def ambient_vector_space(self):
"""
Return the ambient vector space.
.. SEEALSO::
:meth:`ambient_module`
OUTPUT:
The vector space (over the fraction field of the base ring)
where the linear expressions live.
EXAMPLES::
sage: from sage.geometry.linear_expression import LinearExpressionModule
sage: L = LinearExpressionModule(QQ, ('x', 'y', 'z'))
sage: L.ambient_vector_space()
Vector space of dimension 3 over Rational Field
sage: M = LinearExpressionModule(ZZ, ('r', 's'))
sage: M.ambient_module()
Ambient free module of rank 2 over the principal ideal domain Integer Ring
sage: M.ambient_vector_space()
Vector space of dimension 2 over Rational Field
"""
from sage.modules.all import VectorSpace
field = self.base_ring().fraction_field()
return VectorSpace(field, self.ngens())
def fiber(self):
r"""
Return the generic fiber of the vector bundle.
OUTPUT:
A vector space over :meth:`base_ring`.
EXAMPLES::
sage: T_P2 = toric_varieties.P2().sheaves.tangent_bundle()
sage: T_P2.fiber()
Vector space of dimension 2 over Rational Field
"""
from sage.modules.all import VectorSpace
return VectorSpace(self.base_ring(), self.rank())
def vertices(self):
"""
EXAMPLES::
sage: P = polymake.convex_hull([[1,0,0,0], [1,0,0,1], [1,0,1,0], [1,0,1,1], [1,1,0,0], [1,1,0,1], [1,1,1,0], [1,1,1,1]]) # optional - polymake
sage: P.vertices() # optional - polymake
[(1, 0, 0, 0), (1, 0, 0, 1), (1, 0, 1, 0), (1, 0, 1, 1), (1, 1, 0, 0), (1, 1, 0, 1), (1, 1, 1, 0), (1, 1, 1, 1)]
"""
try:
return self.__vertices
except AttributeError:
pass
s = self.cmd('VERTICES')
s = s.rstrip().split('\n')[1:]
if len(s) == 0:
ans = Sequence([], immutable=True)
else:
n = len(s[0].split())
V = VectorSpace(QQ, n)
ans = Sequence((V(x.split()) for x in s), immutable=True)
self.__vertices = ans
return ans
def facets(self):
"""
EXAMPLES::
sage: P = polymake.convex_hull([[1,0,0,0], [1,0,0,1], [1,0,1,0], [1,0,1,1], [1,1,0,0], [1,1,0,1], [1,1,1,0], [1,1,1,1]]) # optional: needs polymake
sage: P.facets() # optional
[(0, 0, 0, 1), (0, 1, 0, 0), (0, 0, 1, 0), (1, 0, 0, -1), (1, 0, -1, 0), (1, -1, 0, 0)]
"""
try:
return self.__facets
except AttributeError:
pass
s = self.cmd('FACETS')
s = s.rstrip().split('\n')[1:]
if len(s) == 0:
ans = Sequence([], immutable=True)
else:
n = len(s[0].split())
V = VectorSpace(QQ, n)
ans = Sequence((V(x.split()) for x in s), immutable=True)
self.__facets = ans
return ans
def right_angle_triangle(w, h):
r"""
TESTS::
sage: from flatsurf import *
sage: R = similarity_surfaces.right_angle_triangle(2, 3)
sage: R
ConeSurface built from 2 polygons
sage: TestSuite(R).run()
"""
from sage.modules.free_module_element import vector
F = Sequence([w, h]).universe()
if not F.is_field():
F = F.fraction_field()
V = VectorSpace(F, 2)
P = ConvexPolygons(F)
s = Surface_list(base_ring=F)
s.add_polygon(P([V((w, 0)), V((-w, h)), V((0, -h))])) # gets label 0
s.add_polygon(P([V((0, h)), V((-w, -h)), V((w, 0))])) # gets label 1
s.change_polygon_gluings(0, [(1, 2), (1, 1), (1, 0)])
s.set_immutable()
return ConeSurface(s)
def __call__(self, program, complex, subcomplex=None, **kwds):
"""
Call a CHomP program to compute the homology of a chain
complex, simplicial complex, or cubical complex.
See :class:`CHomP` for full documentation.
EXAMPLES::
sage: from sage.interfaces.chomp import CHomP
sage: T = cubical_complexes.Torus()
sage: CHomP()('homcubes', T) # indirect doctest, optional - CHomP
{0: 0, 1: Z x Z, 2: Z}
"""
from sage.misc.temporary_file import tmp_filename
from sage.homology.all import CubicalComplex, cubical_complexes
from sage.homology.all import SimplicialComplex, Simplex
from sage.homology.chain_complex import HomologyGroup
from subprocess import Popen, PIPE
from sage.rings.all import QQ, ZZ
from sage.modules.all import VectorSpace, vector
from sage.combinat.free_module import CombinatorialFreeModule
if not have_chomp(program):
raise OSError("Program %s not found" % program)
verbose = kwds.get('verbose', False)
generators = kwds.get('generators', False)
extra_opts = kwds.get('extra_opts', '')
base_ring = kwds.get('base_ring', ZZ)
if extra_opts:
extra_opts = extra_opts.split()
else:
extra_opts = []
# type of complex:
cubical = False
simplicial = False
chain = False
# CHomP seems to have problems with cubical complexes if the
# first interval in the first cube defining the complex is
# degenerate. So replace the complex X with [0,1] x X.
if isinstance(complex, CubicalComplex):
cubical = True
edge = cubical_complexes.Cube(1)
original_complex = complex
complex = edge.product(complex)
if verbose:
print("Cubical complex")
elif isinstance(complex, SimplicialComplex):
simplicial = True
if verbose:
print("Simplicial complex")
else:
chain = True
base_ring = kwds.get('base_ring', complex.base_ring())
if verbose:
print("Chain complex over %s" % base_ring)
if base_ring == QQ:
raise ValueError(
"CHomP doesn't compute over the rationals, only over Z or F_p."
)
if base_ring.is_prime_field():
p = base_ring.characteristic()
extra_opts.append('-p%s' % p)
mod_p = True
else:
mod_p = False
#
# complex
#
try:
data = complex._chomp_repr_()
except AttributeError:
raise AttributeError(
"Complex cannot be converted to use with CHomP.")
datafile = tmp_filename()
with open(datafile, 'w') as f:
f.write(data)
#
# subcomplex
#
if subcomplex is None:
if cubical:
subcomplex = CubicalComplex([complex.n_cells(0)[0]])
elif simplicial:
m = re.search(r'\(([^,]*),', data)
v = int(m.group(1))
subcomplex = SimplicialComplex([[v]])
else:
# replace subcomplex with [0,1] x subcomplex.
if cubical:
subcomplex = edge.product(subcomplex)
#
# generators
#
if generators:
genfile = tmp_filename()
extra_opts.append('-g%s' % genfile)
#
# call program
#
if subcomplex is not None:
try:
sub = subcomplex._chomp_repr_()
except AttributeError:
raise AttributeError(
"Subcomplex cannot be converted to use with CHomP.")
subfile = tmp_filename()
with open(subfile, 'w') as f:
f.write(sub)
else:
subfile = ''
if verbose:
print("Popen called with arguments", end="")
print([program, datafile, subfile] + extra_opts)
print("")
print("CHomP output:")
print("")
# output = Popen([program, datafile, subfile, extra_opts],
cmd = [program, datafile]
if subfile:
cmd.append(subfile)
if extra_opts:
cmd.extend(extra_opts)
output = Popen(cmd, stdout=PIPE).communicate()[0]
if verbose:
print(output)
print("End of CHomP output")
print("")
if generators:
with open(genfile, 'r') as f:
gens = f.read()
if verbose:
print("Generators:")
print(gens)
#
# process output
#
if output.find('ERROR') != -1:
raise RuntimeError('error inside CHomP')
# output contains substrings of one of the forms
# "H_1 = Z", "H_1 = Z_2 + Z", "H_1 = Z_2 + Z^2",
# "H_1 = Z + Z_2 + Z"
if output.find('trivial') != -1:
if mod_p:
return {0: VectorSpace(base_ring, 0)}
else:
return {0: HomologyGroup(0, ZZ)}
d = {}
h = re.compile("^H_([0-9]*) = (.*)$", re.M)
tors = re.compile("Z_([0-9]*)")
#
# homology groups
#
for m in h.finditer(output):
if verbose:
print(m.groups())
# dim is the dimension of the homology group
dim = int(m.group(1))
# hom_str is the right side of the equation "H_n = Z^r + Z_k + ..."
hom_str = m.group(2)
# need to read off number of summands and their invariants
if hom_str.find("0") == 0:
if mod_p:
hom = VectorSpace(base_ring, 0)
else:
hom = HomologyGroup(0, ZZ)
else:
rk = 0
if hom_str.find("^") != -1:
rk_srch = re.search(r'\^([0-9]*)\s?', hom_str)
rk = int(rk_srch.group(1))
rk += len(re.findall(r"(Z$)|(Z\s)", hom_str))
if mod_p:
rk = rk if rk != 0 else 1
if verbose:
print("dimension = %s, rank of homology = %s" %
(dim, rk))
hom = VectorSpace(base_ring, rk)
else:
n = rk
invts = []
for t in tors.finditer(hom_str):
n += 1
invts.append(int(t.group(1)))
for i in range(rk):
invts.append(0)
if verbose:
print(
"dimension = %s, number of factors = %s, invariants = %s"
% (dim, n, invts))
hom = HomologyGroup(n, ZZ, invts)
#
# generators
#
if generators:
if cubical:
g = process_generators_cubical(gens, dim)
if verbose:
print("raw generators: %s" % g)
if g:
module = CombinatorialFreeModule(
base_ring,
original_complex.n_cells(dim),
prefix="",
bracket=True)
basis = module.basis()
output = []
for x in g:
v = module(0)
for term in x:
v += term[0] * basis[term[1]]
output.append(v)
g = output
elif simplicial:
g = process_generators_simplicial(gens, dim, complex)
if verbose:
print("raw generators: %s" % gens)
if g:
module = CombinatorialFreeModule(base_ring,
complex.n_cells(dim),
prefix="",
bracket=False)
basis = module.basis()
output = []
for x in g:
v = module(0)
for term in x:
if complex._is_numeric():
v += term[0] * basis[term[1]]
else:
translate = complex._translation_from_numeric(
)
simplex = Simplex(
[translate[a] for a in term[1]])
v += term[0] * basis[simplex]
output.append(v)
g = output
elif chain:
g = process_generators_chain(gens, dim, base_ring)
if verbose:
print("raw generators: %s" % gens)
if g:
if not mod_p:
# sort generators to match up with corresponding invariant
g = [
_[1]
for _ in sorted(zip(invts, g), key=lambda x: x[0])
]
d[dim] = (hom, g)
else:
d[dim] = hom
else:
d[dim] = hom
if chain:
new_d = {}
diff = complex.differential()
if len(diff) == 0:
return {}
bottom = min(diff)
top = max(diff)
for dim in d:
if complex._degree_of_differential == -1: # chain complex
new_dim = bottom + dim
else: # cochain complex
new_dim = top - dim
if isinstance(d[dim], tuple):
# generators included.
group = d[dim][0]
gens = d[dim][1]
new_gens = []
dimension = complex.differential(new_dim).ncols()
# make sure that each vector is embedded in the
# correct ambient space: pad with a zero if
# necessary.
for v in gens:
v_dict = v.dict()
if dimension - 1 not in v.dict():
v_dict[dimension - 1] = 0
new_gens.append(vector(base_ring, v_dict))
else:
new_gens.append(v)
new_d[new_dim] = (group, new_gens)
else:
new_d[new_dim] = d[dim]
d = new_d
return d
def homology(self, dim=None, **kwds):
r"""
The reduced homology of this cell complex.
:param dim: If None, then return the homology in every
dimension. If ``dim`` is an integer or list, return the
homology in the given dimensions. (Actually, if ``dim`` is
a list, return the homology in the range from ``min(dim)``
to ``max(dim)``.)
:type dim: integer or list of integers or None; optional,
default None
:param base_ring: commutative ring, must be ZZ or a field.
:type base_ring: optional, default ZZ
:param subcomplex: a subcomplex of this simplicial complex.
Compute homology relative to this subcomplex.
:type subcomplex: optional, default empty
:param generators: If ``True``, return generators for the homology
groups along with the groups. NOTE: Since :trac:`6100`, the result
may not be what you expect when not using CHomP since its return
is in terms of the chain complex.
:type generators: boolean; optional, default False
:param cohomology: If True, compute cohomology rather than homology.
:type cohomology: boolean; optional, default False
:param algorithm: The options are 'auto', 'dhsw', 'pari' or 'no_chomp'.
See below for a description of what they mean.
:type algorithm: string; optional, default 'auto'
:param verbose: If True, print some messages as the homology is
computed.
:type verbose: boolean; optional, default False
.. note::
The keyword arguments to this function get passed on to
:meth:``chain_complex`` and its homology.
ALGORITHM:
If ``algorithm`` is set to 'auto' (the default), then use
CHomP if available. (CHomP is available at the web page
http://chomp.rutgers.edu/. It is also an experimental package
for Sage.)
CHomP computes homology, not cohomology, and only works over
the integers or finite prime fields. Therefore if any of
these conditions fails, or if CHomP is not present, or if
``algorithm`` is set to 'no_chomp', go to plan B: if ``self``
has a ``_homology`` method -- each simplicial complex has
this, for example -- then call that. Such a method implements
specialized algorithms for the particular type of cell
complex.
Otherwise, move on to plan C: compute the chain complex of
``self`` and compute its homology groups. To do this: over a
field, just compute ranks and nullities, thus obtaining
dimensions of the homology groups as vector spaces. Over the
integers, compute Smith normal form of the boundary matrices
defining the chain complex according to the value of
``algorithm``. If ``algorithm`` is 'auto' or 'no_chomp', then
for each relatively small matrix, use the standard Sage
method, which calls the Pari package. For any large matrix,
reduce it using the Dumas, Heckenbach, Saunders, and Welker
elimination algorithm: see
:func:`sage.homology.chain_complex.dhsw_snf` for details.
Finally, ``algorithm`` may also be 'pari' or 'dhsw', which
forces the named algorithm to be used regardless of the size
of the matrices and regardless of whether CHomP is available.
As of this writing, CHomP is by far the fastest option,
followed by the 'auto' or 'no_chomp' setting of using the
Dumas, Heckenbach, Saunders, and Welker elimination algorithm
for large matrices and Pari for small ones.
EXAMPLES::
sage: P = delta_complexes.RealProjectivePlane()
sage: P.homology()
{0: 0, 1: C2, 2: 0}
sage: P.homology(base_ring=GF(2))
{0: Vector space of dimension 0 over Finite Field of size 2,
1: Vector space of dimension 1 over Finite Field of size 2,
2: Vector space of dimension 1 over Finite Field of size 2}
sage: S7 = delta_complexes.Sphere(7)
sage: S7.homology(7)
Z
sage: cubical_complexes.KleinBottle().homology(1, base_ring=GF(2))
Vector space of dimension 2 over Finite Field of size 2
If CHomP is installed, Sage can compute generators of homology
groups::
sage: S2 = simplicial_complexes.Sphere(2)
sage: S2.homology(dim=2, generators=True, base_ring=GF(2)) # optional - CHomP
(Vector space of dimension 1 over Finite Field of size 2, [(0, 1, 2) + (0, 1, 3) + (0, 2, 3) + (1, 2, 3)])
When generators are computed, Sage returns a pair for each
dimension: the group and the list of generators. For
simplicial complexes, each generator is represented as a
linear combination of simplices, as above, and for cubical
complexes, each generator is a linear combination of cubes::
sage: S2_cub = cubical_complexes.Sphere(2)
sage: S2_cub.homology(dim=2, generators=True) # optional - CHomP
(Z, [-[[0,1] x [0,1] x [0,0]] + [[0,1] x [0,1] x [1,1]] - [[0,0] x [0,1] x [0,1]] - [[0,1] x [1,1] x [0,1]] + [[0,1] x [0,0] x [0,1]] + [[1,1] x [0,1] x [0,1]]])
"""
from sage.interfaces.chomp import have_chomp, homcubes, homsimpl
from sage.homology.cubical_complex import CubicalComplex
from sage.homology.simplicial_complex import SimplicialComplex
from sage.modules.all import VectorSpace
from sage.homology.chain_complex import HomologyGroup
base_ring = kwds.get('base_ring', ZZ)
cohomology = kwds.get('cohomology', False)
subcomplex = kwds.get('subcomplex', None)
verbose = kwds.get('verbose', False)
algorithm = kwds.get('algorithm', 'auto')
if dim is not None:
if isinstance(dim, (list, tuple)):
low = min(dim) - 1
high = max(dim) + 2
else:
low = dim - 1
high = dim + 2
dims = range(low, high)
else:
dims = None
# try to use CHomP if computing homology (not cohomology) and
# working over Z or F_p, p a prime.
if (algorithm == 'auto' and cohomology is False
and (base_ring == ZZ or
(base_ring.is_prime_field() and base_ring != QQ))):
# homcubes, homsimpl seems fastest if all of homology is computed.
H = None
if isinstance(self, CubicalComplex):
if have_chomp('homcubes'):
if 'subcomplex' in kwds:
del kwds['subcomplex']
H = homcubes(self, subcomplex, **kwds)
elif isinstance(self, SimplicialComplex):
if have_chomp('homsimpl'):
if 'subcomplex' in kwds:
del kwds['subcomplex']
H = homsimpl(self, subcomplex, **kwds)
# now pick off the requested dimensions
if H:
answer = {}
if not dims:
for d in range(self.dimension() + 1):
if base_ring == ZZ:
answer[d] = H.get(d, HomologyGroup(0))
else:
answer[d] = H.get(d, VectorSpace(base_ring, 0))
else:
for d in dims:
if base_ring == ZZ:
answer[d] = H.get(d, HomologyGroup(0))
else:
answer[d] = H.get(d, VectorSpace(base_ring, 0))
if dim is not None:
if not isinstance(dim, (list, tuple)):
if base_ring == ZZ:
answer = answer.get(dim, HomologyGroup(0))
else:
answer = answer.get(dim, VectorSpace(base_ring, 0))
return answer
# Derived classes can implement specialized algorithms using a
# _homology_ method. See SimplicialComplex for one example.
if hasattr(self, '_homology_'):
return self._homology_(dim, **kwds)
C = self.chain_complex(cochain=cohomology,
augmented=True,
dimensions=dims,
**kwds)
if 'subcomplex' in kwds:
del kwds['subcomplex']
answer = C.homology(**kwds)
if isinstance(answer, dict):
if cohomology:
too_big = self.dimension() + 1
if (not ((isinstance(dim, (list, tuple)) and too_big in dim)
or too_big == dim) and too_big in answer):
del answer[too_big]
if -2 in answer:
del answer[-2]
if -1 in answer:
del answer[-1]
for d in range(self.dimension() + 1):
if d not in answer:
if base_ring == ZZ:
answer[d] = HomologyGroup(0)
else:
answer[d] = VectorSpace(base_ring, 0)
if dim is not None:
if isinstance(dim, (list, tuple)):
temp = {}
for n in dim:
temp[n] = answer[n]
answer = temp
else: # just a single dimension
if base_ring == ZZ:
answer = answer.get(dim, HomologyGroup(0))
else:
answer = answer.get(dim, VectorSpace(base_ring, 0))
return answer
def from_flipper(h):
r"""
Build a (half-)translation surface from a flipper pseudo-Anosov.
EXAMPLES::
sage: from flatsurf import *
sage: import flipper # optional - flipper
A torus example::
sage: t1 = (0r,1r,2r) # optional - flipper
sage: t2 = (~0r,~1r,~2r) # optional - flipper
sage: T = flipper.create_triangulation([t1,t2]) # optional - flipper
sage: L1 = T.lamination([1r,0r,1r]) # optional - flipper
sage: L2 = T.lamination([0r,1r,1r]) # optional - flipper
sage: h1 = L1.encode_twist() # optional - flipper
sage: h2 = L2.encode_twist() # optional - flipper
sage: h = h1*h2^(-1r) # optional - flipper
sage: f = h.flat_structure() # optional - flipper
sage: ts = translation_surfaces.from_flipper(h) # optional - flipper
sage: ts # optional - flipper
HalfTranslationSurface built from 2 polygons
sage: TestSuite(ts).run() # optional - flipper
A non-orientable example::
sage: T = flipper.load('SB_4') # optional - flipper
sage: h = T.mapping_class('s_0S_1s_2S_3s_1S_2') # optional - flipper
sage: h.is_pseudo_anosov() # optional - flipper
True
sage: S = translation_surfaces.from_flipper(h) # optional - flipper
sage: TestSuite(S).run() # optional - flipper
sage: S.num_polygons() # optional - flipper
4
sage: from flatsurf.geometry.similarity_surface_generators import flipper_nf_element_to_sage
sage: a = flipper_nf_element_to_sage(h.dilatation()) # optional - flipper
"""
from .surface import surface_list_from_polygons_and_gluings
f = h.flat_structure()
x = next(itervalues(f.edge_vectors)).x
K = flipper_nf_to_sage(x.field)
V = VectorSpace(K, 2)
edge_vectors = {
i: V((flipper_nf_element_to_sage(e.x, K),
flipper_nf_element_to_sage(e.y, K)))
for i, e in iteritems(f.edge_vectors)
}
to_polygon_number = {
k: (i, j)
for i, t in enumerate(f.triangulation) for j, k in enumerate(t)
}
C = ConvexPolygons(K)
polys = []
adjacencies = {}
for i, t in enumerate(f.triangulation):
for j, k in enumerate(t):
adjacencies[(i, j)] = to_polygon_number[~k]
try:
poly = C([edge_vectors[i] for i in tuple(t)])
except ValueError:
raise ValueError("t = {}, edges = {}".format(
t, [edge_vectors[i].n(digits=6) for i in t]))
polys.append(poly)
return HalfTranslationSurface(
surface_list_from_polygons_and_gluings(polys, adjacencies))
def arnoux_yoccoz(genus):
r"""
Construct the Arnoux-Yoccoz surface of genus 3 or greater.
This presentation of the surface follows Section 2.3 of
Joshua P. Bowman's paper "The Complete Family of Arnoux-Yoccoz
Surfaces."
EXAMPLES::
sage: from flatsurf import *
sage: s = translation_surfaces.arnoux_yoccoz(4)
sage: TestSuite(s).run()
sage: s.is_delaunay_decomposed()
True
sage: s = s.canonicalize()
sage: field=s.base_ring()
sage: a = field.gen()
sage: from sage.matrix.constructor import Matrix
sage: m = Matrix([[a,0],[0,~a]])
sage: ss = m*s
sage: ss = ss.canonicalize()
sage: s.cmp(ss) == 0
True
The Arnoux-Yoccoz pseudo-Anosov are known to have (minimal) invariant
foliations with SAF=0::
sage: S3 = translation_surfaces.arnoux_yoccoz(3)
sage: Jxx, Jyy, Jxy = S3.j_invariant()
sage: Jxx.is_zero() and Jyy.is_zero()
True
sage: Jxy
[ 0 2 0]
[ 2 -2 0]
[ 0 0 2]
sage: S4 = translation_surfaces.arnoux_yoccoz(4)
sage: Jxx, Jyy, Jxy = S4.j_invariant()
sage: Jxx.is_zero() and Jyy.is_zero()
True
sage: Jxy
[ 0 2 0 0]
[ 2 -2 0 0]
[ 0 0 2 2]
[ 0 0 2 0]
"""
g = ZZ(genus)
assert g >= 3
x = polygen(AA)
p = sum([x**i for i in range(1, g + 1)]) - 1
cp = AA.common_polynomial(p)
alpha_AA = AA.polynomial_root(cp, RIF(1 / 2, 1))
field = NumberField(alpha_AA.minpoly(), 'alpha', embedding=alpha_AA)
a = field.gen()
V = VectorSpace(field, 2)
p = [None for i in range(g + 1)]
q = [None for i in range(g + 1)]
p[0] = V(((1 - a**g) / 2, a**2 / (1 - a)))
q[0] = V((-a**g / 2, a))
p[1] = V((-(a**(g - 1) + a**g) / 2, (a - a**2 + a**3) / (1 - a)))
p[g] = V((1 + (a - a**g) / 2, (3 * a - 1 - a**2) / (1 - a)))
for i in range(2, g):
p[i] = V(((a - a**i) / (1 - a), a / (1 - a)))
for i in range(1, g + 1):
q[i] = V(((2 * a - a**i - a**(i + 1)) / (2 * (1 - a)),
(a - a**(g - i + 2)) / (1 - a)))
P = ConvexPolygons(field)
s = Surface_list(field)
T = [None] * (2 * g + 1)
Tp = [None] * (2 * g + 1)
from sage.matrix.constructor import Matrix
m = Matrix([[1, 0], [0, -1]])
for i in range(1, g + 1):
# T_i is (P_0,Q_i,Q_{i-1})
T[i] = s.add_polygon(
P(edges=[q[i] - p[0], q[i - 1] - q[i], p[0] - q[i - 1]]))
# T_{g+i} is (P_i,Q_{i-1},Q_{i})
T[g + i] = s.add_polygon(
P(edges=[q[i - 1] - p[i], q[i] - q[i - 1], p[i] - q[i]]))
# T'_i is (P'_0,Q'_{i-1},Q'_i)
Tp[i] = s.add_polygon(m * s.polygon(T[i]))
# T'_{g+i} is (P'_i,Q'_i, Q'_{i-1})
Tp[g + i] = s.add_polygon(m * s.polygon(T[g + i]))
for i in range(1, g):
s.change_edge_gluing(T[i], 0, T[i + 1], 2)
s.change_edge_gluing(Tp[i], 2, Tp[i + 1], 0)
for i in range(1, g + 1):
s.change_edge_gluing(T[i], 1, T[g + i], 1)
s.change_edge_gluing(Tp[i], 1, Tp[g + i], 1)
#P 0 Q 0 is paired with P' 0 Q' 0, ...
s.change_edge_gluing(T[1], 2, Tp[g], 2)
s.change_edge_gluing(Tp[1], 0, T[g], 0)
# P1Q1 is paired with P'_g Q_{g-1}
s.change_edge_gluing(T[g + 1], 2, Tp[2 * g], 2)
s.change_edge_gluing(Tp[g + 1], 0, T[2 * g], 0)
# P1Q0 is paired with P_{g-1} Q_{g-1}
s.change_edge_gluing(T[g + 1], 0, T[2 * g - 1], 2)
s.change_edge_gluing(Tp[g + 1], 2, Tp[2 * g - 1], 0)
# PgQg is paired with Q1P2
s.change_edge_gluing(T[2 * g], 2, T[g + 2], 0)
s.change_edge_gluing(Tp[2 * g], 0, Tp[g + 2], 2)
for i in range(2, g - 1):
# PiQi is paired with Q'_i P'_{i+1}
s.change_edge_gluing(T[g + i], 2, Tp[g + i + 1], 2)
s.change_edge_gluing(Tp[g + i], 0, T[g + i + 1], 0)
s.set_immutable()
return TranslationSurface(s)
