def l_infinity_delaunay_triangulation(self,
triangulated=False,
in_place=False,
limit=None,
direction=None):
r"""
Returns a L-infinity Delaunay triangulation of a surface, or make some
triangle flips to get closer to the Delaunay decomposition.
INPUT:
- ``triangulated`` -- optional (boolean, default ``False``) If true, the
algorithm assumes the surface is already triangulated. It does this
without verification.
- ``in_place`` -- optional (boolean, default ``False``) If true, the
triangulating and the triangle flips are done in place. Otherwise, a
mutable copy of the surface is made.
- ``limit`` -- optional (positive integer) If provided, then at most ``limit``
many diagonal flips will be done.
- ``direction`` -- optional (vector). Used to determine labels when a
pair of triangles is flipped. Each triangle has a unique separatrix
which points in the provided direction or its negation. As such a
vector determines a sign for each triangle. A pair of adjacent
triangles have opposite signs. Labels are chosen so that this sign is
preserved (as a function of labels).
EXAMPLES::
sage: from flatsurf import *
sage: s0 = translation_surfaces.veech_double_n_gon(5)
sage: field = s0.base_ring()
sage: a = field.gen()
sage: m = matrix(field, 2, [2,a,1,1])
sage: s = m*s0
sage: s = s.l_infinity_delaunay_triangulation()
sage: TestSuite(s).run()
sage: s = (m**2)*s0
sage: s = s.l_infinity_delaunay_triangulation()
sage: TestSuite(s).run()
sage: s = (m**3)*s0
sage: s = s.l_infinity_delaunay_triangulation()
sage: TestSuite(s).run()
"""
if not self.is_finite():
raise NotImplementedError(
"no L-infinity Delaunay implemented for infinite surfaces")
if triangulated:
if in_place:
s = self
else:
from flatsurf.geometry.surface import Surface_dict
s = self.__class__(Surface_dict(surface=self, mutable=True))
else:
from flatsurf.geometry.surface import Surface_list
s = self.__class__(
Surface_list(surface=self.triangulate(in_place=in_place),
mutable=True))
if direction is None:
base_ring = self.base_ring()
direction = self.vector_space()(
(base_ring.zero(), base_ring.one()))
else:
assert not direction.is_zero()
triangles = set(s.label_iterator())
if limit is None:
limit = -1
else:
limit = int(limit)
while triangles and limit:
p1 = triangles.pop()
for e1 in range(3):
p2, e2 = s.opposite_edge(p1, e1)
if s._edge_needs_flip_Linfinity(p1, e1, p2, e2):
s.triangle_flip(p1, e1, in_place=True, direction=direction)
triangles.add(p1)
triangles.add(p2)
limit -= 1
return s
def surface_to_xml_string(s, complain=True):
r"""
Convert the surface s into a XML string.
Note that this only encodes the Surface part of the object and not the SimilaritySurface
wrapper.
By default complain=True, and it will print warning messages and raise errors if it doesn't
think you will be able to reconstruct the surface using surface_from_xml_string.
Currently s should be Surface_list (or a wrapped Surface_list). If not and complain=True,
an error message will be printed and the surface will be converted to Surface_list. If
complain=False, the surface will be encoded but you may not be able to recover it.
Also the surface should be defined over the rationals. Otherwise a ValueError is raised, which
can be disabled by setting complain=False.
"""
if not s.is_finite():
raise ValueError("Can only xml encode a finite surface.")
from flatsurf.geometry.similarity_surface import SimilaritySurface
if isinstance(s, SimilaritySurface):
s = s.underlying_surface()
from flatsurf.geometry.surface import Surface_list
if complain:
if not isinstance(s, Surface_list):
# Convert to surface list
print("Warning:surface_to_xml_string is converting to Surface_list before encoding, "+\
" labels will likely be changed. "+\
"If this is not desired, "+
"call surface_to_xml_string with complain=False.")
s = Surface_list(surface=s)
from xml.sax.saxutils import escape
output = []
output.append('<surface>')
# Include the type of surface (in case we care about it in the future)
output.append('<type>')
output.append(escape(repr(type(s))))
output.append('</type>')
from sage.rings.rational_field import QQ
if complain and s.base_ring() != QQ:
raise ValueError("Refusing to encode surface with base_ring!=QQ, "+\
"because we can not guarantee we bring the surface back. "+
"If you would like to encode it anyway, "+
"call surface_to_xml_string with complain=False.")
output.append('<base_ring>')
output.append(escape(repr(s.base_ring())))
output.append('</base_ring>')
output.append('<base_label>')
output.append(escape(repr(s.base_label())))
output.append('</base_label>')
output.append('<polygons>')
for label in s.label_iterator():
output.append('<polygon>')
output.append('<label>')
output.append(escape(repr(label)))
output.append('</label>')
p = s.polygon(label)
for e in range(p.num_edges()):
output.append('<edge>')
v = p.edge(e)
output.append('<x>')
output.append(escape(repr(v[0])))
output.append('</x>')
output.append('<y>')
output.append(escape(repr(v[1])))
output.append('</y>')
output.append('</edge>')
output.append('</polygon>')
output.append('</polygons>')
output.append('<gluings>')
for l, e in s.edge_iterator():
ll, ee = s.opposite_edge(l, e)
if ll < l or (ll == l and ee < e):
continue
output.append('<glue>')
output.append('<l>')
output.append(escape(repr(l)))
output.append('</l>')
output.append('<e>')
output.append(escape(repr(e)))
output.append('</e>')
output.append('<ll>')
output.append(escape(repr(ll)))
output.append('</ll>')
output.append('<ee>')
output.append(escape(repr(ee)))
output.append('</ee>')
output.append('</glue>')
output.append('</gluings>')
output.append('</surface>')
return ''.join(output)
def surface_to_xml_string(s, complain=True):
r"""
Convert the surface s into a XML string.
Note that this only encodes the Surface part of the object and not the SimilaritySurface
wrapper.
By default complain=True, and it will print warning messages and raise errors if it doesn't
think you will be able to reconstruct the surface using surface_from_xml_string.
Currently s should be Surface_list (or a wrapped Surface_list). If not and complain=True,
an error message will be printed and the surface will be converted to Surface_list. If
complain=False, the surface will be encoded but you may not be able to recover it.
Also the surface should be defined over the rationals. Otherwise a ValueError is raised, which
can be disabled by setting complain=False.
"""
if not s.is_finite():
raise ValueError("Can only xml encode a finite surface.")
from flatsurf.geometry.similarity_surface import SimilaritySurface
if isinstance(s,SimilaritySurface):
s=s.underlying_surface()
from flatsurf.geometry.surface import Surface_list
if complain:
if not isinstance(s,Surface_list):
# Convert to surface list
print("Warning:surface_to_xml_string is converting to Surface_list before encoding, "+\
" labels will likely be changed. "+\
"If this is not desired, "+
"call surface_to_xml_string with complain=False.")
s=Surface_list(surface=s)
from xml.sax.saxutils import escape
output=[]
output.append('<surface>')
# Include the type of surface (in case we care about it in the future)
output.append('<type>')
output.append(escape(repr(type(s))))
output.append('</type>')
from sage.rings.rational_field import QQ
if complain and s.base_ring()!=QQ:
raise ValueError("Refusing to encode surface with base_ring!=QQ, "+\
"because we can not guarantee we bring the surface back. "+
"If you would like to encode it anyway, "+
"call surface_to_xml_string with complain=False.")
output.append('<base_ring>')
output.append(escape(repr(s.base_ring())))
output.append('</base_ring>')
output.append('<base_label>')
output.append(escape(repr(s.base_label())))
output.append('</base_label>')
output.append('<polygons>')
for label in s.label_iterator():
output.append('<polygon>')
output.append('<label>')
output.append(escape(repr(label)))
output.append('</label>')
p=s.polygon(label)
for e in xrange(p.num_edges()):
output.append('<edge>')
v=p.edge(e)
output.append('<x>')
output.append(escape(repr(v[0])))
output.append('</x>')
output.append('<y>')
output.append(escape(repr(v[1])))
output.append('</y>')
output.append('</edge>')
output.append('</polygon>')
output.append('</polygons>')
output.append('<gluings>')
for l,e in s.edge_iterator():
ll,ee=s.opposite_edge(l,e)
if ll<l or (ll==l and ee<e):
continue
output.append('<glue>')
output.append('<l>')
output.append(escape(repr(l)))
output.append('</l>')
output.append('<e>')
output.append(escape(repr(e)))
output.append('</e>')
output.append('<ll>')
output.append(escape(repr(ll)))
output.append('</ll>')
output.append('<ee>')
output.append(escape(repr(ee)))
output.append('</ee>')
output.append('</glue>')
output.append('</gluings>')
output.append('</surface>')
return ''.join(output)
def _surface_from_ElementTree(tree):
from flatsurf.geometry.surface import Surface_list
node = tree.find("type")
if node is None:
raise ValueError('Failed to find tag named "node"')
if node.text != repr(Surface_list):
raise NotImplementedError("Currently can only reconstruct from Surface_list. "+\
"Found type "+node.text)
node = tree.find("base_ring")
if node is None:
raise ValueError('Failed to find tag named "base_ring"')
from sage.rings.rational_field import QQ
if node.text == repr(QQ):
base_ring = QQ
else:
raise NotImplementedError("Can only reconstruct when base_ring=QQ. "+\
"Found "+tree.find("base_ring").text)
s = Surface_list(QQ)
# Import the polygons
polygons = tree.find("polygons")
if polygons is None:
raise ValueError('Failed to find tag named "polygons"')
from flatsurf.geometry.polygon import ConvexPolygons
P = ConvexPolygons(base_ring)
for polygon in polygons.findall("polygon"):
node = polygon.find("label")
if node is None:
raise ValueError('Failed to find tag <label> in <polygon>')
label = int(node.text)
edges = []
for edge in polygon.findall("edge"):
node = edge.find("x")
if node is None:
raise ValueError(
'Failed to find tag <x> in <edge> in <polygon> with label='
+ str(label))
x = base_ring(node.text)
node = edge.find("y")
if node is None:
raise ValueError(
'Failed to find tag <y> in <edge> in <polygon> with label='
+ str(label))
y = base_ring(node.text)
edges.append((x, y))
p = P(edges=edges)
s.add_polygon(p, label=label)
if s.num_polygons() == 0:
raise ValueError("Failed to add any polygons.")
# Glue the edges
gluings = tree.find("gluings")
if gluings is None:
raise ValueError('Failed to find tag named "gluings"')
for glue in gluings.findall("glue"):
node = glue.find("l")
if node is None:
raise ValueError('Failed to find tag <l> in <glue>.')
l = int(node.text)
node = glue.find("e")
if node is None:
raise ValueError('Failed to find tag <e> in <glue>.')
e = int(node.text)
node = glue.find("ll")
if node is None:
raise ValueError('Failed to find tag <ll> in <glue>.')
ll = int(node.text)
node = glue.find("ee")
if node is None:
raise ValueError('Failed to find tag <ee> in <glue>.')
ee = int(node.text)
s.change_edge_gluing(l, e, ll, ee)
node = tree.find("base_label")
if node is None:
raise ValueError('Failed to find tag named "base_label"')
base_label = int(node.text)
s.change_base_label(base_label)
# Return the surface:
return s
def _surface_from_ElementTree(tree):
from flatsurf.geometry.surface import Surface_list
node=tree.find("type")
if node is None:
raise ValueError('Failed to find tag named "node"')
if node.text != repr(Surface_list):
raise NotImplementedError("Currently can only reconstruct from Surface_list. "+\
"Found type "+node.text)
node=tree.find("base_ring")
if node is None:
raise ValueError('Failed to find tag named "base_ring"')
from sage.rings.rational_field import QQ
if node.text == repr(QQ):
base_ring=QQ
else:
raise NotImplementedError("Can only reconstruct when base_ring=QQ. "+\
"Found "+tree.find("base_ring").text)
s=Surface_list(QQ)
# Import the polygons
polygons=tree.find("polygons")
if polygons is None:
raise ValueError('Failed to find tag named "polygons"')
from flatsurf.geometry.polygon import Polygons
P=Polygons(base_ring)
for polygon in polygons.findall("polygon"):
node=polygon.find("label")
if node is None:
raise ValueError('Failed to find tag <label> in <polygon>')
label=int(node.text)
edges=[]
for edge in polygon.findall("edge"):
node=edge.find("x")
if node is None:
raise ValueError('Failed to find tag <x> in <edge> in <polygon> with label='+str(label))
x=base_ring(node.text)
node=edge.find("y")
if node is None:
raise ValueError('Failed to find tag <y> in <edge> in <polygon> with label='+str(label))
y=base_ring(node.text)
edges.append((x,y))
p=P(edges=edges)
s.add_polygon(p,label=label)
if s.num_polygons()==0:
raise ValueError("Failed to add any polygons.")
# Glue the edges
gluings=tree.find("gluings")
if gluings is None:
raise ValueError('Failed to find tag named "gluings"')
for glue in gluings.findall("glue"):
node=glue.find("l")
if node is None:
raise ValueError('Failed to find tag <l> in <glue>.')
l=int(node.text)
node=glue.find("e")
if node is None:
raise ValueError('Failed to find tag <e> in <glue>.')
e=int(node.text)
node=glue.find("ll")
if node is None:
raise ValueError('Failed to find tag <ll> in <glue>.')
ll=int(node.text)
node=glue.find("ee")
if node is None:
raise ValueError('Failed to find tag <ee> in <glue>.')
ee=int(node.text)
s.change_edge_gluing(l,e,ll,ee)
node=tree.find("base_label")
if node is None:
raise ValueError('Failed to find tag named "base_label"')
base_label = int(node.text)
s.change_base_label(base_label)
# Return the surface:
return s
