 1/2, 0]]
sage: La.vertices_list()
[[0, 0, 0], [0, 1/2, 1/2], [1/3, 1/3, 1/3], [1/4, 1/2, 1/4]]
sage: Lb.vertices_list()
[[0, 0, 0], [1/3, 1/3, 1/3], [1/2, 1/2, 0], [1/4, 1/2, 1/4]]
::
sage: L,La,Lb = cassaigne_polyhedron(4)
sage: L.vertices_list()
[[0, 0, 0, 0],
[0, 1/3, 1/3, 1/3],
[1/3, 1/3, 1/3, 0],
[1/4, 1/4, 1/4, 1/4],
[1/5, 2/5, 1/5, 1/5],
[1/5, 1/5, 2/5, 1/5]]
::
sage: L,La,Lb = cassaigne_polyhedron(5)
sage: L.vertices_list()
[[0, 0, 0, 0, 0],
[0, 1/4, 1/4, 1/4, 1/4],
[1/4, 1/4, 1/4, 1/4, 0],
[1/6, 1/6, 1/3, 1/6, 1/6],
[1/5, 1/5, 1/5, 1/5, 1/5],
[1/6, 1/3, 1/6, 1/6, 1/6],
[1/7, 2/7, 2/7, 1/7, 1/7],
[1/7, 2/7, 1/7, 2/7, 1/7],
[1/7, 1/7, 2/7, 2/7, 1/7],
[1/6, 1/6, 1/6, 1/3, 1/6]]
"""
from sage.geometry.polyhedron.constructor import Polyhedron
# [-1,7,3,4] represents the inequality 7x_1+3x_2+4x_3>= 1.
positive = [ [0]*i + [1] + [0]*(d-i) for i in range(1, d+1)]
atmostone = [[1] + [-1]*d]
ai_lt_a1d = [[0]+[1]+[0]*(i-2)+[-1]+[0]*(d-i-1)+[1] for i in range(2,d)]
ai_gt_a1 = [[0]+[-1]+[0]*(i-2)+[1]+[0]*(d-i-1)+[0] for i in range(2,d)]
ai_gt_ad = [[0]+[0]+[0]*(i-2)+[1]+[0]*(d-i-1)+[-1] for i in range(2,d)]
a1_gt_ad = [[0]+[1]+[0]*(d-2)+[-1]]
a1_lt_ad = [[0]+[-1]+[0]*(d-2)+[1]]
L = Polyhedron(ieqs=positive + atmostone + ai_lt_a1d + ai_gt_a1 + ai_gt_ad)
La = Polyhedron(ieqs=positive+atmostone+ai_lt_a1d+ai_gt_a1+ai_gt_ad+a1_lt_ad)
Lb = Polyhedron(ieqs=positive+atmostone+ai_lt_a1d+ai_gt_a1+ai_gt_ad+a1_gt_ad)
return L, La, Lb
def verify(self, inequalities, equations):
"""
Compare result to PPL if the base ring is QQ.
This method is for debugging purposes and compares the
computation with another backend if available.
INPUT:
- ``inequalities``, ``equations`` -- see :class:`Hrep2Vrep`.
EXAMPLES::
sage: from sage.geometry.polyhedron.double_description_inhomogeneous import Hrep2Vrep
sage: H = Hrep2Vrep(QQ, 1, [(1,2)], [])
sage: H.verify([(1,2)], [])
"""
from sage.rings.all import QQ
from sage.geometry.polyhedron.constructor import Polyhedron
if self.base_ring is not QQ:
return
P = Polyhedron(vertices=self.vertices, rays=self.rays, lines=self.lines,
base_ring=QQ, ambient_dim=self.dim, backend='ppl')
Q = Polyhedron(ieqs=inequalities, eqns=equations,
base_ring=QQ, ambient_dim=self.dim, backend='ppl')
if (P != Q) or \
(len(self.vertices) != P.n_vertices()) or \
(len(self.rays) != P.n_rays()) or \
(len(self.lines) != P.n_lines()):
print 'incorrect!',
print Q.Vrepresentation()
print P.Hrepresentation()
def is_convergent(n, den):
r"""
TESTS::
sage: import itertools
sage: from surface_dynamics.misc.generalized_multiple_zeta_values import is_convergent
sage: V = FreeModule(ZZ, 3)
sage: va = V((1,0,0)); va.set_immutable()
sage: vb = V((0,1,0)); vb.set_immutable()
sage: vc = V((0,0,1)); vc.set_immutable()
sage: vd = V((1,1,0)); vd.set_immutable()
sage: ve = V((1,0,1)); ve.set_immutable()
sage: vf = V((0,1,1)); vf.set_immutable()
sage: vg = V((1,1,1)); vg.set_immutable()
sage: gens = [va,vb,vc,vd,ve,vf,vg]
sage: N = 0
sage: for p in itertools.product([0,1,2], repeat=7): # optional: mzv
....: if sum(map(bool,p)) == 3 and is_convergent(3, list(zip(gens,p))):
....: print(p)
....: N += 1
(0, 0, 0, 0, 1, 1, 2)
(0, 0, 0, 0, 1, 2, 1)
(0, 0, 0, 0, 1, 2, 2)
(0, 0, 0, 0, 2, 1, 1)
(0, 0, 0, 0, 2, 1, 2)
(0, 0, 0, 0, 2, 2, 1)
(0, 0, 0, 0, 2, 2, 2)
(0, 0, 0, 1, 0, 1, 2)
...
(2, 0, 2, 0, 0, 2, 0)
(2, 0, 2, 2, 0, 0, 0)
(2, 1, 0, 0, 0, 0, 2)
(2, 1, 0, 0, 0, 2, 0)
(2, 2, 0, 0, 0, 0, 2)
(2, 2, 0, 0, 0, 2, 0)
(2, 2, 0, 0, 2, 0, 0)
(2, 2, 2, 0, 0, 0, 0)
sage: print(N) # optional: mzv
125
"""
from sage.geometry.polyhedron.constructor import Polyhedron
from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
assert all(len(v) == n for v, p in den), (n, den)
# TODO: fast code path
R = PolynomialRing(QQ, 'x', n)
x = R.gens()
den_poly = prod(linear_form(R, v)**p for v, p in den)
newton_polytope = Polyhedron(vertices=den_poly.exponents(),
rays=negative_rays(n))
V = newton_polytope.intersection(Polyhedron(rays=[[1] * n])).vertices()
r = max(max(v.vector()) for v in V)
return r > 1
def cone_triangulate(C, hyperplane=None):
r"""
Triangulation of rational cone contained in the positive quadrant.
EXAMPLES::
sage: from surface_dynamics.misc.linalg import cone_triangulate
sage: P = Polyhedron(rays=[(1,0,0),(0,1,0),(1,0,1),(0,1,1)])
sage: list(cone_triangulate(P)) # random
[[(0, 1, 1), (0, 1, 0), (1, 0, 0)], [(0, 1, 1), (1, 0, 1), (1, 0, 0)]]
sage: len(_)
2
sage: rays = [(0, 1, 0, -1, 0, 0),
....: (1, 0, -1, 0, 0, -1),
....: (0, 1, -1, 0, 0, -1),
....: (0, 0, 1, 0, 0, 0),
....: (0, 0, 0, 1, 0, -1),
....: (1, -1, 0, 0, 1, -1),
....: (0, 0, 0, 0, 1, -1),
....: (0, 0, 1, -1, 1, 0),
....: (0, 0, 1, -1, 0, 0),
....: (0, 0, 1, 0, -1, 0),
....: (0, 0, 0, 1, -1, -1),
....: (1, -1, 0, 0, 0, -1),
....: (0, 0, 0, 0, 0, -1)]
sage: P = Polyhedron(rays=rays)
sage: list(cone_triangulate(P, hyperplane=(1, 2, 3, -1, 0, -5))) # random
[[(0, 0, 0, 0, 0, -1),
(0, 0, 0, 0, 1, -1),
(0, 0, 0, 1, -1, -1),
(0, 0, 1, 0, 0, 0),
(0, 1, -1, 0, 0, -1),
(1, -1, 0, 0, 1, -1)],
...
(0, 0, 1, 0, 0, 0),
(0, 1, -1, 0, 0, -1),
(0, 1, 0, -1, 0, 0),
(1, -1, 0, 0, 1, -1),
(1, 0, -1, 0, 0, -1)]]
sage: len(_)
16
"""
rays = [r.vector() for r in C.rays()]
dim = len(rays[0])
if hyperplane is None:
hyperplane = [1] * dim
scalings = [sum(x * h for x, h in zip(r, hyperplane)) for r in rays]
assert all(s > 0 for s in scalings)
normalized_rays = [r / s for r, s in zip(rays, scalings)]
P = Polyhedron(vertices=normalized_rays)
for t in P.triangulate():
simplex = [P.Vrepresentation(i).vector() for i in t]
yield [(r / gcd(r)).change_ring(ZZ) for r in simplex]
def family_two(n, backend=None):
r"""
Return the vector configuration of the simplicial arrangement
`A(n,1)` from the family `\mathcal R(1)` in Grunbaum's list [Gru]_.
The arrangement will have an ``n`` hyperplanes, with ``n`` even, consisting
of the edges of the regular `n/2`-gon and the `n/2` lines of
mirror symmetry.
INPUT:
- ``n`` -- integer. ``n`` `\geq 6`. The number of lines in the arrangement.
- ``backend`` -- string (default = ``None``). The backend to use.
OUTPUT:
A vector configuration.
EXAMPLES::
sage: from cn_hyperarr.infinite_families import *
sage: pf = family_two(8,'normaliz'); pf # optional - pynormaliz
Vector configuration of 8 vectors in dimension 3
The number of lines must be even::
sage: pf3 = family_two(3,'normaliz'); # optional - pynormaliz
Traceback (most recent call last):
...
AssertionError: n must be even
The number of lines must be at least 6::
sage: pf4 = family_two(4,'normaliz') # optional - pynormaliz
Traceback (most recent call last):
...
ValueError: n (=2) must be an integer greater than 2
"""
assert n % 2 == 0, "n must be even"
reg_poly = polytopes.regular_polygon(n / QQ(2), backend='normaliz')
reg_cone = Polyhedron(
rays=[list(v.vector()) + [1] for v in reg_poly.vertices()],
backend=backend)
vecs = [h.A() for h in reg_cone.Hrepresentation()]
z = QQbar.zeta(n)
vecs += [[(z**k).real(), (z**k).imag(), 0] for k in range(n / QQ(2))]
return VectorConfiguration(vecs, backend=backend)
def taut_rays(tri, angle):
# get the extreme rays of the taut cone - note that the returned
# vectors are non-negative because they "point up"
N = edge_equation_matrix_taut(tri, angle)
N = Matrix(N)
dim = N.dimensions()[1]
elem_ieqs = [[0] + list(elem_vector(i, dim)) for i in range(dim)]
N_rows = [v.list() for v in N.rows()]
N_eqns = [[0] + v for v in N_rows]
P = Polyhedron(ieqs = elem_ieqs, eqns = N_eqns)
rays = [ray.vector() for ray in P.rays()]
for ray in rays:
assert all(a.is_integer() for a in ray)
# all of the entries are integers, represented in QQ, so we clean them.
return [vector(ZZ(a) for a in ray) for ray in rays]
def platonic_dodecahedron():
r"""Produce a triple consisting of a polyhedral version of the platonic dodecahedron,
the associated cone surface, and a ConeSurfaceToPolyhedronMap from the surface
to the polyhedron.
EXAMPLES::
sage: from flatsurf.geometry.polyhedra import platonic_dodecahedron
sage: polyhedron,surface,surface_to_polyhedron = platonic_dodecahedron()
sage: TestSuite(surface).run()
r"""
vertices = []
phi = AA(1 + sqrt(5)) / 2
F = NumberField(phi.minpoly(), "phi", embedding=phi)
phi = F.gen()
for x in range(-1, 3, 2):
for y in range(-1, 3, 2):
for z in range(-1, 3, 2):
vertices.append(vector(F, (x, y, z)))
for x in range(-1, 3, 2):
for y in range(-1, 3, 2):
vertices.append(vector(F, (0, x * phi, y / phi)))
vertices.append(vector(F, (y / phi, 0, x * phi)))
vertices.append(vector(F, (x * phi, y / phi, 0)))
scale = AA(2 / sqrt(1 + (phi - 1)**2 + (1 / phi - 1)**2))
p = Polyhedron(vertices=vertices)
s, m = polyhedron_to_cone_surface(p, scaling_factor=scale)
return p, s, m
def platonic_icosahedron():
r"""Produce a triple consisting of a polyhedral version of the platonic icosahedron,
the associated cone surface, and a ConeSurfaceToPolyhedronMap from the surface
to the polyhedron.
EXAMPLES::
sage: from flatsurf.geometry.polyhedra import platonic_icosahedron
sage: polyhedron,surface,surface_to_polyhedron = platonic_icosahedron()
sage: TestSuite(surface).run()
r"""
vertices = []
phi = AA(1 + sqrt(5)) / 2
F = NumberField(phi.minpoly(), "phi", embedding=phi)
phi = F.gen()
for i in range(3):
for s1 in range(-1, 3, 2):
for s2 in range(-1, 3, 2):
p = 3 * [None]
p[i] = s1 * phi
p[(i + 1) % 3] = s2
p[(i + 2) % 3] = 0
vertices.append(vector(F, p))
p = Polyhedron(vertices=vertices)
s, m = polyhedron_to_cone_surface(p)
return p, s, m
def random_interior_point(self, a=10, integer=False):
r"""
Return a random interior point of a polytope.
INPUT:
- ``a`` -- number, amplitude of random deplacement in the direction of each
ray.
- ``integer`` -- bool, whether the output must be with integer
coordinates
EXEMPLES::
sage: from slabbe.combinat import random_interior_point
sage: p = polytopes.hypercube(3)
sage: p = p + vector([20,0,0])
sage: p.center()
(20, 0, 0)
sage: random_interior_point(p) # random
(19.33174562788114, -0.5428002756082744, -0.3568284089832092)
sage: random_interior_point(p) # random
(20.039169786976075, -0.4121594862234468, -0.05623023234688396)
sage: random_interior_point(p, integer=True) # random
(21, 0, 0)
"""
from sage.geometry.polyhedron.constructor import Polyhedron
L = list(self.vertices())
L.extend(a * random() * ray.vector() for ray in self.rays())
P = Polyhedron(L)
return random_interior_point_compact_polytope(P)
def opposite_polyhedron(P, base_ring=None):
r"""Generate the polyhedron whose vertices are oppositve to a given one.
The opposite polyhedron `-P` is defined as the polyhedron such that `x \in -P` if and only if `-x \in P`.
INPUT:
* ``P`` - an object of class Polyhedron.
* ``base_ring`` - (default: that of P). The ``base_ring`` passed to construct `-P`.
OUTPUT:
A polyhedron whose vertices are opposite to those of `P`.
EXAMPLES::
sage: from polyhedron_tools.misc import BoxInfty, opposite_polyhedron
sage: P = BoxInfty([1,1], 0.5)
sage: minusP = opposite_polyhedron(P)
sage: P.plot(aspect_ratio=1) + minusP.plot() # not tested (plot)
TO-DO:
The possibility to receive P in matrix form `(A, b)`.
"""
if base_ring is None:
base_ring = P.base_ring()
return Polyhedron(vertices=[-1 * vector(v) for v in P.vertices_list()],
base_ring=base_ring)
def intersection(self, other):
r"""
The intersection of ``self`` with ``other``.
INPUT:
- ``other`` -- a hyperplane, a polyhedron, or something that
defines a polyhedron
OUTPUT:
A polyhedron.
EXAMPLES::
sage: H.<x,y,z> = HyperplaneArrangements(QQ)
sage: h = x + y + z - 1
sage: h.intersection(x - y)
A 1-dimensional polyhedron in QQ^3 defined as the convex hull of 1 vertex and 1 line
sage: h.intersection(polytopes.cube())
A 2-dimensional polyhedron in QQ^3 defined as the convex hull of 3 vertices
"""
from sage.geometry.polyhedron.base import is_Polyhedron
from sage.geometry.polyhedron.constructor import Polyhedron
if not is_Polyhedron(other):
try:
other = other.polyhedron()
except AttributeError:
other = Polyhedron(other)
return self.polyhedron().intersection(other)
def polyhedron_break_tie(polyhedron, undo=False):
from sage.geometry.polyhedron.constructor import Polyhedron
if polyhedron.equations():
raise NotImplementedError
return Polyhedron(ieqs=[break_tie(tuple(ieq), undo=undo)
for ieq in polyhedron.inequalities()])
def split_polyhedra(dim):
r"""
::
sage: from partitioner import split_polyhedra, repr_pretty_Hrepresentation
sage: for P in split_polyhedra(2):
....: print(repr_pretty_Hrepresentation(P, strict_inequality=True,
....: prefix='s'))
s1 > s0
s0 >= s1
sage: for P in split_polyhedra(3):
....: print(repr_pretty_Hrepresentation(P, strict_inequality=True,
....: prefix='s'))
s1 >= s0, s2 > s1
s2 > s0, s1 >= s2
s2 > s0, s0 > s1
s2 > s1, s0 >= s2
s1 >= s0, s0 >= s2
s1 >= s2, s0 > s1
sage: for P in split_polyhedra(4):
....: print(repr_pretty_Hrepresentation(P, strict_inequality=True,
....: prefix='s'))
s1 >= s0, s2 > s1, s3 > s2
s1 >= s0, s3 > s1, s2 >= s3
s2 >= s0, s3 > s1, s1 >= s2
s2 >= s0, s3 > s2, s1 >= s3
s3 > s0, s2 > s1, s1 >= s3
s3 > s0, s2 >= s3, s1 >= s2
s2 >= s0, s3 > s2, s0 > s1
s3 > s0, s2 >= s3, s0 > s1
s3 > s0, s2 > s1, s0 > s2
s2 > s1, s3 > s2, s0 >= s3
s2 >= s0, s3 > s1, s0 >= s3
s3 > s1, s2 >= s3, s0 > s2
s1 >= s0, s3 > s1, s0 > s2
s3 > s0, s1 >= s3, s0 > s2
s3 > s0, s1 >= s2, s0 > s1
s3 > s1, s1 >= s2, s0 >= s3
s1 >= s0, s3 > s2, s0 >= s3
s3 > s2, s1 >= s3, s0 > s1
s1 >= s0, s2 > s1, s0 >= s3
s2 >= s0, s1 >= s2, s0 >= s3
s2 >= s0, s1 >= s3, s0 > s1
s2 > s1, s1 >= s3, s0 > s2
s1 >= s0, s2 >= s3, s0 > s2
s2 >= s3, s1 >= s2, s0 > s1
"""
from sage.combinat.permutation import Permutations
from sage.geometry.polyhedron.constructor import Polyhedron
return iter(
polyhedron_break_tie(
Polyhedron(
ieqs=[tuple(1 if i==b else (-1 if i==a else 0)
for i in range(dim+1))
for a, b in zip(pi[:-1], pi[1:])]))
for pi in Permutations(dim))
def polytope(self, P=None):
"""
Return a polytope of ``self``.
INPUT:
- ``P`` -- (optional) a space to realize the polytope; default is
the weight lattice realization of the crystal
EXAMPLES::
sage: MV = crystals.infinity.MVPolytopes(['C', 3])
sage: b = MV.module_generators[0].f_string([3,2,3,2,1])
sage: P = b.polytope(); P
A 3-dimensional polyhedron in QQ^3 defined as the convex hull of 6 vertices
sage: P.vertices()
(A vertex at (0, 0, 0),
A vertex at (0, 1, -1),
A vertex at (0, 1, 1),
A vertex at (1, -1, 0),
A vertex at (1, 1, -2),
A vertex at (1, 1, 2))
"""
if P is None:
P = self.parent().weight_lattice_realization()
from sage.geometry.polyhedron.constructor import Polyhedron
return Polyhedron([v.to_vector() for v in self._polytope_vertices(P)])
def line_vertices(self):
r"""
Return the vertices for each line in a 3-d acyclic vector_configuration.
NOTE:
An acyclic vector configuration corresponds to the normal vectors of
a hyperplane arrangement with a selected base region.
OUTPUT:
A list of pairs.
EXAMPLES::
sage: from cn_hyperarr.vector_classes import *
sage: vc = VectorConfiguration([[1,0,0],[0,1,0],[0,0,1],[1,1,0],[0,1,1],[1,1,1]])
sage: vc.line_vertices()
[(0, 1), (1, 2), (2, 3), (0, 4)]
The vectors [-1,0,1] and [1,0,1] are the vertices of the line containing
[0,0,1]::
sage: vc = VectorConfiguration([[0,0,1],[-1,0,1],[1,0,1]])
sage: vc.line_vertices()
[(1, 2)]
"""
assert self.is_acyclic(), "The vector configuration should be acyclic"
nb_pts = self.n_vectors()
cocircuits = self.three_dim_cocircuits()
list_verts = []
for coc in cocircuits:
zero_indices = coc[1]
if len(zero_indices) > 2:
the_cone = Polyhedron(rays=[self[j] for j in zero_indices],
backend=self._backend)
the_verts = tuple([
_ for _ in zero_indices
if not the_cone.relative_interior_contains(self[_])
])
assert len(the_verts) == 2, "problem with the cocircuits"
if the_verts not in list_verts:
list_verts += [the_verts]
return list_verts
def is_union_convex(t):
r"""
Return whether the union of the polyhedrons is convex.
INPUT:
- ``t`` -- list of polyhedron
EXAMPLES::
sage: from slabbe.polyhedron_partition import is_union_convex
sage: h = 1/2
sage: p = Polyhedron([(0,h),(0,1),(h,1)])
sage: q = Polyhedron([(0,0), (0,h), (h,1), (1,1), (1,h), (h,0)])
sage: r = Polyhedron([(h,0), (1,0), (1,h)])
sage: is_union_convex((p,q,r))
True
sage: is_union_convex((p,q))
True
sage: is_union_convex((p,r))
False
Here we need to consider the three at the same time to get a convex
union::
sage: h = 1/5
sage: p = Polyhedron([(0,0),(h,1-h),(0,1)])
sage: q = Polyhedron([(0,1), (h,1-h), (1,1)])
sage: r = Polyhedron([(0,0), (h,1-h), (1,1), (1,0)])
sage: is_union_convex((p,q))
False
sage: is_union_convex((p,r))
False
sage: is_union_convex((q,r))
False
sage: is_union_convex((p,q,r))
True
"""
if not t:
return True
base_ring = t[0].base_ring()
vertices = sum((p.vertices() for p in t), tuple())
r = Polyhedron(vertices, base_ring=base_ring)
return r.volume() == sum(p.volume() for p in t)
def deformation_cone(v):
r"""
Return the deformation cone of the given vector ``v``
EXAMPLES::
sage: from surface_dynamics.misc.linalg import deformation_cone
sage: K.<sqrt2> = QuadraticField(2)
sage: v3 = vector([sqrt2, 1, 1+sqrt2])
sage: P = deformation_cone(v3)
sage: P
A 2-dimensional polyhedron in QQ^3 defined as the convex hull of 1 vertex and 2 rays
sage: P.rays_list()
[[1, 0, 1], [0, 1, 1]]
"""
V = deformation_space(v)
P = Polyhedron(lines=deformation_space(v).basis())
B = Polyhedron(rays=(QQ**V.degree()).basis())
return P.intersection(B)
def _an_element_(self):
"""
Returns a Newton polygon (which is the empty one)
TESTS:
sage: from sage.geometry.newton_polygon import NewtonPolygon
sage: NewtonPolygon._an_element_()
Empty Newton polygon
"""
return self(Polyhedron(base_ring=self.base_ring(), ambient_dim=2))
def _borcherds_product_polyhedron(self, pole_order, prec, verbose=False):
r"""
Construct a polyhedron representing a cone of Heegner divisors. For internal use in the methods borcherds_input_basis() and borcherds_input_Qbasis().
INPUT:
- ``pole_order`` -- pole order
- ``prec`` -- precision
OUTPUT: a tuple consisting of an integral matrix M, a Polyhedron p, and a WeilRepModularFormsBasis X
EXAMPLES::
sage: from weilrep import *
sage: m = ParamodularForms(5)
sage: m._borcherds_product_polyhedron(1/4, 5)[1]
A 2-dimensional polyhedron in QQ^3 defined as the convex hull of 1 vertex and 2 rays
"""
S = self.gram_matrix()
wt = self.input_wt()
w = self.weilrep()
rds = w.rds()
norm_dict = w.norm_dict()
X = w.nearly_holomorphic_modular_forms_basis(wt,
pole_order,
prec,
verbose=verbose)
N = len([g for g in rds if not norm_dict[tuple(g)]])
v_list = w.coefficient_vector_exponents(0,
1,
starting_from=-pole_order,
include_vectors=True)
exp_list = [v[1] for v in v_list]
v_list = [vector(v[0]) for v in v_list]
positive = [None] * len(exp_list)
zero = vector([0] * (len(exp_list) + 1))
M = matrix([
x.coefficient_vector(starting_from=-pole_order, ending_with=0)[:-N]
for x in X
])
vs = M.transpose().kernel().basis()
for i, n in enumerate(exp_list):
ieq = copy(zero)
ieq[i + 1] = 1
for j, m in enumerate(exp_list[:i]):
N = sqrt(m / n)
if N in ZZ:
v1 = v_list[i]
v2 = v_list[j]
ieq[j + 1] = denominator(v1 * N -
v2) == 1 or denominator(v1 * N +
v2) == 1
positive[i] = ieq
p = Polyhedron(ieqs=positive, eqns=[vector([0] + list(v)) for v in vs])
return M, p, X
def deformation_cone(v):
r"""
Return the deformation cone of the given vector ``v``
EXAMPLES::
sage: from surface_dynamics.misc.linalg import deformation_cone
sage: K.<sqrt2> = QuadraticField(2)
sage: v3 = vector([sqrt2, 1, 1+sqrt2])
sage: P = deformation_cone(v3)
sage: P
A 2-dimensional polyhedron in QQ^3 defined as the convex hull of 1 vertex and 2 rays
sage: P.rays_list()
[[1, 0, 1], [0, 1, 1]]
"""
from sage.rings.all import QQ
from sage.geometry.polyhedron.constructor import Polyhedron
V = deformation_space(v)
P = Polyhedron(lines=deformation_space(v).basis())
B = Polyhedron(rays=(QQ**V.degree()).basis())
return P.intersection(B)
def verify(self, vertices, rays, lines):
"""
Compare result to PPL if the base ring is QQ.
This method is for debugging purposes and compares the
computation with another backend if available.
INPUT:
- ``vertices``, ``rays``, ``lines`` -- see :class:`Vrep2Hrep`.
EXAMPLES::
sage: from sage.geometry.polyhedron.double_description_inhomogeneous import Vrep2Hrep
sage: vertices = [(-1/2,0)]
sage: rays = [(-1/2,2/3), (1/2,-1/3)]
sage: lines = []
sage: V2H = Vrep2Hrep(QQ, 2, vertices, rays, lines)
sage: V2H.verify(vertices, rays, lines)
"""
from sage.rings.all import QQ
from sage.geometry.polyhedron.constructor import Polyhedron
if self.base_ring is not QQ:
return
P = Polyhedron(vertices=vertices, rays=rays, lines=lines,
base_ring=QQ, ambient_dim=self.dim)
trivial = [self.base_ring.one()] + [self.base_ring.zero()] * self.dim # always true equation
Q = Polyhedron(ieqs=self.inequalities + [trivial], eqns=self.equations,
base_ring=QQ, ambient_dim=self.dim)
if not P == Q:
print 'incorrect!', P, Q
print Q.Vrepresentation()
print P.Hrepresentation()
def experiment():
rank = 5
level = 4
num_points = 11
client = cbc.CBClient()
liealg = cbd.TypeALieAlgebra(rank, store_fusion=True, exact=True)
rays = []
wts = []
ranks = []
for wt in liealg.get_weights(level):
if wt == tuple([0 for x in range(0, rank)]):
continue
cbb = cbd.SymmetricConformalBlocksBundle(client, liealg, wt,
num_points, level)
if cbb.get_rank() == 0:
continue
divisor = cbb.get_symmetrized_divisor()
if divisor == [
sage.Rational(0) for x in range(0, num_points // 2 - 1)
]:
continue
rays = rays + [divisor]
wts = wts + [wt]
ranks = ranks + [cbb.get_rank()]
#print(wt, cbb.get_rank(), divisor)
p = Polyhedron(rays=rays, base_ring=RationalField(), backend="cdd")
extremal_rays = [list(v.vector()) for v in p.Vrepresentation()]
c = Cone(p)
print("Extremal rays:")
for i in range(0, len(rays)):
ray = rays[i]
wt = wts[i]
rk = ranks[i]
if ray in extremal_rays:
print(rk, wt, ray)
def platonic_tetrahedron():
r"""Produce a triple consisting of a polyhedral version of the platonic tetrahedron,
the associated cone surface, and a ConeSurfaceToPolyhedronMap from the surface
to the polyhedron.
EXAMPLES::
sage: from flatsurf.geometry.polyhedra import platonic_tetrahedron
sage: polyhedron,surface,surface_to_polyhedron = platonic_tetrahedron()
sage: TestSuite(surface).run()
r"""
vertices = []
for x in range(-1, 3, 2):
for y in range(-1, 3, 2):
vertices.append(vector(QQ, (x, y, x * y)))
p = Polyhedron(vertices=vertices)
s, m = polyhedron_to_cone_surface(p, scaling_factor=AA(1 / sqrt(2)))
return p, s, m
def random_interior_point_compact_polytope(self,
uniform='simplex',
integer=False):
r"""
Return a random interior point of a compact polytope.
INPUT:
- ``uniform`` -- ``'points'`` (slow) or ``'simplex'`` (fast), whether
to take the probability uniformly with respect to the set of integral
points or with respect to the simplexes.
- ``integer`` -- bool, whether the output must be with integer
coordinates
EXEMPLES::
sage: from slabbe.combinat import random_interior_point_compact_polytope
sage: p = polytopes.hypercube(3)
sage: p = p + vector([30,20,10])
sage: p.center()
(30, 20, 10)
sage: random_interior_point_compact_polytope(p) # random
(19.33174562788114, -0.5428002756082744, -0.3568284089832092)
sage: random_interior_point_compact_polytope(p) # random
(20.039169786976075, -0.4121594862234468, -0.05623023234688396)
sage: random_interior_point_compact_polytope(p, integer=True) # random
(30, 19, 9)
"""
assert self.is_compact(), "input is not compact"
from sage.geometry.polyhedron.constructor import Polyhedron
triangulation = self.triangulate()
T = []
for simplex_indices in self.triangulate():
simplex_vertices = [self.Vrepresentation(i) for i in simplex_indices]
simplex = Polyhedron(simplex_vertices)
T.append(simplex)
if uniform == 'points':
v = [p.integral_points_count() for p in T] # slow
i = non_uniform_randint(v)
elif uniform == 'simplex':
i = randrange(len(T))
else:
raise ValueError('unknown value for uniform(={})'.format(uniform))
return random_interior_point_simplex(T[i], integer=integer)
def permutahedron(self, point=None):
r"""
Return the permutahedron of ``self``.
This is the convex hull of the point ``point`` in the weight
basis under the action of ``self`` on the underlying vector
space `V`.
INPUT:
- ``point`` -- optional, a point given by its coordinates in
the weight basis (default is `(1, 1, 1, \ldots)`)
.. NOTE::
The result is expressed in the root basis coordinates.
EXAMPLES::
sage: W = ReflectionGroup(['A',3]) # optional - gap3
sage: W.permutahedron() # optional - gap3
A 3-dimensional polyhedron in QQ^3 defined as the convex hull
of 24 vertices
sage: W = ReflectionGroup(['A',3],['B',2]) # optional - gap3
sage: W.permutahedron() # optional - gap3
A 5-dimensional polyhedron in QQ^5 defined as the convex hull of 192 vertices
TESTS::
sage: W = ReflectionGroup(['A',3]) # optional - gap3
sage: W.permutahedron([3,5,8]) # optional - gap3
A 3-dimensional polyhedron in QQ^3 defined as the convex hull
of 24 vertices
"""
n = self.rank()
weights = self.fundamental_weights()
if point is None:
point = [1] * n
v = sum(point[i] * wt for i, wt in enumerate(weights))
from sage.geometry.polyhedron.constructor import Polyhedron
return Polyhedron(vertices=[v*w.to_matrix() for w in self])
def platonic_octahedron():
r"""Produce a triple consisting of a polyhedral version of the platonic octahedron,
the associated cone surface, and a ConeSurfaceToPolyhedronMap from the surface
to the polyhedron.
EXAMPLES::
sage: from flatsurf.geometry.polyhedra import platonic_octahedron
sage: polyhedron,surface,surface_to_polyhedron = platonic_octahedron()
sage: TestSuite(surface).run()
r"""
vertices = []
for i in range(3):
temp = vector(QQ, [1 if k == i else 0 for k in range(3)])
for j in range(-1, 3, 2):
vertices.append(j * temp)
octahedron = Polyhedron(vertices=vertices)
surface,surface_to_octahedron = \
polyhedron_to_cone_surface(octahedron,scaling_factor=AA(sqrt(2)))
return octahedron, surface, surface_to_octahedron
def plot_walls(self, walls):
r"""
Plot ``walls``, i.e. 2-d cones, and their labels.
Ray generators must be specified during construction or using
:meth:`set_rays` before calling this method and these specified
ray generators will be used in conjunction with
:meth:`~sage.geometry.cone.ConvexRationalPolyhedralCone.ambient_ray_indices`
of ``walls``.
INPUT:
- ``walls`` -- a list of 2-d cones.
OUTPUT:
- a plot.
EXAMPLES::
sage: quadrant = Cone([(1,0), (0,1)])
sage: from sage.geometry.toric_plotter import ToricPlotter
sage: tp = ToricPlotter(dict(), 2, quadrant.rays())
sage: tp.plot_walls([quadrant])
Graphics object consisting of 2 graphics primitives
Let's also check that the truncating polyhedron is functioning
correctly::
sage: tp = ToricPlotter({"mode": "box"}, 2, quadrant.rays())
sage: tp.plot_walls([quadrant])
Graphics object consisting of 2 graphics primitives
"""
result = Graphics()
if not walls or not self.show_walls:
return result
rays = self.rays
extra_options = self.extra_options
mode = self.mode
alpha = self.wall_alpha
colors = color_list(self.wall_color, len(walls))
zorder = self.wall_zorder
if mode == "box":
if self.dimension <= 2:
ieqs = [(self.xmax, -1, 0), (- self.xmin, 1, 0),
(self.ymax, 0, -1), (- self.ymin, 0, 1)]
else:
ieqs = [(self.xmax, -1, 0, 0), (- self.xmin, 1, 0, 0),
(self.ymax, 0, -1, 0), (- self.ymin, 0, 1, 0),
(self.zmax, 0, 0, -1), (- self.zmin, 0, 0, 1)]
box = Polyhedron(ieqs=ieqs, base_ring=RDF)
for wall, color in zip(walls, colors):
result += box.intersection(wall.polyhedron()).render_solid(
alpha=alpha, color=color, zorder=zorder, **extra_options)
elif mode == "generators":
origin = self.origin
for wall, color in zip(walls, colors):
vertices = [rays[i] for i in wall.ambient_ray_indices()]
vertices.append(origin)
result += Polyhedron(vertices=vertices, base_ring=RDF).render_solid(
alpha=alpha, color=color, zorder=zorder, **extra_options)
label_sectors = []
round = mode == "round"
for wall, color in zip(walls, colors):
S = wall.linear_subspace()
lsd = S.dimension()
if lsd == 0: # Strictly convex wall
r1, r2 = (rays[i] for i in wall.ambient_ray_indices())
elif lsd == 1: # wall is a half-plane
for i, ray in zip(wall.ambient_ray_indices(), wall.rays()):
if ray in S:
r1 = rays[i]
else:
r2 = rays[i]
if round:
# Plot one "extra" sector
result += sector(- r1, r2,
alpha=alpha, color=color, zorder=zorder, **extra_options)
else: # wall is a plane
r1, r2 = S.basis()
r1 = vector(RDF, r1)
r1 = r1 / r1.norm() * self.radius
r2 = vector(RDF, r2)
r2 = r2 / r2.norm() * self.radius
if round:
# Plot three "extra" sectors
result += sector(r1, - r2,
alpha=alpha, color=color, zorder=zorder, **extra_options)
result += sector(- r1, r2,
alpha=alpha, color=color, zorder=zorder, **extra_options)
result += sector(- r1, - r2,
alpha=alpha, color=color, zorder=zorder, **extra_options)
label_sectors.append([r1, r2])
if round:
result += sector(r1, r2,
alpha=alpha, color=color, zorder=zorder, **extra_options)
result += self.plot_labels(self.wall_label,
[sum(label_sector) / 3 for label_sector in label_sectors])
return result
def RandomTriangulation(n, embed=False, base_ring=QQ):
"""
Returns a random triangulation on n vertices.
A triangulation is a planar graph all of whose faces are
triangles (3-cycles).
The graph is built by independently generating `n` points
uniformly at random on the surface of a sphere, finding the
convex hull of those points, and then returning the 1-skeleton
of that polyhedron.
INPUT:
- ``n`` -- number of vertices (recommend `n \ge 3`)
- ``embed`` -- (optional, default ``False``) whether to use the
stereographic point projections to draw the graph.
- ``base_ring`` -- (optional, default ``QQ``) specifies the field over
which to do the intermediate computations. The default setting is slower,
but works for any input; one can instead use ``RDF``, but this occasionally
fails due to loss of precision, as mentioned on :trac:10276.
EXAMPLES::
sage: g = graphs.RandomTriangulation(10)
sage: g.is_planar()
True
sage: g.num_edges() == 3*g.order() - 6
True
TESTS::
sage: for i in range(10):
....: g = graphs.RandomTriangulation(30,embed=True)
....: assert g.is_planar() and g.size() == 3*g.order()-6
"""
from sage.misc.prandom import normalvariate
from sage.geometry.polyhedron.constructor import Polyhedron
from sage.rings.real_double import RDF
# this function creates a random unit vector in R^3
def rand_unit_vec():
vec = [normalvariate(0, 1) for k in range(3)]
mag = sum([x * x for x in vec]) ** 0.5
return [x / mag for x in vec]
# generate n unit vectors at random
points = [rand_unit_vec() for k in range(n)]
# find their convex hull
P = Polyhedron(vertices=points, base_ring=base_ring)
# extract the 1-skeleton
g = P.vertex_graph()
g.rename('Planar triangulation on {} vertices'.format(n))
if embed:
from sage.geometry.polyhedron.plot import ProjectionFuncStereographic
from sage.modules.free_module_element import vector
proj = ProjectionFuncStereographic([0, 0, 1])
ppoints = [proj(vector(x)) for x in points]
g.set_pos({i: ppoints[i] for i in range(len(points))})
return g
def induced_out_partition(self, trans, ieq):
r"""
Returns the output partition obtained as the induction of the given
transformation on the domain given by an inequality.
Note: the output partition corresponds to the arrival partition in
the domain, not the initial one.
INPUT:
- ``trans`` -- a function: polyhedron -> polyhedron
- ``ieq`` -- list, an inequality. An entry equal to "[-1,7,3,4]"
represents the inequality 7x_1+3x_2+4x_3>= 1.
OUTPUT:
dict of polyhedron partitions with keys giving the return time
EXAMPLES::
sage: from slabbe import PolyhedronPartition, rotation_mod
sage: h = 1/3
sage: p = Polyhedron([(0,h),(0,1),(h,1)])
sage: q = Polyhedron([(0,0), (0,h), (h,1), (h,0)])
sage: r = Polyhedron([(h,1), (1,1), (1,h), (h,0)])
sage: s = Polyhedron([(h,0), (1,0), (1,h)])
sage: P = PolyhedronPartition({0:p, 1:q, 2:r, 3:s})
sage: u = rotation_mod(0, 1/3, 1, QQ)
sage: ieq = [h, -1, 0] # x0 <= h
sage: P.induced_out_partition(u, ieq)
{3: Polyhedron partition of 4 atoms with 4 letters}
::
sage: P = PolyhedronPartition({0:p, 1:q, 2:r, 3:s})
sage: ieq2 = [1/2, -1, 0] # x0 <= 1/2
sage: d = P.induced_out_partition(u, ieq2)
sage: d
{1: Polyhedron partition of 2 atoms with 2 letters,
2: Polyhedron partition of 3 atoms with 3 letters,
3: Polyhedron partition of 4 atoms with 4 letters}
sage: Q = PolyhedronPartition(d[1].atoms()+d[2].atoms()+d[3].atoms())
sage: Q.is_pairwise_disjoint()
True
::
sage: P = PolyhedronPartition({0:p, 1:q, 2:r, 3:s})
sage: ieq3 = [-1/2, 1, 0] # x0 >= 1/2
sage: P.induced_out_partition(u, ieq3)
{2: Polyhedron partition of 3 atoms with 3 letters,
3: Polyhedron partition of 4 atoms with 4 letters}
It is an error if the induced region is empty::
sage: P = PolyhedronPartition({0:p, 1:q, 2:r, 3:s})
sage: ieq4 = [-1/2, -1, 0] # x0 <= -1/2
sage: P.induced_out_partition(u, ieq4)
Traceback (most recent call last):
...
ValueError: Inequality An inequality (-2, 0) x - 1 >= 0 does
not intersect P (=Polyhedron partition of 4 atoms with 4
letters)
The whole domain::
sage: P = PolyhedronPartition({0:p, 1:q, 2:r, 3:s})
sage: ieq5 = [1/2, 1, 0] # x0 >= -1/2
sage: P.induced_out_partition(u, ieq5)
{1: Polyhedron partition of 4 atoms with 4 letters}
An irrational rotation::
sage: z = polygen(QQ, 'z') #z = QQ['z'].0 # same as
sage: K = NumberField(z**2-z-1, 'phi', embedding=RR(1.6))
sage: phi = K.gen()
sage: h = 1/phi^2
sage: p = Polyhedron([(0,h),(0,1),(h,1)])
sage: q = Polyhedron([(0,0), (0,h), (h,1), (h,0)])
sage: r = Polyhedron([(h,1), (1,1), (1,h), (h,0)])
sage: s = Polyhedron([(h,0), (1,0), (1,h)])
sage: P = PolyhedronPartition({0:p, 1:q, 2:r, 3:s}, base_ring=K)
sage: u = rotation_mod(0, 1/phi, 1, K)
sage: ieq = [phi^-4, -1, 0] # x0 <= phi^-4
sage: d = P.induced_out_partition(u, ieq)
sage: d
{5: Polyhedron partition of 6 atoms with 6 letters,
8: Polyhedron partition of 9 atoms with 9 letters}
"""
# good side of the hyperplane
half = Polyhedron(ieqs=[ieq])
half_part = PolyhedronPartition([half])
# the other side of the hyperplane
other_half = Polyhedron(ieqs=[[-a for a in ieq]])
other_half_part = PolyhedronPartition([other_half])
# initial refinement
P = self.refinement(half_part)
if len(P) == 0:
raise ValueError("Inequality {} does not intersect P "
"(={})".format(half.inequalities()[0], self))
level = 1
ans = {}
P = P.apply_transformation(trans)
while len(P):
P_returned = P.refinement(half_part)
if P_returned:
ans[level] = P_returned
# for what is remaining we do:
P = P.refinement(other_half_part)
P = P.refinement(self)
P = P.apply_transformation(trans)
level += 1
return ans