Source code for flatsurf.geometry.chamanara

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.
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#        Copyright (C) 2013-2019 W. Patrick Hooper
#                      2013-2019 Vincent Delecroix
#                           2023 Julian Rüth
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from flatsurf.geometry.surface import OrientedSimilaritySurface
from flatsurf.geometry.minimal_cover import MinimalTranslationCover
from flatsurf.geometry.lazy import LazyRelabeledSurface
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(0) Polygon(vertices=[(0, 0), (1, 0), (-1, 2), (-1, 1)]) """ if lab not in ZZ: raise KeyError(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(LazyRelabeledSurface): def __init__(self, alpha): super().__init__(MinimalTranslationCover(ChamanaraSurface(alpha)))
[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)