r"""
Chamanara's surfaces which depend on a parameter `\alpha` less than one.
REFERENCES:
- Defined as `X_\alpha` in Chamanara, Reza, "Affine automorphism groups of
surfaces of infinite type", City University of New York, 2002.
.. jupyter-execute::
:hide-code:
# Allow jupyter-execute blocks in this module to contain doctests
import jupyter_doctest_tweaks
EXAMPLES:
.. jupyter-execute::
sage: from flatsurf import translation_surfaces
sage: s = translation_surfaces.chamanara(1/2)
sage: s.plot()
...Graphics object consisting of 129 graphics primitives
"""
# ********************************************************************
# This file is part of sage-flatsurf.
#
# Copyright (C) 2013-2019 W. Patrick Hooper
# 2013-2019 Vincent Delecroix
# 2023 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 flatsurf.geometry.surface import OrientedSimilaritySurface
from flatsurf.geometry.minimal_cover import MinimalTranslationCover
from sage.rings.integer_ring import ZZ
[docs]def ChamanaraPolygon(alpha):
from sage.categories.fields import Fields
field = alpha.parent()
if field not in Fields():
ValueError("The value of alpha must lie in a field.")
if alpha <= 0 or alpha >= 1:
ValueError("The value of alpha must be between zero and one.")
# The value of x is $\sum_{n=0}^\infty \alpha^n$.
x = 1 / (1 - alpha)
from flatsurf import Polygon
return Polygon(edges=[(1, 0), (-x, x), (0, -1), (x - 1, 1 - x)])
[docs]class ChamanaraSurface(OrientedSimilaritySurface):
r"""
The Chamanara surface $X_{\alpha}$.
EXAMPLES::
sage: from flatsurf.geometry.chamanara import ChamanaraSurface
sage: S = ChamanaraSurface(1/2); S
Chamanara surface with parameter 1/2
TESTS::
sage: TestSuite(S).run()
"""
def __init__(self, alpha):
self._p = ChamanaraPolygon(alpha)
field = alpha.parent()
if not field.is_field():
field = field.fraction_field()
self.rename("Chamanara surface with parameter {}".format(alpha))
from flatsurf.geometry.categories import DilationSurfaces
super().__init__(
field,
category=DilationSurfaces()
.Oriented()
.InfiniteType()
.Compact()
.WithoutBoundary()
.Connected()
.Rational(),
)
[docs] def is_dilation_surface(self, positive=False):
r"""
Return whether this surface is a dilation surface, overrides
:meth:`flatsurf.geometry.categories.similarity_surfaces.SimilaritySurfaces.ParentMethods.is_dilation_surface`.
EXAMPLES::
sage: from flatsurf.geometry.chamanara import ChamanaraSurface
sage: S = ChamanaraSurface(1/2)
sage: S.is_dilation_surface(positive=True)
False
sage: S.is_dilation_surface(positive=False)
True
"""
return not positive
[docs] def is_cone_surface(self):
r"""
Return whether this surface is a cone surfaces, overrides
:meth:`flatsurf.geometry.categories.similarity_surfaces.SimilaritySurfaces.ParentMethods.is_cone_surface`.
EXAMPLES::
sage: from flatsurf.geometry.chamanara import ChamanaraSurface
sage: S = ChamanaraSurface(1/2)
sage: S.is_cone_surface()
False
"""
return False
[docs] def is_translation_surface(self, positive=True):
r"""
Return whether this surfaces is a (half-)translation surface, overrides
:meth:`flatsurf.geometry.categories.similarity_surfaces.SimilaritySurfaces.ParentMethods.is_translation_surface`.
EXAMPLES::
sage: from flatsurf.geometry.chamanara import ChamanaraSurface
sage: S = ChamanaraSurface(1/2)
sage: S.is_translation_surface(positive=True)
False
sage: S.is_translation_surface(positive=False)
False
"""
return False
[docs] def labels(self):
r"""
Return the labels used to identify the polygons that make up this
surface, i.e., the integers.
This overrides
:meth:`flatsurf.geometry.categories.polygonal_surfaces.PolygonalSurfaces.ParentMethods.labels`
EXAMPLES::
sage: from flatsurf.geometry.chamanara import ChamanaraSurface
sage: S = ChamanaraSurface(1/2)
sage: S.labels()
(0, 1, -1, 2, -2, 3, -3, 4, -4, 5, -5, 6, -6, 7, -7, 8, …)
"""
from flatsurf.geometry.surface import LabelsFromView
return LabelsFromView(self, ZZ, finite=False)
[docs] def roots(self):
r"""
Return a label in each connected component of this surface.
This overrides
:meth:`flatsurf.geometry.categories.polygonal_surfaces.PolygonalSurfaces.ParentMethods.roots`.
EXAMPLES::
sage: from flatsurf.geometry.chamanara import ChamanaraSurface
sage: S = ChamanaraSurface(1/2)
sage: S.roots()
(0,)
"""
return (ZZ(0),)
[docs] def is_mutable(self):
r"""
Return whether this surface is mutable which it is not.
EXAMPLES::
sage: from flatsurf.geometry.chamanara import ChamanaraSurface
sage: S = ChamanaraSurface(1/2)
sage: S.is_mutable()
False
"""
return False
[docs] def polygon(self, lab):
r"""
EXAMPLES::
sage: from flatsurf import translation_surfaces
sage: C = translation_surfaces.chamanara(1/2)
sage: C.polygon('a')
Traceback (most recent call last):
...
ValueError: invalid label 'a'
"""
if lab not in ZZ:
raise ValueError("invalid label {!r}".format(lab))
return self._p
[docs] def opposite_edge(self, p, e):
if e == 0 or e == 2:
return 1 - p, e
elif e == 1:
if p < 0:
return p + 1, 3
elif p > 1:
return p - 1, 3
else:
# p==0 or p==1
return 1 - p, 1
else:
# e==3
if p <= 0:
return p - 1, 1
else:
# p>=1
return p + 1, 1
def __hash__(self):
r"""
Return a hash value for this surface that is compatible with equality testing.
EXAMPLES::
sage: from flatsurf.geometry.chamanara import ChamanaraSurface
sage: hash(ChamanaraSurface(1/2)) == hash(ChamanaraSurface(1/2))
True
"""
return hash((self._p, self.base_ring()))
[docs] def graphical_surface(self, **kwds):
adjacencies = [(0, 1)]
for i in range(8):
adjacencies.append((-i, 3))
adjacencies.append((i + 1, 3))
return super().graphical_surface(adjacencies=adjacencies, **kwds)
def __eq__(self, other):
r"""
Return whether this surface is indistinguishable from ``other``.
See :meth:`SimilaritySurfaces.FiniteType._test_eq_surface` for details
on this notion of equality.
EXAMPLES::
sage: from flatsurf import translation_surfaces
sage: C = translation_surfaces.chamanara(1/2)
sage: C == C
True
sage: D = translation_surfaces.chamanara(1/3)
sage: C == D
False
"""
if not isinstance(other, ChamanaraSurface):
return False
return self._p == other._p and self.base_ring() == other.base_ring()
[docs]def chamanara_half_dilation_surface(alpha, n=None):
r"""
Return Chamanara's surface thought of as a Half Dilation surface.
EXAMPLES::
sage: from flatsurf.geometry.chamanara import chamanara_half_dilation_surface
sage: s = chamanara_half_dilation_surface(1/2)
sage: TestSuite(s).run()
"""
if n is not None:
import warnings
warnings.warn(
"the n keyword of chamanara_half_dilation_surface() is not supported anymore; it will be removed in a future version of sage-flatsurf"
)
return ChamanaraSurface(alpha)
[docs]class ChamanaraTranslationSurface(MinimalTranslationCover):
def __init__(self, alpha):
MinimalTranslationCover.__init__(self, ChamanaraSurface(alpha))
self._refine_category_(self.category().Compact())
[docs] def graphical_surface(self, **kwds):
label = self.root()
adjacencies = [(label, 1)]
for i in range(8):
adjacencies.append((label, 3))
label = self.opposite_edge(label, 3)[0]
label = self.root()
label = self.opposite_edge(label, 1)[0]
for i in range(8):
adjacencies.append((label, 3))
label = self.opposite_edge(label, 3)[0]
return super().graphical_surface(adjacencies=adjacencies, **kwds)
[docs]def chamanara_surface(alpha, n=None):
r"""
Return Chamanara's surface thought of as a translation surface.
EXAMPLES::
sage: from flatsurf.geometry.chamanara import chamanara_surface
sage: s = chamanara_surface(1/2)
sage: TestSuite(s).run()
"""
if n is not None:
import warnings
warnings.warn(
"the n keyword of chamanara_half_dilation_surface() is not supported anymore; it will be removed in a future version of sage-flatsurf"
)
return ChamanaraTranslationSurface(alpha)