r"""
Conversion of sage-flatsurf objects to libflatsurf/pyflatsurf and vice versa.
Ideally, there should be no need to call the functionality in this module
directly. Interaction with libflatsurf/pyflatsurf should be handled
transparently by the sage-flatsurf objects. Even for authors of sage-flatsurf
there should essentially never be a need to use this module directly since
objects should provide a ``_pyflatsurf()`` method that returns a
:class:`Conversion` to libflatsurf/pyflatsurf.
EXAMPLES::
sage: from flatsurf import translation_surfaces
sage: from flatsurf.geometry.pyflatsurf_conversion import FlatTriangulationConversion # random output due to deprecation warnings in cppyy
sage: S = translation_surfaces.veech_double_n_gon(5).triangulate()
sage: conversion = FlatTriangulationConversion.to_pyflatsurf(S) # optional: pyflatsurf
sage: conversion # optional: pyflatsurf
Conversion from Triangulation of Translation Surface in H_2(2) built from 2 regular pentagons to FlatTriangulationCombinatorial(...) with vectors ...
"""
# ********************************************************************
# This file is part of sage-flatsurf.
#
# Copyright (C) 2019 Vincent Delecroix
# 2019-2024 Julian Rüth
#
# sage-flatsurf is free software: you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation, either version 2 of the License, or
# (at your option) any later version.
#
# sage-flatsurf is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU General Public License for more details.
#
# You should have received a copy of the GNU General Public License
# along with sage-flatsurf. If not, see <https://www.gnu.org/licenses/>.
# ********************************************************************
from sage.misc.cachefunc import cached_method
from flatsurf.features import pyflatsurf_feature, pyeantic_feature
if pyeantic_feature.is_present():
pyeantic_feature.fix_unwrap_intrusive_ptr()
[docs]class Conversion:
r"""
Generic base class for a conversion from sage-flatsurf to pyflatsurf.
INPUT:
- ``domain`` -- the domain of this conversion, can be ``None``, when the
domain cannot be represented that easily with a sage-flatsurf object.
- ``codomain`` -- the codomain of this conversion, can be ``None``, when
the codomain cannot be represented that easily with
libflatsurf/pyflatsurf object.
EXAMPLES::
sage: from flatsurf import translation_surfaces
sage: from flatsurf.geometry.pyflatsurf_conversion import FlatTriangulationConversion
sage: S = translation_surfaces.veech_double_n_gon(5).triangulate()
sage: conversion = FlatTriangulationConversion.to_pyflatsurf(S) # optional: pyflatsurf
TESTS::
sage: from flatsurf.geometry.pyflatsurf_conversion import Conversion
sage: isinstance(conversion, Conversion) # optional: pyflatsurf
True
"""
[docs] def __init__(self, domain, codomain):
self._domain = domain
self._codomain = codomain
[docs] @classmethod
def to_pyflatsurf(cls, domain, codomain=None):
r"""
Return a :class:`Conversion` from ``domain`` to the ``codomain``.
EXAMPLES::
sage: from flatsurf import translation_surfaces
sage: from flatsurf.geometry.pyflatsurf_conversion import FlatTriangulationConversion
sage: S = translation_surfaces.veech_double_n_gon(5).triangulate()
sage: conversion = FlatTriangulationConversion.to_pyflatsurf(S) # optional: pyflatsurf
"""
raise NotImplementedError(
"this converter does not implement conversion to pyflatsurf yet"
)
[docs] @classmethod
def from_pyflatsurf(cls, codomain, domain=None):
r"""
Return a :class:`Conversion` from ``domain`` to ``codomain``.
EXAMPLES::
sage: from flatsurf import translation_surfaces
sage: from flatsurf.geometry.pyflatsurf_conversion import FlatTriangulationConversion
sage: S = translation_surfaces.veech_double_n_gon(5).triangulate()
sage: conversion = FlatTriangulationConversion.to_pyflatsurf(S) # optional: pyflatsurf
sage: FlatTriangulationConversion.from_pyflatsurf(conversion.codomain()) # optional: pyflatsurf
Conversion from ...
"""
raise NotImplementedError(
"this converter does not implement conversion from pyflatsurf yet"
)
[docs] @classmethod
def to_pyflatsurf_from_elements(cls, elements, codomain=None):
r"""
Return a :class:`Conversion` that converts the sage-flatsurf
``elements`` to ``codomain``.
EXAMPLES::
sage: from flatsurf import translation_surfaces
sage: from flatsurf.geometry.pyflatsurf_conversion import RingConversion
sage: conversion = RingConversion.to_pyflatsurf_from_elements([1, 2, 3]) # optional: gmpxxyy # random output due to cppyy deprecation warnings
sage: conversion # optional: gmpxxyy
Conversion from Integer Ring to __gmp_expr<__mpz_struct[1],__mpz_struct[1]>
"""
from sage.all import Sequence
return cls.to_pyflatsurf(
domain=Sequence(elements).universe(), codomain=codomain
)
[docs] def domain(self):
r"""
Return the domain of this conversion, a sage-flatsurf object.
EXAMPLES::
sage: from flatsurf import translation_surfaces
sage: from flatsurf.geometry.pyflatsurf_conversion import FlatTriangulationConversion
sage: S = translation_surfaces.veech_double_n_gon(5).triangulate()
sage: conversion = FlatTriangulationConversion.to_pyflatsurf(S) # optional: pyflatsurf
sage: conversion.domain() # optional: pyflatsurf
Triangulation of Translation Surface in H_2(2) built from 2 regular pentagons
"""
if self._domain is not None:
return self._domain
raise NotImplementedError(
f"{type(self).__name__} does not implement domain() yet"
)
[docs] def codomain(self):
r"""
Return the codomain of this conversion, a pyflatsurf object.
EXAMPLES::
sage: from flatsurf import translation_surfaces
sage: from flatsurf.geometry.pyflatsurf_conversion import FlatTriangulationConversion
sage: S = translation_surfaces.veech_double_n_gon(5).triangulate()
sage: conversion = FlatTriangulationConversion.to_pyflatsurf(S) # optional: pyflatsurf
sage: conversion.codomain() # optional: pyflatsurf
FlatTriangulationCombinatorial(vertices = ...) with vectors {...}
"""
if self._codomain is not None:
return self._codomain
raise NotImplementedError(
f"{type(self).__name__} does not implement codomain() yet"
)
[docs] def __call__(self, x):
r"""
Return the conversion at an element of :meth:`domain` and return the
corresponding pyflatsurf object.
EXAMPLES::
sage: from flatsurf import translation_surfaces
sage: from flatsurf.geometry.pyflatsurf_conversion import FlatTriangulationConversion
sage: from flatsurf.geometry.surface_objects import SurfacePoint
sage: S = translation_surfaces.veech_double_n_gon(5).triangulate()
sage: conversion = FlatTriangulationConversion.to_pyflatsurf(S) # optional: pyflatsurf
sage: p = SurfacePoint(S, (0, 2), (0, 1/2))
sage: conversion(p) # optional: pyflatsurf
((-1/4*a^2 + 1/2*a + 1/2 ~ 0.54654802), (1/4*a^2 - 3/4 ~ 0.15450850), (1/4 ~ 0.25000000)) in (-6, 8, 9)
"""
raise NotImplementedError(
f"{type(self).__name__} does not implement a mapping of elements yet"
)
[docs] def section(self, y):
r"""
Return the conversion of an element of :meth:`codomain` and return the
corresponding sage-flatsurf object.
This is the inverse of :meth:`__call__`.
EXAMPLES::
sage: from flatsurf import translation_surfaces
sage: from flatsurf.geometry.pyflatsurf_conversion import FlatTriangulationConversion
sage: from flatsurf.geometry.surface_objects import SurfacePoint
sage: S = translation_surfaces.veech_double_n_gon(5).triangulate()
sage: conversion = FlatTriangulationConversion.to_pyflatsurf(S) # optional: pyflatsurf
sage: p = SurfacePoint(S, (0, 2), (0, 1/2))
sage: q = conversion(p) # optional: pyflatsurf
sage: conversion.section(q) # optional: pyflatsurf
Point (0, 1/2) of polygon (0, 2)
"""
raise NotImplementedError(
f"{type(self).__name__} does not implement a section yet"
)
[docs] def __repr__(self):
r"""
Return a printable representation of this conversion.
EXAMPLES::
sage: from flatsurf import translation_surfaces
sage: from flatsurf.geometry.pyflatsurf_conversion import FlatTriangulationConversion
sage: from flatsurf.geometry.surface_objects import SurfacePoint
sage: S = translation_surfaces.veech_double_n_gon(5).triangulate()
sage: FlatTriangulationConversion.to_pyflatsurf(S) # optional: pyflatsurf
Conversion from Triangulation of Translation Surface in H_2(2) built from 2 regular pentagons to FlatTriangulationCombinatorial(...) with vectors ...
"""
codomain = self.codomain()
if hasattr(codomain, "__cpp_name__"):
codomain = codomain.__cpp_name__
return f"Conversion from {self.domain()} to {codomain}"
[docs]class RingConversion(Conversion):
r"""
A conversion between a SageMath ring and a C/C++ ring.
EXAMPLES::
sage: from flatsurf import translation_surfaces
sage: from flatsurf.geometry.pyflatsurf_conversion import RingConversion
sage: conversion = RingConversion.to_pyflatsurf_from_elements([1, 2, 3]) # optional: gmpxxyy
TESTS::
sage: from flatsurf.geometry.pyflatsurf_conversion import RingConversion
sage: isinstance(conversion, RingConversion) # optional: gmpxxyy
True
"""
@staticmethod
def _ring_conversions():
r"""
Return the available ring conversion types.
EXAMPLES::
sage: from flatsurf.geometry.pyflatsurf_conversion import RingConversion
sage: conversions = RingConversion._ring_conversions()
sage: list(conversions) # random output, depends on the installed packages
[<class 'flatsurf.geometry.pyflatsurf_conversion.RingConversion_eantic'>]
"""
from flatsurf.features import (
pyeantic_feature,
pyexactreal_feature,
gmpxxyy_feature,
)
if gmpxxyy_feature.is_present():
yield RingConversion_gmp
yield RingConversion_int
if pyeantic_feature.is_present():
yield RingConversion_eantic
yield RingConversion_algebraic
if pyexactreal_feature.is_present():
yield RingConversion_exactreal
@classmethod
def _create_conversion(cls, domain=None, codomain=None):
r"""
Return a conversion from ``domain`` to ``codomain``.
Return ``None`` if this conversion cannot handle this
``domain``/``codomain`` combination.
At least one of ``domain`` and ``codomain`` must not be ``None``.
INPUT:
- ``domain`` -- a SageMath ring (default: ``None``)
- ``codomain`` -- a ring that pyflatsurf understands (default: ``None``)
EXAMPLES::
sage: from flatsurf.geometry.pyflatsurf_conversion import RingConversion_eantic
sage: RingConversion_eantic._create_conversion(QuadraticField(2)) # optional: pyeantic
Conversion from Number Field in a with defining polynomial x^2 - 2 with a = 1.414213562373095? to NumberField(a^2 - 2, [...])
"""
raise NotImplementedError(
f"{cls.__name__} does not implement _create_conversion() yet"
)
@classmethod
def _deduce_codomain_from_codomain_elements(cls, elements):
r"""
Given elements from pyflatsurf, deduce which pyflatsurf ring they live
in.
Return ``None`` if no (single) conversion can handle these elements.
INPUT:
- ``elements`` -- any sequence of objects that pyflatsurf understands
EXAMPLES::
sage: from flatsurf.geometry.pyflatsurf_conversion import RingConversion, RingConversion_eantic
sage: conversion = RingConversion.to_pyflatsurf(QuadraticField(2)) # optional: pyeantic
sage: element = conversion.codomain().gen() # optional: pyeantic
sage: RingConversion_eantic._deduce_codomain_from_codomain_elements([element]) # optional: pyeantic
NumberField(a^2 - 2, [...])
"""
raise NotImplementedError(
f"{cls.__name__} does not implement _deduce_codomain_from_codomain_elements() yet"
)
@classmethod
def _deduce_codomain_from_domain_elements(cls, elements):
r"""
Given elements from sage-flatsurf, deduce which pyflatsurf ring they
live in.
Return ``None`` if no (single) conversion can handle these elements.
INPUT:
- ``elements`` -- any sequence of objects that sage-flatsurf understands
EXAMPLES::
sage: from flatsurf.geometry.pyflatsurf_conversion import RingConversion, RingConversion_eantic
sage: K.<a> = QuadraticField(2)
sage: RingConversion_eantic._deduce_codomain_from_domain_elements([a]) # optional: pyeantic
NumberField(a^2 - 2, [...])
"""
from sage.all import Sequence
conversion = cls._create_conversion(domain=Sequence(elements).universe())
if conversion is None:
return None
return conversion.codomain()
[docs] @classmethod
def to_pyflatsurf(cls, domain, codomain=None):
r"""
Return a :class:`RingConversion` that converts the SageMath ring ``domain``
to something that libflatsurf/pyflatsurf can understand.
INPUT:
- ``domain`` -- a ring
- ``codomain`` -- a C/C++ type or ``None`` (default: ``None``); if
``None``, the corresponding type is constructed.
EXAMPLES::
sage: from flatsurf import translation_surfaces
sage: from flatsurf.geometry.pyflatsurf_conversion import RingConversion
sage: RingConversion.to_pyflatsurf(QQ) # optional: gmpxxyy
Conversion from Rational Field to __gmp_expr<__mpq_struct[1],__mpq_struct[1]>
"""
for conversion_type in RingConversion._ring_conversions():
conversion = conversion_type._create_conversion(
domain=domain, codomain=codomain
)
if conversion is not None:
return conversion
raise NotImplementedError(
f"cannot determine pyflatsurf ring corresponding to {domain} yet"
)
[docs] @classmethod
def from_pyflatsurf_from_flat_triangulation(cls, flat_triangulation, domain=None):
r"""
Return a :class:`RingConversion` that can map ``domain`` to the ring
over which ``flat_triangulation`` is defined.
INPUT:
- ``flat_triangulation`` -- a libflatsurf ``FlatTriangulation``
- ``domain`` -- a SageMath ring, or ``None`` (default: ``None``); if
``None``, the ring is determined automatically.
EXAMPLES::
sage: from flatsurf import translation_surfaces
sage: from flatsurf.geometry.pyflatsurf_conversion import FlatTriangulationConversion, RingConversion
sage: S = translation_surfaces.veech_double_n_gon(5).triangulate()
sage: flat_triangulation = FlatTriangulationConversion.to_pyflatsurf(S).codomain() # optional: pyflatsurf
sage: conversion = RingConversion.from_pyflatsurf_from_flat_triangulation(flat_triangulation) # optional: pyflatsurf
Note that this conversion does not roundtrip back to the same SageMath
ring. An e-antic ring only has a single variable name but a SageMath
number field has a variable name and a (potentially different) variable
name in the defining polynomial::
sage: conversion.domain() is S.base_ring() # optional: pyflatsurf
False
sage: conversion.domain() # optional: pyflatsurf
Number Field in a with defining polynomial x^4 - 5*x^2 + 5 with a = 1.902113032590308?
sage: S.base_ring()
Number Field in a with defining polynomial y^4 - 5*y^2 + 5 with a = 1.902113032590308?
We can explicitly specify the domain to create the same conversion again::
sage: conversion = RingConversion.from_pyflatsurf_from_flat_triangulation(flat_triangulation, domain=S.base_ring()) # optional: pyflatsurf
sage: conversion.domain() is S.base_ring() # optional: pyflatsurf
True
"""
return cls.from_pyflatsurf_from_elements(
[
flat_triangulation.fromHalfEdge(he).x()
for he in flat_triangulation.halfEdges()
]
+ [
flat_triangulation.fromHalfEdge(he).y()
for he in flat_triangulation.halfEdges()
],
domain=domain,
)
[docs] @classmethod
def from_pyflatsurf_from_elements(cls, elements, domain=None):
r"""
Return a :class:`RingConversion` that can map ``domain`` to the ring of
``elements``.
INPUT:
- ``elements`` -- a sequence of ring elements that libflatsurf/pyflatsurf understands
- ``domain`` -- a SageMath ring, or ``None`` (default: ``None``); if
``None``, the ring is determined automatically.
EXAMPLES::
sage: from flatsurf.geometry.pyflatsurf_conversion import RingConversion
sage: import gmpxxyy # optional: gmpxxyy
sage: conversion = RingConversion.from_pyflatsurf_from_elements([gmpxxyy.mpz()]) # optional: gmpxxyy
"""
for conversion_type in RingConversion._ring_conversions():
codomain = conversion_type._deduce_codomain_from_codomain_elements(elements)
if codomain is None:
continue
conversion = conversion_type._create_conversion(
domain=domain, codomain=codomain
)
if conversion is not None:
return conversion
raise NotImplementedError(
f"cannot determine a SageMath ring for {elements} yet"
)
[docs] @classmethod
def to_pyflatsurf_from_elements(cls, elements, codomain=None):
r"""
Return a :class:`RingConversion` than can map from the ``elements`` to
the ``codomain``.
INPUT:
- ``elements`` -- a sequence of SageMath ring elements
- ``codomain`` -- a libflatsurf parent ring or ``None`` (default:
``None``); if ``None``, the ring is determined automatically.
EXAMPLES::
sage: from flatsurf.geometry.pyflatsurf_conversion import RingConversion
sage: import cppyy # optional: gmpxxyy
sage: conversion = RingConversion.to_pyflatsurf_from_elements([1]) # optional: gmpxxyy
"""
from sage.all import Sequence
for conversion_type in RingConversion._ring_conversions():
deduced_codomain = codomain
if deduced_codomain is None:
deduced_codomain = (
conversion_type._deduce_codomain_from_domain_elements(elements)
)
if deduced_codomain is None:
continue
conversion = conversion_type._create_conversion(
domain=Sequence(elements).universe(), codomain=deduced_codomain
)
if conversion is not None:
return conversion
raise NotImplementedError(
f"cannot determine a pyflatsurf ring for {elements} yet"
)
[docs] @classmethod
def from_pyflatsurf(cls, codomain, domain=None):
r"""
Return a :class:`RingConversion` that maps ``domain`` to ``codomain``.
INPUT:
- ``codomain`` -- a libflatsurf/pyflatsurf type or ring
- ``domain`` -- a SageMath ring, or ``None`` (default: ``None``); if
``None``, the ring is determined automatically.
EXAMPLES::
sage: from flatsurf.geometry.pyflatsurf_conversion import RingConversion
sage: conversion = RingConversion.to_pyflatsurf(QQ) # optional: gmpxxyy
sage: RingConversion.from_pyflatsurf(conversion.codomain()) # optional: gmpxxyy
Conversion from Rational Field to __gmp_expr<__mpq_struct[1],__mpq_struct[1]>
"""
for conversion_type in RingConversion._ring_conversions():
conversion = conversion_type._create_conversion(
domain=domain, codomain=codomain
)
if conversion is not None:
return conversion
raise NotImplementedError(
f"cannot determine pyflatsurf ring corresponding to {codomain} yet"
)
def _vectors(self):
r"""
Return the pyflatsurf ``Vectors`` parent over the codomain of this
conversion.
EXAMPLES::
sage: from flatsurf.geometry.pyflatsurf_conversion import RingConversion
sage: conversion = RingConversion.to_pyflatsurf(QQ) # optional: pyflatsurf
sage: conversion._vectors() # optional: pyflatsurf
Flatsurf Vectors over Rational Field
"""
from pyflatsurf.vector import Vectors
return Vectors(self.codomain())
[docs]class RingConversion_eantic(RingConversion):
r"""
A conversion from a SageMath number field to an e-antic real embedded number field.
EXAMPLES::
sage: from flatsurf.geometry.pyflatsurf_conversion import RingConversion
sage: conversion = RingConversion.to_pyflatsurf(domain=QuadraticField(2)) # optional: pyeantic
sage: from flatsurf.geometry.pyflatsurf_conversion import RingConversion_eantic
sage: isinstance(conversion, RingConversion_eantic) # optional: pyeantic
True
"""
@classmethod
def _create_conversion(cls, domain=None, codomain=None):
r"""
Implements :meth:`RingConversion._create_conversion`.
EXAMPLES::
sage: from flatsurf.geometry.pyflatsurf_conversion import RingConversion_eantic
sage: RingConversion_eantic._create_conversion(domain=QuadraticField(3)) # optional: pyeantic
Conversion from Number Field in a with defining polynomial x^2 - 3 with a = 1.732050807568878? to NumberField(a^2 - 3, [...])
"""
if domain is None and codomain is None:
raise ValueError("at least one of domain and codomain must be set")
if domain is None:
from pyeantic import RealEmbeddedNumberField
renf = RealEmbeddedNumberField(codomain)
domain = renf.number_field
from sage.all import NumberFields, QQ, RR
if domain not in NumberFields():
return None
if domain is QQ:
# GMP should handle the rationals
return None
if not domain.embeddings(RR):
raise NotImplementedError(
"cannot determine pyflatsurf ring for not real-embedded number fields yet"
)
if not domain.is_absolute():
raise NotImplementedError(
"cannot determine pyflatsurf ring for a relative number field since there are no relative fields in e-antic yet"
)
if codomain is None:
from pyeantic import RealEmbeddedNumberField
renf = RealEmbeddedNumberField(domain)
import pyeantic.cppyy_eantic
codomain = pyeantic.cppyy_eantic.unwrap_intrusive_ptr(renf.renf)
return RingConversion_eantic(domain, codomain)
[docs] def __call__(self, x):
r"""
Return the image of ``x`` under this conversion.
EXAMPLES::
sage: from flatsurf.geometry.pyflatsurf_conversion import RingConversion
sage: domain = QuadraticField(2)
sage: conversion = RingConversion.to_pyflatsurf(domain) # optional: pyeantic
sage: conversion(domain.gen()) # optional: pyeantic
(a ~ 1.4142136)
"""
import sage.structure.element
# pylint: disable=c-extension-no-member
parent = sage.structure.element.parent(x)
# pylint: enable=c-extension-no-member
if parent is not self.domain():
raise ValueError(
f"argument must be in the domain of this conversion but {x} is in {parent} and not in {self.domain()}"
)
return self._pyrenf()(list(x)).renf_elem
@cached_method
def _pyrenf(self):
r"""
Return the pyeantic ``RealEmbeddedNumberField`` that wraps the codomain
of this conversion.
EXAMPLES::
sage: from flatsurf import translation_surfaces
sage: from flatsurf.geometry.pyflatsurf_conversion import RingConversion
sage: domain = QuadraticField(2)
sage: conversion = RingConversion.to_pyflatsurf(domain) # optional: pyeantic
sage: conversion._pyrenf() # optional: pyeantic
Real Embedded Number Field in a with defining polynomial x^2 - 2 with a = 1.414213562373095?
"""
from pyeantic import RealEmbeddedNumberField
return RealEmbeddedNumberField(self.codomain())
[docs] def section(self, y):
r"""
Return the preimage of ``y`` under this conversion.
EXAMPLES::
sage: from flatsurf import translation_surfaces
sage: from flatsurf.geometry.pyflatsurf_conversion import RingConversion
sage: domain = QuadraticField(2)
sage: conversion = RingConversion.to_pyflatsurf(domain) # optional: pyeantic
sage: gen = conversion(domain.gen()) # optional: pyeantic
sage: conversion.section(gen) # optional: pyeantic
a
"""
return self.domain()(self._pyrenf()(y))
@classmethod
def _deduce_codomain_from_codomain_elements(cls, elements):
r"""
Given elements from e-antic, deduce the e-antic ring they live in.
This implements :meth:`RingConversion._deduce_codomain_from_codomain_elements`.
Return ``None`` if this is not an e-antic element or if no single
e-antic ring can handle them.
INPUT:
- ``elements`` -- a sequence of e-antic elements or something else
EXAMPLES::
sage: from flatsurf.geometry.pyflatsurf_conversion import RingConversion, RingConversion_eantic
sage: conversion = RingConversion.to_pyflatsurf(QuadraticField(2)) # optional: pyeantic
sage: element = conversion.codomain().gen() # optional: pyeantic
sage: RingConversion_eantic._deduce_codomain_from_codomain_elements([element]) # optional: pyeantic
NumberField(a^2 - 2, [...])
sage: RingConversion_eantic._deduce_codomain_from_codomain_elements([element, 2*element]) # optional: pyeantic
NumberField(a^2 - 2, [...])
sage: import cppyy # optional: pyeantic
sage: RingConversion_eantic._deduce_codomain_from_codomain_elements([element, cppyy.gbl.eantic.renf_elem_class()]) # optional: pyeantic
NumberField(a^2 - 2, [...])
"""
import pyeantic
ring = None
for element in elements:
if not isinstance(element, pyeantic.eantic.renf_elem_class):
return None
import pyeantic.cppyy_eantic
element_ring = pyeantic.cppyy_eantic.unwrap_intrusive_ptr(element.parent())
if ring is None or ring.degree() == 1:
ring = element_ring
elif element_ring != ring and element_ring.degree() != 1:
return None
return ring
[docs]class RingConversion_algebraic(RingConversion):
r"""
A conversion from the algebraic numbers in SageMath ``AA`` to an e-antic
real embedded number field.
EXAMPLES:
There is no general notion of algebraic numbers in libflatsurf yet, so we
can deduce such a conversion for a finite number of elements that span a
finite number field::
sage: from flatsurf.geometry.pyflatsurf_conversion import RingConversion
sage: conversion = RingConversion.to_pyflatsurf(domain=AA) # optional: pyeantic
Traceback (most recent call last):
...
NotImplementedError: ...
sage: conversion = RingConversion.to_pyflatsurf_from_elements(elements=[AA(1), AA(sqrt(2))]) # optional: pyeantic
TESTS::
sage: from flatsurf.geometry.pyflatsurf_conversion import RingConversion_algebraic
sage: isinstance(conversion, RingConversion_algebraic) # optional: pyeantic
True
"""
[docs] def __init__(self, domain, codomain):
super().__init__(domain=domain, codomain=codomain)
self._eantic_conversion = RingConversion_eantic._create_conversion(
codomain=self.codomain()
)
@classmethod
def _create_conversion(cls, domain=None, codomain=None):
r"""
Implements :meth:`RingConversion._create_conversion`.
EXAMPLES:
Since there is no generic algebraic numbers in libflatsurf, we need to
know the libflatsurf number field upfront::
sage: from flatsurf.geometry.pyflatsurf_conversion import RingConversion_algebraic
sage: from pyeantic import RealEmbeddedNumberField # optional: pyeantic
sage: RingConversion_algebraic._create_conversion(domain=AA, codomain=RealEmbeddedNumberField(QuadraticField(2)).renf) # optional: pyeantic
Conversion from Algebraic Real Field to NumberField(a^2 - 2, [...])
"""
if codomain is None:
return None
from sage.all import AA
if domain is not AA:
return None
return RingConversion_algebraic(domain, codomain)
[docs] def __call__(self, x):
r"""
Return the image of ``x`` under this conversion.
EXAMPLES::
sage: from flatsurf.geometry.pyflatsurf_conversion import RingConversion
sage: conversion = RingConversion.to_pyflatsurf_from_elements(elements=[AA(sqrt(2))]) # optional: pyeantic
sage: conversion(AA(sqrt(2))) # optional: pyeantic
(a ~ 1.4142136)
"""
return self._eantic_conversion(self._eantic_conversion.domain()(x))
@classmethod
def _deduce_codomain_from_domain_elements(cls, elements):
r"""
Return an e-antic number field that contains the algebraic SageMath
``elements``.
EXAMPLES::
sage: from flatsurf.geometry.pyflatsurf_conversion import RingConversion_algebraic
sage: RingConversion_algebraic._deduce_codomain_from_domain_elements([AA(sqrt(2)), AA(sqrt(3))]) # optional: pyeantic
NumberField(a^4 - 4*a^2 + 1, [...])
"""
from sage.all import AA
if not all(element.parent() is AA for element in elements):
return None
from sage.all import number_field_elements_from_algebraics, NumberField
number_field, elements, embedding = number_field_elements_from_algebraics(
elements, embedded=True
)
return RingConversion_eantic._create_conversion(number_field).codomain()
@classmethod
def _deduce_codomain_from_codomain_elements(cls, elements):
r"""
Return an e-antic number field that contains the e-antic ``elements``.
EXAMPLES::
sage: from flatsurf.geometry.pyflatsurf_conversion import RingConversion_algebraic
sage: from pyeantic import RealEmbeddedNumberField # optional: pyeantic
sage: RingConversion_algebraic._deduce_codomain_from_codomain_elements([RealEmbeddedNumberField(QuadraticField(2)).renf.gen()]) # optional: pyeantic
NumberField(a^2 - 2, [...])
"""
return RingConversion_eantic._deduce_codomain_from_codomain_elements(elements)
[docs]class RingConversion_exactreal(RingConversion):
r"""
A conversion from a pyexactreal SageMath ring to a exact-real ring.
EXAMPLES::
sage: from flatsurf.geometry.pyflatsurf_conversion import RingConversion
sage: from pyexactreal import ExactReals # optional: pyexactreal
sage: conversion = RingConversion.to_pyflatsurf(domain=ExactReals(QQ)) # optional: pyexactreal
Traceback (most recent call last):
...
NotImplementedError: cannot deduce the exact real module that corresponds to this generic ring of exact reals since there is no generic exact-real ring without a fixed set of generators in libexactreal yet
sage: from pyexactreal import QQModule, RealNumber # optional: pyexactreal
sage: M = QQModule(RealNumber.rational(1)) # optional: pyexactreal
sage: conversion = RingConversion.to_pyflatsurf(domain=ExactReals(QQ), codomain=M) # optional: pyexactreal
sage: from flatsurf.geometry.pyflatsurf_conversion import RingConversion_exactreal
sage: isinstance(conversion, RingConversion_exactreal) # optional: pyexactreal
True
"""
@classmethod
def _create_conversion(cls, domain=None, codomain=None):
r"""
Implements :meth:`RingConversion._create_conversion`.
EXAMPLES::
sage: from flatsurf.geometry.pyflatsurf_conversion import RingConversion_exactreal
sage: from pyexactreal import QQModule, RealNumber, ExactReals # optional: pyexactreal
sage: M = QQModule(RealNumber.rational(1)) # optional: pyexactreal
sage: RingConversion_exactreal._create_conversion(domain=ExactReals(QQ), codomain=M) # optional: pyexactreal
Conversion from Real Numbers as (Rational Field)-Module to ℚ-Module(1)
"""
if domain is None and codomain is None:
raise ValueError("at least one of domain and codomain must be set")
if domain is None:
# TODO: Does this work when the codomain.ring() is ZZ or QQ?
domain_base_conversion = RingConversion.from_pyflatsurf(
codomain=codomain.ring().parameters
)
from pyexactreal import ExactReals
domain = ExactReals(domain_base_conversion.domain())
if codomain is None:
from pyeantic.real_embedded_number_field import RealEmbeddedNumberField
# TODO: Add the other base rings.
if isinstance(domain.base_ring(), RealEmbeddedNumberField):
import pyexactreal
codomain = pyexactreal.exactreal.Module[
pyexactreal.exactreal.NumberField
]
else:
raise NotImplementedError(
"cannot deduce the exact real module that corresponds to this generic ring of exact reals since there is no generic exact-real ring without a fixed set of generators in libexactreal yet"
)
return RingConversion_exactreal(domain, codomain)
[docs] def __call__(self, x):
r"""
Return the image of ``x`` under this conversion.
EXAMPLES::
sage: from flatsurf.geometry.pyflatsurf_conversion import RingConversion_exactreal
sage: from pyexactreal import QQModule, RealNumber, ExactReals # optional: pyexactreal
sage: domain = ExactReals(QQ) # optional: pyexactreal
sage: M = QQModule(RealNumber.rational(1)) # optional: pyexactreal
sage: conversion = RingConversion_exactreal._create_conversion(domain=domain, codomain=M) # optional: pyexactreal
sage: conversion(domain(1)) # optional: pyexactreal
1
"""
import sage.structure.element
# pylint: disable=c-extension-no-member
parent = sage.structure.element.parent(x)
# pylint: enable=c-extension-no-member
if parent is not self.domain():
raise ValueError(
f"argument must be in the domain of this conversion but {x} is in {parent} and not in {self.domain()}"
)
return x._backend
@classmethod
def _deduce_codomain_from_domain_elements(cls, elements):
r"""
Return an exact-real module that contains the SageMath pyexactreal ``elements``.
INPUT:
- ``elements`` -- a sequence of pyexactreal elements that SageMath understands.
EXAMPLES::
sage: from flatsurf.geometry.pyflatsurf_conversion import RingConversion_exactreal
sage: from pyexactreal import QQModule, RealNumber, ExactReals # optional: pyexactreal
sage: domain = ExactReals(QQ) # optional: pyexactreal
sage: RingConversion_exactreal._deduce_codomain_from_domain_elements([domain.random_element()]) # optional: pyexactreal
ℚ-Module(ℝ(...))
sage: RingConversion_exactreal._deduce_codomain_from_domain_elements([domain.one(), domain.random_element()]) # optional: pyexactreal
ℚ-Module(1, ℝ(...))
"""
codomain = None
if any(not hasattr(element, "_backend") for element in elements):
return None
return RingConversion_exactreal._deduce_codomain_from_codomain_elements(
[element._backend for element in elements]
)
def _vectors(self):
r"""
Return the pyflatsurf parent of vectors over the codomain of this conversion.
EXAMPLES::
sage: from flatsurf.geometry.pyflatsurf_conversion import RingConversion_exactreal
sage: from pyexactreal import QQModule, RealNumber, ExactReals # optional: pyexactreal
sage: domain = ExactReals(QQ) # optional: pyexactreal
sage: M = QQModule(RealNumber.rational(1)) # optional: pyexactreal
sage: conversion = RingConversion_exactreal._create_conversion(domain=domain, codomain=M) # optional: pyexactreal
sage: conversion._vectors() # optional: pyexactreal
Flatsurf Vectors over Real Numbers as (Rational Field)-Module
"""
from pyflatsurf.vector import Vectors
from pyexactreal.exact_reals import ExactReals
from pyeantic.real_embedded_number_field import RealEmbeddedNumberField
if isinstance(self.domain(), ExactReals):
return Vectors(self.domain())
# TODO: Add the other base rings.
if isinstance(self.domain().base_ring(), RealEmbeddedNumberField):
return Vectors(ExactReals(self.domain().base_ring().number_field))
raise NotImplementedError
@classmethod
def _deduce_codomain_from_codomain_elements(cls, elements):
r"""
Return an exact-real module that contains the exact-real elements.
INPUT:
- ``elements`` -- a sequence of exact-real module elements
EXAMPLES::
sage: from flatsurf.geometry.pyflatsurf_conversion import RingConversion_exactreal
sage: from pyexactreal import QQModule, RealNumber, ExactReals # optional: pyexactreal
sage: domain = ExactReals(QQ) # optional: pyexactreal
sage: RingConversion_exactreal._deduce_codomain_from_codomain_elements([domain.random_element()._backend]) # optional: pyexactreal
ℚ-Module(ℝ(...))
sage: RingConversion_exactreal._deduce_codomain_from_codomain_elements([domain.one()._backend, domain.random_element()._backend]) # optional: pyexactreal
ℚ-Module(1, ℝ(...))
"""
module = None
for element in elements:
if not element.__class__.__name__.startswith("Element<"):
return None
element_module = element.module()
if module is None:
module = element_module
module = module.span(module, element_module)
return module
[docs] def section(self, y):
r"""
Return the preimage of ``y`` under this conversion.
EXAMPLES::
sage: from flatsurf.geometry.pyflatsurf_conversion import RingConversion_exactreal
sage: from pyexactreal import QQModule, RealNumber, ExactReals # optional: pyexactreal
sage: domain = ExactReals(QQ) # optional: pyexactreal
sage: M = QQModule(RealNumber.random()) # optional: pyexactreal
sage: conversion = RingConversion_exactreal._create_conversion(domain=domain, codomain=M) # optional: pyexactreal
sage: conversion.section(M.gen(0R)) # optional: pyexactreal
ℝ(...)
"""
return self.domain()(y)
[docs]class RingConversion_int(RingConversion):
r"""
Conversion between SageMath integers and machine long long integers.
EXAMPLES::
sage: from flatsurf import translation_surfaces
sage: from flatsurf.geometry.pyflatsurf_conversion import RingConversion
sage: conversion = RingConversion.to_pyflatsurf(domain=int) # optional: gmpxxyy
sage: from flatsurf.geometry.pyflatsurf_conversion import RingConversion_int
sage: isinstance(conversion, RingConversion_int) # optional: gmpxxyy
True
"""
@classmethod
def _create_conversion(cls, domain=None, codomain=None):
r"""
Implements :meth:`RingConversion._create_conversion`.
EXAMPLES::
sage: from flatsurf.geometry.pyflatsurf_conversion import RingConversion_int
sage: RingConversion_int._create_conversion(domain=int) # optional: gmpxxyy
Conversion from <class 'int'> to long long
"""
if domain is None and codomain is None:
raise ValueError("at least one of domain and codomain must be set")
import cppyy
longlong = getattr(cppyy.gbl, "long long")
if domain in [None, int] and codomain in [None, longlong]:
return RingConversion_int(int, longlong)
return None
@classmethod
def _deduce_codomain_from_codomain_elements(cls, elements):
r"""
Given long longs, return the long long type.
This implements :meth:`RingConversion._deduce_codomain_from_codomain_elements`.
Return ``None`` if these are not long long element.
INPUT:
- ``element`` -- a long long or something else
EXAMPLES::
sage: from flatsurf.geometry.pyflatsurf_conversion import RingConversion, RingConversion_int
sage: conversion = RingConversion.to_pyflatsurf(int) # optional: gmpxxyy
sage: element = conversion.codomain()(1R) # optional: gmpxxyy
sage: RingConversion_int._deduce_codomain_from_codomain_elements([element]) # optional: gmpxxyy
<class 'cppyy.gbl.long long'>
"""
import cppyy
longlong = getattr(cppyy.gbl, "long long")
for element in elements:
if not isinstance(element, longlong):
return None
return longlong
[docs] def __call__(self, x):
r"""
Return the image of ``x`` under this conversion.
EXAMPLES::
sage: from flatsurf.geometry.pyflatsurf_conversion import RingConversion
sage: domain = int
sage: conversion = RingConversion.to_pyflatsurf(int) # optional: gmpxxyy
sage: conversion(1R) # optional: gmpxxyy
1
"""
if not isinstance(x, int):
raise ValueError("argument must be an int")
return self.codomain()(x)
[docs] def section(self, y):
r"""
Return the preimage of ``y`` under this conversion.
EXAMPLES::
sage: from flatsurf import translation_surfaces
sage: from flatsurf.geometry.pyflatsurf_conversion import RingConversion
sage: domain = int
sage: conversion = RingConversion.to_pyflatsurf(domain) # optional: gmpxxyy
sage: y = conversion(3R) # optional: gmpxxyy
sage: conversion.section(y) # optional: gmpxxyy
3
"""
return self.domain()(str(y))
[docs]class RingConversion_gmp(RingConversion):
r"""
Conversion between SageMath integers and rationals and GMP mpz and mpq.
EXAMPLES::
sage: from flatsurf import translation_surfaces
sage: from flatsurf.geometry.pyflatsurf_conversion import RingConversion
sage: conversion = RingConversion.to_pyflatsurf(domain=ZZ) # optional: gmpxxyy
sage: from flatsurf.geometry.pyflatsurf_conversion import RingConversion_gmp
sage: isinstance(conversion, RingConversion_gmp) # optional: gmpxxyy
True
"""
@classmethod
def _create_conversion(cls, domain=None, codomain=None):
r"""
Implements :meth:`RingConversion._create_conversion`.
EXAMPLES::
sage: from flatsurf.geometry.pyflatsurf_conversion import RingConversion_gmp
sage: RingConversion_gmp._create_conversion(domain=ZZ) # optional: gmpxxyy
Conversion from Integer Ring to __gmp_expr<__mpz_struct[1],__mpz_struct[1]>
"""
if domain is None and codomain is None:
raise ValueError("at least one of domain and codomain must be set")
from sage.all import ZZ, QQ
import cppyy
import gmpxxyy
if domain is None:
if codomain == cppyy.gbl.mpz_class:
domain = ZZ
elif codomain is cppyy.gbl.mpq_class:
domain = QQ
else:
return None
if codomain is None:
if domain is ZZ:
codomain = cppyy.gbl.mpz_class
elif domain is QQ:
codomain = cppyy.gbl.mpq_class
else:
return None
if domain is ZZ and codomain is cppyy.gbl.mpz_class:
return RingConversion_gmp(domain, codomain)
if domain is QQ and codomain is cppyy.gbl.mpq_class:
return RingConversion_gmp(domain, codomain)
return None
@classmethod
def _deduce_codomain_from_codomain_elements(cls, elements):
r"""
Given elements from GMP, return the GMP ring they live in.
This implements :meth:`RingConversion._deduce_codomain_from_codomain_elements`.
Return ``None`` if they're not (compatible) GMP elements.
INPUT:
- ``elements`` -- a sequence of GMP elements or something else
EXAMPLES::
sage: from flatsurf.geometry.pyflatsurf_conversion import RingConversion, RingConversion_gmp
sage: conversion = RingConversion.to_pyflatsurf(ZZ) # optional: gmpxxyy
sage: element = conversion.codomain()(1) # optional: gmpxxyy
sage: RingConversion_gmp._deduce_codomain_from_codomain_elements([element]) # optional: gmpxxyy
<class cppyy.gbl.__gmp_expr<__mpz_struct[1],__mpz_struct[1]> at 0x...>
"""
import cppyy
import gmpxxyy
ring = None
for element in elements:
if isinstance(element, cppyy.gbl.mpz_class):
if ring is None or ring is cppyy.gbl.mpz_class:
ring = cppyy.gbl.mpz_class
else:
return None
if isinstance(element, cppyy.gbl.mpq_class):
if ring is None or ring is cppyy.gbl.mpq_class:
ring = cppyy.gbl.mpq_class
else:
return None
return ring
[docs] def __call__(self, x):
r"""
Return the image of ``x`` under this conversion.
EXAMPLES::
sage: from flatsurf.geometry.pyflatsurf_conversion import RingConversion
sage: domain = QQ
sage: conversion = RingConversion.to_pyflatsurf(QQ) # optional: gmpxxyy
sage: x = 1/3
sage: y = conversion(x) # optional: gmpxxyy
sage: conversion.section(y) # optional: gmpxxyy
1/3
"""
import sage.structure.element
# pylint: disable=c-extension-no-member
parent = sage.structure.element.parent(x)
# pylint: enable=c-extension-no-member
if parent is not self.domain():
raise ValueError(
f"argument must be in the domain of this conversion but {x} is in {parent} and not in {self.domain()}"
)
return self.codomain()(str(x))
[docs] def section(self, y):
r"""
Return the preimage of ``y`` under this conversion.
EXAMPLES::
sage: from flatsurf import translation_surfaces
sage: from flatsurf.geometry.pyflatsurf_conversion import RingConversion
sage: domain = QQ
sage: conversion = RingConversion.to_pyflatsurf(domain) # optional: gmpxxyy
sage: y = conversion(1/3) # optional: gmpxxyy
sage: conversion.section(y) # optional: gmpxxyy
1/3
"""
return self.domain()(str(y))
[docs]class VectorSpaceConversion(Conversion):
r"""
Converts vectors in a SageMath vector space into libflatsurf ``Vector<T>``\s.
EXAMPLES::
sage: from flatsurf.geometry.pyflatsurf_conversion import VectorSpaceConversion
sage: conversion = VectorSpaceConversion.to_pyflatsurf(QQ^2) # optional: pyflatsurf
"""
[docs] def __init__(self, domain, codomain, ring_conversion=None):
if ring_conversion is None:
ring_conversion = RingConversion.to_pyflatsurf(domain.base_ring())
self._ring_conversion = ring_conversion
super().__init__(domain, codomain)
[docs] @classmethod
def to_pyflatsurf(cls, domain, codomain=None, ring_conversion=None):
r"""
Return a :class:`Conversion` from ``domain`` to ``codomain``.
INPUT:
- ``domain`` -- a SageMath free module
- ``codomain`` -- a libflatsurf ``Vector<T>`` type or ``None``
(default: ``None``); if ``None``, the type is determined
automatically.
EXAMPLES::
sage: from flatsurf.geometry.pyflatsurf_conversion import VectorSpaceConversion
sage: conversion = VectorSpaceConversion.to_pyflatsurf(QQ^2) # optional: pyflatsurf
"""
pyflatsurf_feature.require()
if codomain is None:
if ring_conversion is None:
ring_conversion = RingConversion.to_pyflatsurf(domain.base_ring())
from pyflatsurf.vector import Vectors
codomain = Vectors(ring_conversion.domain()).Vector
return VectorSpaceConversion(domain, codomain, ring_conversion=ring_conversion)
[docs] @classmethod
def from_pyflatsurf_from_elements(cls, elements, domain=None, ring_conversion=None):
r"""
Return a :class:`Conversion` that converts the pyflatsurf ``elements``
into vectors in ``domain``.
INPUT:
- ``domain`` -- a SageMath ``VectorSpace`` or ``None``; if ``None``, it
is determined automatically.
EXAMPLES::
sage: from flatsurf.geometry.pyflatsurf_conversion import VectorSpaceConversion
sage: codomain = VectorSpaceConversion.to_pyflatsurf(QQ^2).codomain() # optional: pyflatsurf
sage: VectorSpaceConversion.from_pyflatsurf_from_elements([codomain()]) # optional: pyflatsurf
Conversion from Vector space of dimension 2 over Rational Field to flatsurf::Vector<__gmp_expr<__mpq_struct[1],__mpq_struct[1]>...>
"""
if domain is None:
ring_conversion = RingConversion.from_pyflatsurf_from_elements(
[element.x() for element in elements]
+ [element.y() for element in elements]
)
from sage.all import VectorSpace
domain = VectorSpace(ring_conversion.domain(), 2)
return VectorSpaceConversion.to_pyflatsurf(
domain=domain, ring_conversion=ring_conversion
)
[docs] @classmethod
def to_pyflatsurf_from_elements(cls, elements, codomain=None):
r"""
Return a conversion that can convert the SageMath ``elements`` to ``codomain``.
EXAMPLES::
sage: from flatsurf.geometry.pyflatsurf_conversion import VectorSpaceConversion
sage: VectorSpaceConversion.to_pyflatsurf_from_elements([vector((1, 2))]) # optional: pyflatsurf
Conversion from Ambient free module of rank 2 over the principal ideal domain Integer Ring to ...
sage: VectorSpaceConversion.to_pyflatsurf_from_elements([vector((1, 2)), vector((1/2, 2/3))]) # optional: pyflatsurf
Conversion from Vector space of dimension 2 over Rational Field to ...
"""
ring_conversion = RingConversion.to_pyflatsurf_from_elements(
[v[0] for v in elements] + [v[1] for v in elements]
)
from sage.all import Sequence
return cls.to_pyflatsurf(
domain=Sequence(elements).universe(),
codomain=codomain or ring_conversion._vectors(),
ring_conversion=ring_conversion,
)
@cached_method
def _vectors(self):
r"""
Return the pyflatsurf ``Vectors`` helper for the codomain of this conversion.
EXAMPLES::
sage: from flatsurf.geometry.pyflatsurf_conversion import VectorSpaceConversion
sage: conversion = VectorSpaceConversion.to_pyflatsurf(QQ^2) # optional: pyflatsurf
sage: conversion._vectors() # optional: pyflatsurf
Flatsurf Vectors over Rational Field
"""
return self._ring_conversion._vectors()
[docs] def __call__(self, vector):
r"""
Return a ``Vector<T>`` corresponding to the SageMath ``vector``.
EXAMPLES::
sage: from flatsurf.geometry.pyflatsurf_conversion import VectorSpaceConversion
sage: conversion = VectorSpaceConversion.to_pyflatsurf(QQ^2) # optional: pyflatsurf
sage: conversion(vector(QQ, [1, 2])) # optional: pyflatsurf
(1, 2)
"""
return self._vectors()(
[self._ring_conversion(coordinate) for coordinate in vector]
).vector
[docs] def section(self, vector):
r"""
Return the SageMath vector corresponding to the pyflatsurf ``vector``.
EXAMPLES::
sage: from flatsurf.geometry.pyflatsurf_conversion import VectorSpaceConversion
sage: conversion = VectorSpaceConversion.to_pyflatsurf(QQ^2) # optional: pyflatsurf
sage: v = vector(QQ, [1, 2]) # optional: pyflatsurf
sage: conversion.section(conversion(v)) == v # optional: pyflatsurf
True
"""
return self.domain()(
[
self._ring_conversion.section(vector.x()),
self._ring_conversion.section(vector.y()),
]
)
[docs]class FlatTriangulationConversion(Conversion):
r"""
Converts a sage-flatsurf surface to a ``FlatTriangulation`` object and
vice-versa.
EXAMPLES::
sage: from flatsurf import translation_surfaces
sage: from flatsurf.geometry.pyflatsurf_conversion import FlatTriangulationConversion
sage: S = translation_surfaces.veech_double_n_gon(5).triangulate()
sage: conversion = FlatTriangulationConversion.to_pyflatsurf(S) # optional: pyflatsurf
"""
[docs] def __init__(self, domain, codomain, label_to_half_edge):
r"""
EXAMPLES::
sage: from flatsurf import translation_surfaces
sage: from flatsurf.geometry.pyflatsurf_conversion import FlatTriangulationConversion
sage: S = translation_surfaces.square_torus().triangulate()
sage: conversion = FlatTriangulationConversion.to_pyflatsurf(S) # optional: pyflatsurf
sage: isinstance(conversion, FlatTriangulationConversion) # optional: pyflatsurf
True
"""
super().__init__(domain=domain, codomain=codomain)
self._label_to_half_edge = label_to_half_edge
self._half_edge_to_label = {
half_edge: label for (label, half_edge) in label_to_half_edge.items()
}
[docs] @classmethod
def to_pyflatsurf(cls, domain, codomain=None):
r"""
Return a :class:`Conversion` from ``domain`` to the ``codomain``.
INPUT:
- ``domain`` -- a sage-flatsurf surface
- ``codomain`` -- a ``FlatTriangulation`` or ``None`` (default:
``None``); if ``None``, the corresponding ``FlatTriangulation`` is
constructed.
.. NOTE:
The ``codomain``, if given, must be indistinguishable from the
codomain that this method would construct automatically.
EXAMPLES::
sage: from flatsurf import translation_surfaces
sage: from flatsurf.geometry.pyflatsurf_conversion import FlatTriangulationConversion
sage: S = translation_surfaces.veech_double_n_gon(5).triangulate()
sage: conversion = FlatTriangulationConversion.to_pyflatsurf(S) # optional: pyflatsurf
"""
pyflatsurf_feature.require()
from flatsurf.geometry.categories import TranslationSurfaces
if domain not in TranslationSurfaces():
raise TypeError("domain must be a translation surface")
if not domain.is_finite_type():
raise ValueError("domain must be finite")
if not domain.is_triangulated():
raise ValueError("domain must be triangulated")
if codomain is None:
vertex_permutation = cls._pyflatsurf_vertex_permutation(domain)
vectors = cls._pyflatsurf_vectors(domain)
from pyflatsurf.factory import make_surface
codomain = make_surface(vertex_permutation, vectors)
return FlatTriangulationConversion(
domain, codomain, cls._pyflatsurf_labels(domain)
)
@classmethod
def _pyflatsurf_labels(cls, domain):
r"""
Return a mapping of the edges of the polygons of ``domain`` to half
edges numbered compatibly with libflatsurf/pyflatsurf, i.e., by
consecutive integers such that the opposite of an edge has the negative
of that edge's label.
EXAMPLES::
sage: from flatsurf import translation_surfaces
sage: from flatsurf.geometry.pyflatsurf_conversion import FlatTriangulationConversion
sage: S = translation_surfaces.veech_double_n_gon(5).triangulate()
sage: FlatTriangulationConversion._pyflatsurf_labels(S)
{((0, 0), 0): 1,
((0, 0), 1): 2,
((0, 0), 2): 3,
((0, 1), 0): -3,
((0, 1), 1): 5,
((0, 1), 2): 6,
((0, 2), 0): -6,
((0, 2), 1): 8,
((0, 2), 2): 9,
((1, 0), 0): -1,
((1, 0), 1): -2,
((1, 0), 2): 4,
((1, 1), 0): -4,
((1, 1), 1): -5,
((1, 1), 2): 7,
((1, 2), 0): -7,
((1, 2), 1): -8,
((1, 2), 2): -9}
"""
labels = {}
for half_edge, opposite_half_edge in domain.gluings():
if half_edge in labels:
continue
labels[half_edge] = len(labels) // 2 + 1
labels[opposite_half_edge] = -labels[half_edge]
return labels
@classmethod
def _pyflatsurf_vectors(cls, domain):
r"""
Return the vectors of the positive half edges in ``domain`` in order.
EXAMPLES::
sage: from flatsurf import translation_surfaces
sage: from flatsurf.geometry.pyflatsurf_conversion import FlatTriangulationConversion
sage: S = translation_surfaces.veech_double_n_gon(5).triangulate()
sage: FlatTriangulationConversion._pyflatsurf_vectors(S) # optional: pyflatsurf
[(1, 0),
((1/2*a^2 - 3/2 ~ 0.30901699), (1/2*a ~ 0.95105652)),
((-1/2*a^2 + 1/2 ~ -1.3090170), (-1/2*a ~ -0.95105652)),
((1/2*a^2 - 1/2 ~ 1.3090170), (1/2*a ~ 0.95105652)),
((-1/2*a^2 + 1 ~ -0.80901699), (1/2*a^3 - 3/2*a ~ 0.58778525)),
((-1/2 ~ -0.50000000), (-1/2*a^3 + 1*a ~ -1.5388418)),
((1/2 ~ 0.50000000), (1/2*a^3 - 1*a ~ 1.5388418)),
((-1/2*a^2 + 1 ~ -0.80901699), (-1/2*a^3 + 3/2*a ~ -0.58778525)),
((1/2*a^2 - 3/2 ~ 0.30901699), (-1/2*a ~ -0.95105652))]
"""
labels = cls._pyflatsurf_labels(domain)
vectors = [None] * (len(labels) // 2)
for (polygon, edge), half_edge in labels.items():
if half_edge < 0:
continue
vectors[half_edge - 1] = domain.polygon(polygon).edge(edge)
vector_conversion = VectorSpaceConversion.to_pyflatsurf_from_elements(vectors)
return [vector_conversion(vector) for vector in vectors]
@classmethod
def _pyflatsurf_vertex_permutation(cls, domain):
r"""
Return the permutation of half edges around vertices of ``domain`` in
cycle notation.
The permutation uses integers, as provided by
:meth:`_pyflatsurf_labels`.
INPUT:
- ``domain`` -- a sage-flatsurf surface
EXAMPLES::
sage: from flatsurf import translation_surfaces
sage: from flatsurf.geometry.pyflatsurf_conversion import FlatTriangulationConversion
sage: S = translation_surfaces.veech_double_n_gon(5).triangulate()
sage: FlatTriangulationConversion._pyflatsurf_vertex_permutation(S)
[[1, -3, -6, -9, 8, 6, -5, 4, 2, -1, -4, -7, 9, -8, 7, 5, 3, -2]]
"""
pyflatsurf_labels = cls._pyflatsurf_labels(domain)
vertex_permutation = {}
for polygon, edge in domain.edges():
pyflatsurf_edge = pyflatsurf_labels[(polygon, edge)]
next_edge = (edge + 1) % len(domain.polygon(polygon).vertices())
pyflatsurf_next_edge = pyflatsurf_labels[(polygon, next_edge)]
vertex_permutation[pyflatsurf_next_edge] = -pyflatsurf_edge
return cls._cycle_decomposition(vertex_permutation)
@classmethod
def _cycle_decomposition(self, permutation):
r"""
Return a permutation in cycle notation.
EXAMPLES::
sage: from flatsurf.geometry.pyflatsurf_conversion import FlatTriangulationConversion
sage: FlatTriangulationConversion._cycle_decomposition({})
[]
sage: FlatTriangulationConversion._cycle_decomposition({1: 1, -1: -1})
[[1], [-1]]
sage: FlatTriangulationConversion._cycle_decomposition({1: 2, 2: 1, 3: 4, 4: 5, 5: 3})
[[1, 2], [3, 4, 5]]
"""
cycles = []
elements = set(permutation.keys())
while elements:
cycle = [elements.pop()]
while True:
cycle.append(permutation[cycle[-1]])
if cycle[-1] == cycle[0]:
cycle.pop()
cycles.append(cycle)
break
elements.remove(cycle[-1])
return cycles
[docs] @classmethod
def from_pyflatsurf(cls, codomain, domain=None):
r"""
Return a :class:`Conversion` from ``domain`` to ``codomain``.
INPUT:
- ``codomain`` -- a ``FlatTriangulation``
- ``domain`` -- a sage-flatsurf surface or ``None`` (default: ``None``); if
``None``, the corresponding surface is constructed.
.. NOTE:
The ``domain``, if given, must be indistinguishable from the domain
that this method would construct automatically.
EXAMPLES::
sage: from flatsurf import translation_surfaces
sage: from flatsurf.geometry.pyflatsurf_conversion import FlatTriangulationConversion
sage: S = translation_surfaces.veech_double_n_gon(5).triangulate()
sage: conversion = FlatTriangulationConversion.to_pyflatsurf(S) # optional: pyflatsurf
sage: FlatTriangulationConversion.from_pyflatsurf(conversion.codomain()) # optional: pyflatsurf
Conversion from ...
"""
pyflatsurf_feature.require()
from bidict import bidict
half_edge_to_polygon_edge = bidict()
for (a, b, c) in codomain.faces():
label = (a.id(), b.id(), c.id())
half_edge_to_polygon_edge[a] = (label, 0)
half_edge_to_polygon_edge[b] = (label, 1)
half_edge_to_polygon_edge[c] = (label, 2)
if domain is None:
ring_conversion = RingConversion.from_pyflatsurf_from_flat_triangulation(
codomain
)
from flatsurf import MutableOrientedSimilaritySurface, Polygon
domain = MutableOrientedSimilaritySurface(ring_conversion.domain())
from sage.all import VectorSpace
vector_conversion = VectorSpaceConversion.to_pyflatsurf(
VectorSpace(ring_conversion.domain(), 2)
)
for (a, b, c) in codomain.faces():
vectors = [
codomain.fromHalfEdge(a),
codomain.fromHalfEdge(b),
codomain.fromHalfEdge(c),
]
vectors = [vector_conversion.section(vector) for vector in vectors]
triangle = Polygon(edges=vectors)
label = (a.id(), b.id(), c.id())
domain.add_polygon(triangle, label=label)
for half_edge, (polygon, edge) in half_edge_to_polygon_edge.items():
opposite = half_edge_to_polygon_edge[-half_edge]
domain.glue((polygon, edge), opposite)
domain.set_immutable()
return FlatTriangulationConversion(
domain,
codomain,
{
(label, edge): half_edge.id()
for (
(label, edge),
half_edge,
) in half_edge_to_polygon_edge.inverse.items()
},
)
[docs] @cached_method
def ring_conversion(self):
r"""
Return the conversion that maps the base ring of the domain of this
conversion to the base ring of the codomain of this conversion.
EXAMPLES::
sage: from flatsurf import translation_surfaces
sage: from flatsurf.geometry.pyflatsurf_conversion import FlatTriangulationConversion
sage: from flatsurf.geometry.surface_objects import SurfacePoint
sage: S = translation_surfaces.veech_double_n_gon(5).triangulate()
sage: conversion = FlatTriangulationConversion.to_pyflatsurf(S) # optional: pyflatsurf
sage: conversion.ring_conversion() # optional: pyflatsurf
Conversion from Number Field in a with defining polynomial y^4 - 5*y^2 + 5 with a = 1.902113032590308? to NumberField(a^4 - 5*a^2 + 5, [...])
"""
return RingConversion.to_pyflatsurf(domain=self.domain().base_ring())
[docs] @cached_method
def vector_space_conversion(self):
r"""
Return the conversion maps two-dimensional vectors over the base ring
of the domain to ``Vector<T>`` for the codomain.
EXAMPLES::
sage: from flatsurf import translation_surfaces
sage: from flatsurf.geometry.pyflatsurf_conversion import FlatTriangulationConversion
sage: from flatsurf.geometry.surface_objects import SurfacePoint
sage: S = translation_surfaces.veech_double_n_gon(5).triangulate()
sage: conversion = FlatTriangulationConversion.to_pyflatsurf(S) # optional: pyflatsurf
sage: conversion.vector_space_conversion() # optional: pyflatsurf
Conversion from Vector space of dimension 2 over Number Field in a with defining polynomial y^4 - 5*y^2 + 5 with a = 1.902113032590308? to flatsurf::Vector<eantic::renf_elem_class>
"""
from sage.all import VectorSpace
return VectorSpaceConversion.to_pyflatsurf(
VectorSpace(self.ring_conversion().domain(), 2)
)
[docs] def __call__(self, x):
r"""
Return the image of ``x`` under this conversion.
INPUT:
- ``x`` -- an object defined on the domain, e.g., a
:class:`flatsurf.geometry.surface_objects.SurfacePoint`
EXAMPLES::
sage: from flatsurf import translation_surfaces
sage: from flatsurf.geometry.pyflatsurf_conversion import FlatTriangulationConversion
sage: from flatsurf.geometry.surface_objects import SurfacePoint
sage: S = translation_surfaces.veech_double_n_gon(5).triangulate()
sage: conversion = FlatTriangulationConversion.to_pyflatsurf(S) # optional: pyflatsurf
We map a point::
sage: p = SurfacePoint(S, (0, 2), (0, 1/2))
sage: conversion(p) # optional: pyflatsurf
((-1/4*a^2 + 1/2*a + 1/2 ~ 0.54654802), (1/4*a^2 - 3/4 ~ 0.15450850), (1/4 ~ 0.25000000)) in (-6, 8, 9)
We map a half edge::
sage: conversion(((0, 0), 0)) # optional: pyflatsurf
1
"""
from flatsurf.geometry.surface_objects import SurfacePoint
if isinstance(x, SurfacePoint):
return self._image_point(x)
if isinstance(x, tuple) and len(x) == 2:
return self._image_half_edge(*x)
raise NotImplementedError(
f"cannot map {type(x)} from sage-flatsurf to pyflatsurf yet"
)
[docs] def section(self, y):
r"""
Return the preimage of ``y`` under this conversion.
INPUT:
- ``y`` -- an object defined in the codomain, e.g., a pyflatsurf
``Point``
EXAMPLES::
sage: from flatsurf import translation_surfaces
sage: from flatsurf.geometry.pyflatsurf_conversion import FlatTriangulationConversion
sage: from flatsurf.geometry.surface_objects import SurfacePoint
sage: S = translation_surfaces.veech_double_n_gon(5).triangulate()
sage: conversion = FlatTriangulationConversion.to_pyflatsurf(S) # optional: pyflatsurf
We roundtrip a point::
sage: p = SurfacePoint(S, (0, 2), (0, 1/2))
sage: q = conversion(p) # optional: pyflatsurf
sage: conversion.section(q) == p # optional: pyflatsurf
True
We roundtrip a half edge::
sage: half_edge = conversion(((0, 0), 0)) # optional: pyflatsurf
sage: conversion.section(half_edge) # optional: pyflatsurf
((0, 0), 0)
"""
import pyflatsurf
if isinstance(y, pyflatsurf.flatsurf.Point[type(self.codomain())]):
return self._preimage_point(y)
if isinstance(y, pyflatsurf.flatsurf.HalfEdge):
return self._preimage_half_edge(y)
raise NotImplementedError(
f"cannot compute the preimage of a {type(y)} in sage-flatsurf yet"
)
def _image_point(self, p):
r"""
Return the image of the :class:`SurfacePoint` ``p`` under this conversion.
This is a helper method for :meth:`__call__`.
EXAMPLES::
sage: from flatsurf import translation_surfaces
sage: from flatsurf.geometry.pyflatsurf_conversion import FlatTriangulationConversion
sage: from flatsurf.geometry.surface_objects import SurfacePoint
sage: S = translation_surfaces.veech_double_n_gon(5).triangulate()
sage: conversion = FlatTriangulationConversion.to_pyflatsurf(S) # optional: pyflatsurf
sage: p = SurfacePoint(S, (0, 2), (0, 1/2))
sage: conversion._image_point(p) # optional: pyflatsurf
((-1/4*a^2 + 1/2*a + 1/2 ~ 0.54654802), (1/4*a^2 - 3/4 ~ 0.15450850), (1/4 ~ 0.25000000)) in (-6, 8, 9)
"""
if p.surface() is not self.domain():
raise ValueError("point is not a point in the domain of this conversion")
label = next(iter(p.labels()))
coordinates = next(iter(p.coordinates(label)))
import pyflatsurf
return pyflatsurf.flatsurf.Point[type(self.codomain())](
self.codomain(),
self((label, 0)),
self.vector_space_conversion()(
coordinates - p.parent().polygon(label).vertex(0)
),
)
def _preimage_point(self, q):
r"""
Return the preimage of the point ``q`` in the domain of this conversion.
This is a helper method for :meth:`section`.
EXAMPLES::
sage: from flatsurf import translation_surfaces
sage: from flatsurf.geometry.pyflatsurf_conversion import FlatTriangulationConversion
sage: from flatsurf.geometry.surface_objects import SurfacePoint
sage: S = translation_surfaces.veech_double_n_gon(5).triangulate()
sage: conversion = FlatTriangulationConversion.to_pyflatsurf(S) # optional: pyflatsurf
sage: p = SurfacePoint(S, (0, 2), (0, 1/2))
sage: q = conversion(p) # optional: pyflatsurf
sage: conversion._preimage_point(q) # optional: pyflatsurf
Point (0, 1/2) of polygon (0, 2)
"""
face = q.face()
label, edge = self.section(face)
coordinates = self.vector_space_conversion().section(q.vector(face))
coordinates += self.domain().polygon(label).vertex(edge)
from flatsurf.geometry.surface_objects import SurfacePoint
return SurfacePoint(self.domain(), label, coordinates)
def _image_half_edge(self, label, edge):
r"""
Return the half edge that ``edge`` of polygon ``label`` maps to under this conversion.
This is a helper method for :meth:`__call__`.
INPUT:
- ``label`` -- an arbitrary polygon label in the :meth:`domain`
- ``edge`` -- an integer, the identifier of an edge in the polygon ``label``
EXAMPLES::
sage: from flatsurf import translation_surfaces
sage: from flatsurf.geometry.pyflatsurf_conversion import FlatTriangulationConversion
sage: from flatsurf.geometry.surface_objects import SurfacePoint
sage: S = translation_surfaces.veech_double_n_gon(5).triangulate()
sage: conversion = FlatTriangulationConversion.to_pyflatsurf(S) # optional: pyflatsurf
sage: conversion._image_half_edge((0, 0), 0) # optional: pyflatsurf
1
"""
import pyflatsurf
return pyflatsurf.flatsurf.HalfEdge(self._label_to_half_edge[(label, edge)])
def _preimage_half_edge(self, half_edge):
r"""
Return the preimage of the ``half_edge`` in the domain of this
conversion as a pair ``(label, edge)``.
This is a helper method for :meth:`section`.
EXAMPLES::
sage: from flatsurf import translation_surfaces
sage: from flatsurf.geometry.pyflatsurf_conversion import FlatTriangulationConversion
sage: from flatsurf.geometry.surface_objects import SurfacePoint
sage: S = translation_surfaces.veech_double_n_gon(5).triangulate()
sage: conversion = FlatTriangulationConversion.to_pyflatsurf(S) # optional: pyflatsurf
sage: import pyflatsurf # optional: pyflatsurf
sage: conversion._preimage_half_edge(pyflatsurf.flatsurf.HalfEdge(1R)) # optional: pyflatsurf
((0, 0), 0)
"""
return self._half_edge_to_label[half_edge.id()]
[docs]def to_pyflatsurf(S):
r"""
Given S a translation surface from sage-flatsurf return a
flatsurf::FlatTriangulation from libflatsurf/pyflatsurf.
"""
return FlatTriangulationConversion.to_pyflatsurf(S.triangulate()).codomain()
[docs]def sage_ring(surface):
r"""
Return the SageMath ring over which the pyflatsurf surface ``surface`` can
be constructed in sage-flatsurf.
EXAMPLES::
sage: from flatsurf import translation_surfaces
sage: from flatsurf.geometry.pyflatsurf_conversion import to_pyflatsurf, sage_ring # optional: pyflatsurf
sage: S = to_pyflatsurf(translation_surfaces.veech_double_n_gon(5)) # optional: pyflatsurf # random output due to matplotlib warnings with some combinations of setuptools and matplotlib
sage: sage_ring(S) # optional: pyflatsurf
Number Field in a with defining polynomial x^4 - 5*x^2 + 5 with a = 1.902113032590308?
"""
from sage.all import Sequence
vectors = [surface.fromHalfEdge(e.positive()) for e in surface.edges()]
return Sequence(
[to_sage_ring(v.x()) for v in vectors] + [to_sage_ring(v.y()) for v in vectors]
).universe()
[docs]def to_sage_ring(x):
r"""
Given a coordinate of a flatsurf::Vector, return a SageMath element from
which :meth:`from_pyflatsurf` can eventually construct a translation surface.
EXAMPLES::
sage: from flatsurf.geometry.pyflatsurf_conversion import to_sage_ring # optional: pyflatsurf
sage: to_sage_ring(1R).parent() # optional: pyflatsurf
Integer Ring
GMP coordinate types::
sage: import cppyy # optional: pyflatsurf
sage: import pyeantic # optional: pyflatsurf
sage: to_sage_ring(cppyy.gbl.mpz_class(1)).parent() # optional: pyflatsurf
Integer Ring
sage: to_sage_ring(cppyy.gbl.mpq_class(1, 2)).parent() # optional: pyflatsurf
Rational Field
e-antic coordinate types::
sage: import pyeantic # optional: pyflatsurf
sage: K = pyeantic.eantic.renf_class.make("a^3 - 3*a + 1", "a", "0.34 +/- 0.01", 64R) # optional: pyflatsurf
sage: to_sage_ring(K.gen()).parent() # optional: pyflatsurf
Number Field in a with defining polynomial x^3 - 3*x + 1 with a = 0.3472963553338607?
exact-real coordinate types::
sage: from pyexactreal import QQModule, RealNumber # optional: pyflatsurf
sage: M = QQModule(RealNumber.random()) # optional: pyflatsurf
sage: to_sage_ring(M.gen(0R)).parent() # optional: pyflatsurf
Real Numbers as (Rational Field)-Module
"""
from flatsurf.features import cppyy_feature
cppyy_feature.require()
import cppyy
import gmpxxyy
def maybe_type(t):
try:
return t()
except AttributeError:
# The type constructed by t might not exist because the required C++ library has not been loaded.
return None
from sage.all import QQ, ZZ
if type(x) is int:
return ZZ(x)
elif type(x) is maybe_type(lambda: cppyy.gbl.mpz_class):
return ZZ(str(x))
elif type(x) is maybe_type(lambda: cppyy.gbl.mpq_class):
return QQ(str(x))
elif type(x) is maybe_type(lambda: cppyy.gbl.eantic.renf_elem_class):
from pyeantic import RealEmbeddedNumberField
real_embedded_number_field = RealEmbeddedNumberField(x.parent())
return real_embedded_number_field.number_field(real_embedded_number_field(x))
elif type(x) is maybe_type(
lambda: cppyy.gbl.exactreal.Element[cppyy.gbl.exactreal.IntegerRing]
):
from pyexactreal import ExactReals
return ExactReals(ZZ)(x)
elif type(x) is maybe_type(
lambda: cppyy.gbl.exactreal.Element[cppyy.gbl.exactreal.RationalField]
):
from pyexactreal import ExactReals
return ExactReals(QQ)(x)
elif type(x) is maybe_type(
lambda: cppyy.gbl.exactreal.Element[cppyy.gbl.exactreal.NumberField]
):
from pyexactreal import ExactReals
return ExactReals(x.module().ring().parameters)(x)
else:
raise NotImplementedError(
f"unknown coordinate ring for element {x} which is a {type(x)}"
)
[docs]def from_pyflatsurf(T):
r"""
Given T a flatsurf::FlatTriangulation from libflatsurf/pyflatsurf, return a
sage-flatsurf translation surface.
EXAMPLES::
sage: from flatsurf import translation_surfaces
sage: from flatsurf.geometry.pyflatsurf_conversion import to_pyflatsurf, from_pyflatsurf # optional: pyflatsurf
sage: S = translation_surfaces.veech_double_n_gon(5) # optional: pyflatsurf
sage: T = from_pyflatsurf(to_pyflatsurf(S)) # optional: pyflatsurf
sage: T # optional: pyflatsurf
Translation Surface in H_2(2) built from 6 isosceles triangles
TESTS::
sage: from flatsurf.geometry.categories import TranslationSurfaces
sage: T in TranslationSurfaces() # optional: pyflatsurf
True
Verify that #137 has been resolved::
sage: from flatsurf import polygons, MutableOrientedSimilaritySurface
sage: from flatsurf.geometry.gl2r_orbit_closure import GL2ROrbitClosure
sage: from flatsurf.geometry.pyflatsurf_conversion import from_pyflatsurf
sage: P = polygons.regular_ngon(10)
sage: S = MutableOrientedSimilaritySurface(P.base_ring())
sage: S.add_polygon(P)
0
sage: for i in range(5): S.glue((0, i), (0, 5+i))
sage: S.set_immutable()
sage: M = S
sage: X = GL2ROrbitClosure(M) # optional: pyflatsurf
sage: D0 = list(X.decompositions(2))[2] # optional: pyflatsurf
sage: T0 = D0.triangulation() # optional: pyflatsurf
sage: from_pyflatsurf(T0) # optional: pyflatsurf
Translation Surface in H_2(1^2) built from 2 isosceles triangles and 6 triangles
"""
return FlatTriangulationConversion.from_pyflatsurf(T).domain()