def test_base_point(self):
f = self.f1
CPF = ComplexPathFactory(f, -1)
self.assertAlmostEqual(CPF.base_point, -1)
CPF = ComplexPathFactory(f, -2)
self.assertAlmostEqual(CPF.base_point, -2)
def test_path(self):
CPF = ComplexPathFactory(self.f1, base_point=-2, kappa=1)
b = CPF.discriminant_points[0]
R = CPF.radius(b)
self.assertAlmostEqual(R, 1.0)
# straight line path from (-2,1) to (1,-2). this intersects the circle
# of radius 1 about the origin at (-1,0) goes around to (0,-1).
#
# this test goes below the circle
z0 = -2 + 1.j
z1 = 1 - 2.j
gamma = CPF.path(z0, z1)
segments = gamma.segments
self.assertEqual(segments[0], ComplexLine(z0,-1))
self.assertEqual(segments[1], ComplexArc(1,0,-pi,pi/2))
self.assertEqual(segments[2], ComplexLine(-1.j,z1))
# straight line path from (-1,2) to (2,-1). this intersects the circle
# of radius 1 about the origin at (0,1) goes around to (1,0).
#
# this test goes below the circle
z0 = -1 + 2.j
z1 = 2 - 1.j
gamma = CPF.path(z0, z1)
segments = gamma.segments
self.assertEqual(segments[0], ComplexLine(z0,1.j))
self.assertEqual(segments[1], ComplexArc(1,0,pi/2,-pi/2))
self.assertEqual(segments[2], ComplexLine(1,z1))
def test_path(self):
CPF = ComplexPathFactory(self.f1, base_point=-2, kappa=1)
b = CPF.discriminant_points[0]
R = CPF.radius(b)
self.assertAlmostEqual(R, 1.0)
# straight line path from (-2,1) to (1,-2). this intersects the circle
# of radius 1 about the origin at (-1,0) goes around to (0,-1).
#
# this test goes below the circle
z0 = -2 + 1.j
z1 = 1 - 2.j
gamma = CPF.path(z0, z1)
segments = gamma.segments
self.assertEqual(segments[0], ComplexLine(z0, -1))
self.assertEqual(segments[1], ComplexArc(1, 0, -pi, pi / 2))
self.assertEqual(segments[2], ComplexLine(-1.j, z1))
# straight line path from (-1,2) to (2,-1). this intersects the circle
# of radius 1 about the origin at (0,1) goes around to (1,0).
#
# this test goes below the circle
z0 = -1 + 2.j
z1 = 2 - 1.j
gamma = CPF.path(z0, z1)
segments = gamma.segments
self.assertEqual(segments[0], ComplexLine(z0, 1.j))
self.assertEqual(segments[1], ComplexArc(1, 0, pi / 2, -pi / 2))
self.assertEqual(segments[2], ComplexLine(1, z1))
def test_closest_discriminant_point(self):
f = self.f3
CPF = ComplexPathFactory(f, -2, kappa=1.0)
# [-I, -1, 1, I]
discriminant_points = CPF.discriminant_points
b = CPF.closest_discriminant_point(-2)
self.assertEqual(b, discriminant_points[1])
b = CPF.closest_discriminant_point(-2j)
self.assertEqual(b, discriminant_points[0])
b = CPF.closest_discriminant_point(-2 + 0.5j)
self.assertEqual(b, discriminant_points[1])
b = CPF.closest_discriminant_point(-2 + 0.5j)
self.assertEqual(b, discriminant_points[1])
b = CPF.closest_discriminant_point(-2 + 0.5j)
self.assertEqual(b, discriminant_points[1])
b = CPF.closest_discriminant_point(-2 + 1.9j)
self.assertEqual(b, discriminant_points[1])
b = CPF.closest_discriminant_point(-2 + 2.1j)
self.assertEqual(b, discriminant_points[3])
def test_closest_discriminant_point(self):
f = self.f3
CPF = ComplexPathFactory(f, -2, kappa=1.0)
# [-I, -1, 1, I]
discriminant_points = CPF.discriminant_points
b = CPF.closest_discriminant_point(-2)
self.assertEqual(b, discriminant_points[1])
b = CPF.closest_discriminant_point(-2j)
self.assertEqual(b, discriminant_points[0])
b = CPF.closest_discriminant_point(-2 + 0.5j)
self.assertEqual(b, discriminant_points[1])
b = CPF.closest_discriminant_point(-2 + 0.5j)
self.assertEqual(b, discriminant_points[1])
b = CPF.closest_discriminant_point(-2 + 0.5j)
self.assertEqual(b, discriminant_points[1])
b = CPF.closest_discriminant_point(-2 + 1.9j)
self.assertEqual(b, discriminant_points[1])
b = CPF.closest_discriminant_point(-2 + 2.1j)
self.assertEqual(b, discriminant_points[3])
def test_path_to_discriminant_point_abel_ordering(self):
# the way abel complex paths are constructed ensure that all points
# along the path to bi lie above all discriminant points below bi and
# above all discriminant points below bi. this will test that scenario
#
# note that this particular test doesn't really work for paths to
# co-linear points
f = self.f3
CPF = ComplexPathFactory(f, -2, kappa=0.5)
# 1.j lies above all other discriminant points. make sure the points
# along the path all lie above the other discriminant point
#
# ignore the base point in these tests
path = CPF.path_to_discriminant_point(1.j)
s = numpy.linspace(0,1,32)
path_points = path(s)
centered_path_points = path_points - CPF.base_point
centered_b0 = CPF.discriminant_points_complex[0] - CPF.base_point
b0_angles = numpy.angle(centered_path_points - centered_b0)
self.assertTrue(all(b0_angles[1:] >= 0))
centered_b1 = CPF.discriminant_points_complex[1] - CPF.base_point
b1_angles = numpy.angle(centered_path_points - centered_b1)
self.assertTrue(all(b1_angles[1:] >= 0))
centered_b2 = CPF.discriminant_points_complex[2] - CPF.base_point
b2_angles = numpy.angle(centered_path_points - centered_b2)
self.assertTrue(all(b2_angles[1:] >= 0))
# -1.j lies below all other discriminant points. make sure the points
# along the path all lie below the other discriminant point
path = CPF.path_to_discriminant_point(-1.j)
s = numpy.linspace(0,1,32)
path_points = path(s)
centered_path_points = path_points - CPF.base_point
b1_angles = numpy.angle(centered_path_points - centered_b1)
self.assertTrue(all(b1_angles[1:] <= 0))
b2_angles = numpy.angle(centered_path_points - centered_b2)
self.assertTrue(all(b2_angles[1:] <= 0))
centered_b3 = CPF.discriminant_points_complex[3] - CPF.base_point
b3_angles = numpy.angle(centered_path_points - centered_b3)
self.assertTrue(all(b3_angles[1:] <= 0))
def test_path_to_discriminant_point_abel_ordering(self):
# the way abel complex paths are constructed ensure that all points
# along the path to bi lie above all discriminant points below bi and
# above all discriminant points below bi. this will test that scenario
#
# note that this particular test doesn't really work for paths to
# co-linear points
f = self.f3
CPF = ComplexPathFactory(f, -2, kappa=0.5)
# 1.j lies above all other discriminant points. make sure the points
# along the path all lie above the other discriminant point
#
# ignore the base point in these tests
path = CPF.path_to_discriminant_point(1.j)
s = numpy.linspace(0, 1, 32)
path_points = path(s)
centered_path_points = path_points - CPF.base_point
centered_b0 = CPF.discriminant_points_complex[0] - CPF.base_point
b0_angles = numpy.angle(centered_path_points - centered_b0)
self.assertTrue(all(b0_angles[1:] >= 0))
centered_b1 = CPF.discriminant_points_complex[1] - CPF.base_point
b1_angles = numpy.angle(centered_path_points - centered_b1)
self.assertTrue(all(b1_angles[1:] >= 0))
centered_b2 = CPF.discriminant_points_complex[2] - CPF.base_point
b2_angles = numpy.angle(centered_path_points - centered_b2)
self.assertTrue(all(b2_angles[1:] >= 0))
# -1.j lies below all other discriminant points. make sure the points
# along the path all lie below the other discriminant point
path = CPF.path_to_discriminant_point(-1.j)
s = numpy.linspace(0, 1, 32)
path_points = path(s)
centered_path_points = path_points - CPF.base_point
b1_angles = numpy.angle(centered_path_points - centered_b1)
self.assertTrue(all(b1_angles[1:] <= 0))
b2_angles = numpy.angle(centered_path_points - centered_b2)
self.assertTrue(all(b2_angles[1:] <= 0))
centered_b3 = CPF.discriminant_points_complex[3] - CPF.base_point
b3_angles = numpy.angle(centered_path_points - centered_b3)
self.assertTrue(all(b3_angles[1:] <= 0))
def test_discriminant_points(self):
# these should be in the proper order
f = self.f1
CPF = ComplexPathFactory(f, -1)
discriminant_points = CPF.discriminant_points
self.assertItemsEqual(discriminant_points, [QQbar(0)])
f = self.f2
CPF = ComplexPathFactory(f, -2)
discriminant_points = CPF.discriminant_points
self.assertItemsEqual(discriminant_points, map(QQbar, [-I, I]))
f = self.f3
CPF = ComplexPathFactory(f, -2)
discriminant_points = CPF.discriminant_points
self.assertItemsEqual(discriminant_points,
map(QQbar,[-I, -1, 1, I]))
def test_discriminant_points(self):
# these should be in the proper order
f = self.f1
CPF = ComplexPathFactory(f, -1)
discriminant_points = CPF.discriminant_points
six.assertCountEqual(self, discriminant_points, [QQbar(0)])
f = self.f2
CPF = ComplexPathFactory(f, -2)
discriminant_points = CPF.discriminant_points
six.assertCountEqual(self, discriminant_points,
[QQbar(z) for z in [-I, I]])
f = self.f3
CPF = ComplexPathFactory(f, -2)
discriminant_points = CPF.discriminant_points
six.assertCountEqual(self, discriminant_points,
[QQbar(z) for z in [-I, -1, 1, I]])
def __init__(self, riemann_surface, base_point=None, base_sheets=None,
kappa=3./5.):
r"""Initialize the Riemann surface path factory.
The user can manually supply a base point in the complex x-plane as
well as an ordering of the base sheets at the base point. The base
point must be a regular point of the curve.
A "relaxation parameter" `kappa` is used when determining the radii of
the "bounding circles" around each discriminant point of the curve.
Paths in the complex plane inside of these bounding circles need to be
constructed with care since the y-roots of the curve are (possibly)
beginning to converge to one another.
Parameters
----------
RS : RiemannSurface
The Riemann surface.
kappa : double
A relaxation factor between 0 and 1.
base_point : complex
An optional user-defined base x-point for the mondromy group
and Riemann surface.
base_sheets : complex, list
An optional user-defined ordered list of y-roots above the
base point.
Returns
-------
None
"""
self.riemann_surface = riemann_surface
self.complex_path_factory = ComplexPathFactory(
riemann_surface.f, base_point=base_point, kappa=kappa)
# now that the compelx path factory is built we have a base point
# (either provided or chosen). we now need to determine the base
# sheets.
#
# if not provided then compute some. otherwise, check they are roots
if not base_sheets:
x,y = self.riemann_surface.f.parent().gens()
p = self.riemann_surface.f(self.base_point, y).univariate_polynomial()
roots = p.roots(CDF, multiplicities=False)
base_sheets = numpy.array(roots, dtype=numpy.complex)
else:
f = self.riemann_surface.f
for sheet in base_sheets:
value = f(self.base_point, sheet)
if abs(value) > 1e-15:
raise ValueError('Base sheets %s do not lie above base '
'point %s'%(base_sheets, self.base_point))
self._base_sheets = base_sheets
# skeleton is computed when first used
self._skeleton = None
def test_intersection_points_and_avoiding_arc(self):
CPF = ComplexPathFactory(self.f1, base_point=-2, kappa=1)
b = CPF.discriminant_points[0]
R = CPF.radius(b)
self.assertAlmostEqual(R, 1.0)
# straight line path from (-2,1) to (1,-2). this intersects the circle
# of radius 1 about the origin at (-1,0) goes around to (0,-1).
z0 = -2 + 1.j
z1 = 1 - 2.j
w0, w1 = CPF.intersection_points(z0, z1, b, R)
self.assertAlmostEqual(w0, -1)
self.assertAlmostEqual(w1, -1.j)
arc = CPF.avoiding_arc(w0, w1, b, R)
self.assertAlmostEqual(arc.R, 1)
self.assertAlmostEqual(arc.w, 0)
self.assertAlmostEqual(arc.theta, pi)
self.assertAlmostEqual(arc.dtheta, pi / 2)
def test_intersection_points_and_avoiding_arc(self):
CPF = ComplexPathFactory(self.f1, base_point=-2, kappa=1)
b = CPF.discriminant_points[0]
R = CPF.radius(b)
self.assertAlmostEqual(R, 1.0)
# straight line path from (-2,1) to (1,-2). this intersects the circle
# of radius 1 about the origin at (-1,0) goes around to (0,-1).
z0 = -2 + 1.j
z1 = 1 - 2.j
w0,w1 = CPF.intersection_points(z0, z1, b, R)
self.assertAlmostEqual(w0, -1)
self.assertAlmostEqual(w1, -1.j)
arc = CPF.avoiding_arc(w0, w1, b, R)
self.assertAlmostEqual(arc.R, 1)
self.assertAlmostEqual(arc.w, 0)
self.assertAlmostEqual(arc.theta, pi)
self.assertAlmostEqual(arc.dtheta, pi/2)
def test_kappa_error(self):
# tests that an error is raised if the base point lies in a bounding
# circle of a discriminant poin
f = self.f2
with self.assertRaises(ValueError):
CPF = ComplexPathFactory(f, 0.5j, kappa=1.0)
f = self.f3
with self.assertRaises(ValueError):
CPF = ComplexPathFactory(f, -1, kappa=1.0)
with self.assertRaises(ValueError):
CPF = ComplexPathFactory(f, -1.5, kappa=1.0)
CPF = ComplexPathFactory(f, -2, kappa=1.0)
with self.assertRaises(ValueError):
CPF = ComplexPathFactory(f, -1, kappa=0.5)
with self.assertRaises(ValueError):
CPF = ComplexPathFactory(f, -1.25, kappa=0.5)
CPF = ComplexPathFactory(f, -1.5, kappa=0.5)
def test_path_to_discriminant_point(self):
#
f = self.f1
CPF = ComplexPathFactory(f, -2, kappa=1.0)
path = CPF.path_to_discriminant_point(0)
self.assertEqual(path, ComplexLine(-2,-1))
#
f = self.f2
CPF = ComplexPathFactory(f, -1, kappa=1.0)
path = CPF.path_to_discriminant_point(-1.j)
z = -sqrt(2.0)/2.0 - (1-sqrt(2.0)/2.0)*1.j
w = path(1.0)
self.assertAlmostEqual(z,w)
path = CPF.path_to_discriminant_point(1.j)
z = z.conjugate()
w = path(1.0)
self.assertAlmostEqual(z,w)
# distance between 4th roots of unity = sqrt(2)
# kappa = 0.5 ==> radius = sqrt(2)/4.0
f = self.f3
CPF = ComplexPathFactory(f, -2, kappa=0.5)
path = CPF.path_to_discriminant_point(-1.j)
alpha = 1.0-sqrt(10.0)/20.0
z = (2*alpha-2) - alpha*1.0j
w = path(1.0)
self.assertAlmostEqual(z,w)
path = CPF.path_to_discriminant_point(1.j)
z = z.conjugate()
w = path(1.0)
self.assertAlmostEqual(z,w)
class RiemannSurfacePathFactory(object):
r"""Factory class for constructing paths on the Riemann surface.
Attributes
----------
RS : RiemannSurface
The Riemann surface.
complex_path_factory : ComplexPathFactory
A factory object for determining how to navigate the complex
x-plane. How to construct paths avoiding, encircling, and
approaching discriminant points.
skeleton : Skeleton
An object for determining how to navigate the complex y-plane. Computes
the branching structure and homology of the surface. Defines how to
swap sheets.
_base_sheets : list
The ordered sheets of the surface at the base point.
_monodromy_group : list
The monodromy group of the curve. Used to compute the homology.
Methods
-------
.. autosummary::
base_point
base_sheets
base_place
discriminant_points
closest_discriminant_point
branch_points
path_to_place
monodromy_group
monodromy_path
a_cycles
b_cycles
c_cycles
show_paths
RiemannSurfacePath_from_cycle
RiemannSurfacePath_from_complex_path
_path_to_discriminant_place
_path_to_regular_
"""
@property
def base_point(self):
return self.complex_path_factory.base_point
@property
def base_sheets(self):
return self._base_sheets
@property
def base_place(self):
return RegularPlace(self, self.base_point, self.base_sheets[0])
@property
def branch_points(self):
return self.monodromy_group()[0]
@property
def discriminant_points(self):
return self.complex_path_factory.discriminant_points
@property
def skeleton(self):
if not self._skeleton:
mon = (self.base_point, self.base_sheets) + \
self.monodromy_group()
self._skeleton = Skeleton(self.riemann_surface, mon)
return self._skeleton
def __init__(self,
riemann_surface,
base_point=None,
base_sheets=None,
kappa=3. / 5.):
r"""Initialize the Riemann surface path factory.
The user can manually supply a base point in the complex x-plane as
well as an ordering of the base sheets at the base point. The base
point must be a regular point of the curve.
A "relaxation parameter" `kappa` is used when determining the radii of
the "bounding circles" around each discriminant point of the curve.
Paths in the complex plane inside of these bounding circles need to be
constructed with care since the y-roots of the curve are (possibly)
beginning to converge to one another.
Parameters
----------
RS : RiemannSurface
The Riemann surface.
kappa : double
A relaxation factor between 0 and 1.
base_point : complex
An optional user-defined base x-point for the mondromy group
and Riemann surface.
base_sheets : complex, list
An optional user-defined ordered list of y-roots above the
base point.
Returns
-------
None
"""
self.riemann_surface = riemann_surface
self.complex_path_factory = ComplexPathFactory(riemann_surface.f,
base_point=base_point,
kappa=kappa)
# now that the compelx path factory is built we have a base point
# (either provided or chosen). we now need to determine the base
# sheets.
#
# if not provided then compute some. otherwise, check they are roots
if not base_sheets:
x, y = self.riemann_surface.f.parent().gens()
p = self.riemann_surface.f(self.base_point,
y).univariate_polynomial()
roots = p.roots(CDF, multiplicities=False)
base_sheets = numpy.array(roots, dtype=numpy.complex)
else:
f = self.riemann_surface.f
for sheet in base_sheets:
value = f(self.base_point, sheet)
if abs(value) > 1e-15:
raise ValueError('Base sheets %s do not lie above base '
'point %s' %
(base_sheets, self.base_point))
self._base_sheets = base_sheets
# skeleton is computed when first used
self._skeleton = None
def __repr__(self):
s = 'Path Factory for %s' % (self.riemann_surface)
return s
def show_paths(self, *args, **kwds):
r"""Plots all of the monodromy paths of the curve.
Parameters
----------
args : list
See :func:`ComplexPathFactory.show_paths`
kwds : dict
See :func:`ComplexPathFactory.show_paths`
Returns
-------
plt : Sage plot
A plot of the monodromy paths.
"""
plt = self.complex_path_factory.show_paths(*args, **kwds)
return plt
def closest_discriminant_point(self, x, **kwds):
r"""Returns the discriminant point closest to `x`.
See Also
--------
:func:`ComplexPathFactory.closest_discriminant_point`
"""
b = self.complex_path_factory.closest_discriminant_point(x)
return b
def path_to_place(self, P):
r"""Returns a path to a specified place `P` on the Riemann surface.
Parameters
----------
P : complex tuple
A place on the Riemann surface.
See Also
--------
:func:`PathFactory._path_to_infinite_place`
:func:`PathFactory._path_to_discriminant_place`
:func:`PathFactory._path_to_regular_place`
"""
if P.is_infinite():
return self._path_to_infinite_place(P)
elif P.is_discriminant():
return self._path_to_discriminant_place(P)
else:
return self._path_to_regular_place(P)
def _path_to_infinite_place(self, P):
r"""Returns a path to a place at an infintiy of the surface.
A place at infinity is one where the `x`-projection of the place is the
point `x = \infty` of the complex Riemann sphere. An infinite place is
a type of discirminant place.
Parameters
----------
P : Place
The target infinite place.
Returns
-------
gamma : RiemannSurfacePath
A path from the base place to the place at infinity.
"""
# determine a place Q from where we can reach the target place P. first,
# pick an appropriate x-point over which a Q is chosen
x0 = self.base_point
if numpy.real(x0) < 0:
xa = 5 * x0 # move away from the origin
else:
xa = -5 # arbitrary choice away from the origin
# next, determine an appropriate y-part from where we can reach the
# place at infinity
p = P.puiseux_series
center, coefficient, ramification_index = p.xdata
ta = CC(xa / coefficient).nth_root(abs(ramification_index))
ta = ta if ramification_index > 0 else 1 / ta
p.extend_to_t(ta)
ya = complex(p.eval_y(ta))
# construct the place Q and compute the path going from P0 to Q
Q = self.riemann_surface(xa, ya)
gamma_P0_to_Q = self.path_to_place(Q)
# construct the path going from Q to P
xend = complex(gamma_P0_to_Q.get_x(1.0))
yend = array(gamma_P0_to_Q.get_y(1.0), dtype=complex)
gamma_x = ComplexRay(xend)
gamma_Q_to_P = RiemannSurfacePathPuiseux(self.riemann_surface, gamma_x,
yend)
gamma = gamma_P0_to_Q + gamma_Q_to_P
return gamma
def _path_to_discriminant_place(self, P):
r"""Returns a path to a discriminant place on the surface.
A "discriminant" place :math:`P` is a place on the Riemann
surface where Puiseux series are required to determine the x-
and y-projections of the place.
Parameters
----------
P : Place
A place on the Riemann surface whose x-projection is a discriminant
point of the curve.
Returns
-------
gamma : RiemannSurfacePath
A path from the base place to the discriminant place `P`.
"""
# compute a valid y-value at x=a=b-R, where b is the
# discriminant point center of the series and R is the radius of
# bounding circle around b, such that analytically continuing
# from that (x,y) to x=b will reach the designated place
p = P.puiseux_series
center, coefficient, ramification_index = p.xdata
R = self.complex_path_factory.radius(center)
a = center - R
t = (-R / coefficient)**(1.0 / ramification_index)
# p.coerce_to_numerical()
p.extend_to_t(t)
y = p.eval_y(t)
# construct a path going from the base place to this regular
# place to the left of the target discriminant point
P1 = self.riemann_surface(a, y)
gamma1 = self._path_to_regular_place(P1)
# construct the RiemannSurfacePath going from this regular place to the
# discriminant point.
xend = complex(gamma1.get_x(1.0))
yend = array(gamma1.get_y(1.0), dtype=complex)
gamma_x = ComplexLine(complex(a), complex(center))
segment = RiemannSurfacePathPuiseux(self.riemann_surface, gamma_x,
yend)
gamma = gamma1 + segment
return gamma
def _path_to_regular_place(self, P):
r"""Returns a path to a regular place on the surface.
A "regular" place :math:`P` is a place on the Riemann surface
where the x-projection of the place is not a discriminant point
of the curve.
Parameters
----------
P : Place
A place on the Riemann surface whose x-projection is a regular
point of the curve.
Returns
-------
gamma : RiemannSurfacePath
A path from the base place to the regular place `P`.
"""
# construct the reverse path from the place to the base x-point in
# order to determine which sheet the place is on
P0 = self.base_place
gamma_x = self.complex_path_factory.path(P0.x, P.x)
gamma = self.RiemannSurfacePath_from_complex_path(gamma_x)
# determine the index of the sheet (sheet_index) of the base place
# continues to the y-component of the target
base_sheets = self.base_sheets
end_sheets = array(gamma.get_y(1.0), dtype=complex)
end_diffs = abs(end_sheets - complex(P.y))
if min(end_diffs) > 1.0e-8:
raise ValueError('Error in constructing Abel path: end of regular '
'path does not match with target place.')
sheet_index = numpy.argmin(end_diffs)
# if this sheet index is zero (the base sheet) then simply return the
# constructed path gamma. otherwise, construct the branch switching
# path
if sheet_index != 0:
ypath = self.skeleton.ypath_from_base_to_sheet(sheet_index)
gamma_swap = self.RiemannSurfacePath_from_cycle(ypath)
ordered_base_sheets = numpy.array(gamma_swap.get_y(1.0),
dtype=complex)
# take the new ordering of branch points and construct the
# path to the target place. there is an inherent check that
# the y-values above the base point match
gamma_rest = self.RiemannSurfacePath_from_complex_path(
gamma_x, y0=ordered_base_sheets)
gamma = gamma_swap + gamma_rest
# sanity check
yend = gamma.get_y(1.0)[0]
if numpy.abs(yend - numpy.complex(P.y)) > 1.0e-8:
raise ValueError('Error in constructing Abel path.')
return gamma
def _path_near_discriminant_point(self, P):
"""Returns a path to a place `P` where the x-coordinate is a
discriminant point on the surface.
"""
raise NotImplementedError()
@cached_method
def monodromy_group(self):
"""Returns the monodromy group of the algebraic curve defining the
Riemann surface.
Returns
-------
list : list
A list containing the base point, base sheets, branch points, and
corresponding monodromy group elements.
"""
x0 = self.base_point
y0 = self.base_sheets
# compute the monodromy element for each discriminant point. if
# it's the identity permutation, don't add to monodromy group
branch_points = []
permutations = []
for bi in self.discriminant_points:
gamma = self.monodromy_path(bi)
# compute the end fibre and the corresponding permutation
yend = array(gamma.get_y(1.0), dtype=complex)
phi = matching_permutation(y0, yend)
# add the point to the monodromy group if the permutation is
# not the identity.
if not phi.is_identity():
branch_points.append(bi)
permutations.append(phi)
# compute the monodromy element of the point at infinity
gamma_x = self.complex_path_factory.monodromy_path_infinity()
gamma = self.RiemannSurfacePath_from_complex_path(gamma_x, x0, y0)
yend = array(gamma.get_y(1.0), dtype=complex)
phi_oo = matching_permutation(y0, yend)
# sanity check: the product of the finite branch point permutations
# should be equal to the inverse of the permutation at infinity
phi_prod = reduce(lambda phi1, phi2: phi2 * phi1, permutations)
phi_prod_all = phi_prod * phi_oo
if not phi_prod_all.is_identity():
raise ValueError('Contradictory permutation at infinity.')
# add infinity if it's a branch point
if not phi_oo.is_identity():
branch_points.append(infinity)
permutations.append(phi_oo)
return branch_points, permutations
def monodromy_path(self, bi, nrots=1):
"""Returns the monodromy path around the discriminant point `bi`.
Parameters
----------
bi : complex
A discriminant point of the curve.
nrots : optional, integer
The number of times to rotate around `bi`. (Default 1).
Returns
-------
RiemannSurfacePath
The path encircling the branch point `bi`.
"""
gammax = self.complex_path_factory.monodromy_path(bi, nrots=nrots)
gamma = self.RiemannSurfacePath_from_complex_path(gammax)
return gamma
def a_cycles(self):
"""Returns the a-cycles on the Riemann surface.
Returns
-------
list, RiemannSurfacePath
A list of the a-cycles of the Riemann surface as
:class:`RiemannSurfacePath` objects.
"""
cycles = self.skeleton.a_cycles()
paths = map(self.RiemannSurfacePath_from_cycle, cycles)
return paths
def b_cycles(self):
"""Returns the b-cycles on the Riemann surface.
Returns
-------
list, RiemannSurfacePath
A list of the b-cycles of the Riemann surface as
:class:`RiemannSurfacePath` objects.
"""
cycles = self.skeleton.b_cycles()
paths = map(self.RiemannSurfacePath_from_cycle, cycles)
return paths
def c_cycles(self):
"""Returns the c-cycles of the Riemann surface and the linear
combination matrix defining the a- and b-cycles from the c-cycles.
The a- and b- cycles of the Riemann surface are formed from linear
combinations of the c-cycles. These linear combinations are obtained
from the :py::meth:`linear_combinations` method.
.. note::
It may be computationally more efficient to integrate over
the (necessary) c-cycles and take linear combinations of the
results than to integrate over the a- and b-cycles
separately. Sometimes the column rank of the linear
combination matrix (that is, the number of c-cycles used to
construct a- and b-cycles) is lower than the size of the
homology group and sometimes the c-cycles are simpler and
shorter than the homology cycles.
Returns
-------
list, RiemannSurfacePath
A list of the c-cycles of the Riemann surface as
:class:`RiemannSurfacePath` objects.
"""
# prune the linear combinations matrix of the columns that don't
# contribute to a c-cycles
cycles, linear_combinations = self.skeleton.c_cycles()
ncols = linear_combinations.shape[1]
indices = [
j for j in range(ncols)
if not (linear_combinations[:, j] == 0).all()
]
c_cycles = [cycles[i] for i in indices]
linear_combinations = linear_combinations[:, indices]
paths = map(self.RiemannSurfacePath_from_cycle, c_cycles)
return paths, linear_combinations
def RiemannSurfacePath_from_cycle(self, cycle):
"""Constructs a :class:`RiemannSurfacePath` object from a list of
x-path data.
Parameters
----------
ypath : list
A list fo tuples defining a y-path. The output of
`self.a_cycles()`, `self.b_cycles()`, etc.
"""
segments = []
for bi, nrots in cycle:
gammax_i = self.complex_path_factory.monodromy_path(bi,
nrots=nrots)
segments.extend(gammax_i.segments)
gammax = ComplexPath(segments)
gamma = self.RiemannSurfacePath_from_complex_path(gammax)
return gamma
def RiemannSurfacePath_from_complex_path(self,
complex_path,
x0=None,
y0=None):
r"""Constructs a :class:`RiemannSurfacePath` object from x-path data.
Parameters
----------
complex_path : ComplexPath
A complex path.
x0 : complex (default `self.base_point`)
The starting x-point of the path.
y0 : complex list (default `self.base_sheets`)
The starting ordering of the y-sheets.
Returns
-------
RiemannSurfacePath
A path on the Riemann surface with the prescribed x-path.
"""
if x0 is None:
x0 = self.base_point
if y0 is None:
y0 = self.base_sheets
# coerce and assert that x0,y0 lies on the path and curve
x0 = complex(x0)
y0 = array(y0, dtype=complex)
if abs(x0 - complex_path(0)) > 1e-7:
raise ValueError('The point %s is not at the start of the '
'ComplexPath %s' % (x0, complex_path))
f = self.riemann_surface.f
curve_error = [abs(complex(f(x0, y0k))) for y0k in y0]
if max(curve_error) > 1e-7:
raise ValueError('The fibre %s above %s does not lie on the '
'curve %s' % (y0.tolist(), x0, f))
# build a list of path segments from each tuple of xdata. build a line
# segment or arc depending on xdata input
x0_segment = x0
y0_segment = y0
segments = []
for segment_x in complex_path.segments:
# for each segment determine if we're far enough away to use Smale
# alpha theory paths or if we have to use Puiseux. we add a
# relazation factor to the radius to account for monodromy paths
xend = segment_x(1.0)
b = self.complex_path_factory.closest_discriminant_point(xend)
R = self.complex_path_factory.radius(b)
if abs(xend - b) > 0.9 * R:
segment = RiemannSurfacePathSmale(self.riemann_surface,
segment_x, y0_segment)
else:
segment = RiemannSurfacePathPuiseux(self.riemann_surface,
segment_x, y0_segment)
# determine the starting place of the next segment
x0_segment = segment.get_x(1.0)
y0_segment = segment.get_y(1.0)
segments.append(segment)
# build the entire path from the path segments
gamma = RiemannSurfacePath(self.riemann_surface, complex_path, y0,
segments)
return gamma
def test_monodromy_path_infinity(self):
f = self.f3
CPF = ComplexPathFactory(f, -3, kappa=0.5)
gamma = CPF.monodromy_path_infinity()
self.assertAlmostEqual(gamma(0), CPF.base_point)
self.assertAlmostEqual(gamma(1.0), CPF.base_point)
def __init__(self,
riemann_surface,
base_point=None,
base_sheets=None,
kappa=3. / 5.):
r"""Initialize the Riemann surface path factory.
The user can manually supply a base point in the complex x-plane as
well as an ordering of the base sheets at the base point. The base
point must be a regular point of the curve.
A "relaxation parameter" `kappa` is used when determining the radii of
the "bounding circles" around each discriminant point of the curve.
Paths in the complex plane inside of these bounding circles need to be
constructed with care since the y-roots of the curve are (possibly)
beginning to converge to one another.
Parameters
----------
RS : RiemannSurface
The Riemann surface.
kappa : double
A relaxation factor between 0 and 1.
base_point : complex
An optional user-defined base x-point for the mondromy group
and Riemann surface.
base_sheets : complex, list
An optional user-defined ordered list of y-roots above the
base point.
Returns
-------
None
"""
self.riemann_surface = riemann_surface
self.complex_path_factory = ComplexPathFactory(riemann_surface.f,
base_point=base_point,
kappa=kappa)
# now that the compelx path factory is built we have a base point
# (either provided or chosen). we now need to determine the base
# sheets.
#
# if not provided then compute some. otherwise, check they are roots
if not base_sheets:
x, y = self.riemann_surface.f.parent().gens()
p = self.riemann_surface.f(self.base_point,
y).univariate_polynomial()
roots = p.roots(CDF, multiplicities=False)
base_sheets = numpy.array(roots, dtype=numpy.complex)
else:
f = self.riemann_surface.f
for sheet in base_sheets:
value = f(self.base_point, sheet)
if abs(value) > 1e-15:
raise ValueError('Base sheets %s do not lie above base '
'point %s' %
(base_sheets, self.base_point))
self._base_sheets = base_sheets
# skeleton is computed when first used
self._skeleton = None
def test_kappa(self):
f = self.f1
CPF = ComplexPathFactory(f, -1, kappa=1.0)
self.assertAlmostEqual(CPF.radius(0), 0.5)
CPF = ComplexPathFactory(f, -1, kappa=0.5)
self.assertAlmostEqual(CPF.radius(0), 0.25)
# two disc points \pm I: distance between them = 2
f = self.f2
CPF = ComplexPathFactory(f, -1, kappa=1.0)
self.assertAlmostEqual(CPF.radius(-1.j), 1.0)
self.assertAlmostEqual(CPF.radius(1.j), 1.0)
CPF = ComplexPathFactory(f, -1, kappa=0.5)
self.assertAlmostEqual(CPF.radius(-1.j), 0.5)
self.assertAlmostEqual(CPF.radius(1.j), 0.5)
# four disc points at 4th roots of unity: distance between them =
# sqrt(2)
f = self.f3
CPF = ComplexPathFactory(f, -2, kappa=1.0)
self.assertAlmostEqual(CPF.radius(-1.j), sqrt(2.0) / 2)
self.assertAlmostEqual(CPF.radius(-1), sqrt(2.0) / 2)
self.assertAlmostEqual(CPF.radius(1), sqrt(2.0) / 2)
self.assertAlmostEqual(CPF.radius(1.j), sqrt(2.0) / 2)
CPF = ComplexPathFactory(f, -2, kappa=0.5)
self.assertAlmostEqual(CPF.radius(-1.j), sqrt(2.0) / 4)
self.assertAlmostEqual(CPF.radius(-1), sqrt(2.0) / 4)
self.assertAlmostEqual(CPF.radius(1), sqrt(2.0) / 4)
self.assertAlmostEqual(CPF.radius(1.j), sqrt(2.0) / 4)
def test_kappa(self):
f = self.f1
CPF = ComplexPathFactory(f, -1, kappa=1.0)
self.assertAlmostEqual(CPF.radius(0), 0.5)
CPF = ComplexPathFactory(f, -1, kappa=0.5)
self.assertAlmostEqual(CPF.radius(0), 0.25)
# two disc points \pm I: distance between them = 2
f = self.f2
CPF = ComplexPathFactory(f, -1, kappa=1.0)
self.assertAlmostEqual(CPF.radius(-1.j), 1.0)
self.assertAlmostEqual(CPF.radius(1.j), 1.0)
CPF = ComplexPathFactory(f, -1, kappa=0.5)
self.assertAlmostEqual(CPF.radius(-1.j), 0.5)
self.assertAlmostEqual(CPF.radius(1.j), 0.5)
# four disc points at 4th roots of unity: distance between them =
# sqrt(2)
f = self.f3
CPF = ComplexPathFactory(f, -2, kappa=1.0)
self.assertAlmostEqual(CPF.radius(-1.j), sqrt(2.0)/2)
self.assertAlmostEqual(CPF.radius(-1), sqrt(2.0)/2)
self.assertAlmostEqual(CPF.radius(1), sqrt(2.0)/2)
self.assertAlmostEqual(CPF.radius(1.j), sqrt(2.0)/2)
CPF = ComplexPathFactory(f, -2, kappa=0.5)
self.assertAlmostEqual(CPF.radius(-1.j), sqrt(2.0)/4)
self.assertAlmostEqual(CPF.radius(-1), sqrt(2.0)/4)
self.assertAlmostEqual(CPF.radius(1), sqrt(2.0)/4)
self.assertAlmostEqual(CPF.radius(1.j), sqrt(2.0)/4)
class RiemannSurfacePathFactory(object):
r"""Factory class for constructing paths on the Riemann surface.
Attributes
----------
RS : RiemannSurface
The Riemann surface.
complex_path_factory : ComplexPathFactory
A factory object for determining how to navigate the complex
x-plane. How to construct paths avoiding, encircling, and
approaching discriminant points.
skeleton : Skeleton
An object for determining how to navigate the complex y-plane. Computes
the branching structure and homology of the surface. Defines how to
swap sheets.
_base_sheets : list
The ordered sheets of the surface at the base point.
_monodromy_group : list
The monodromy group of the curve. Used to compute the homology.
Methods
-------
.. autosummary::
base_point
base_sheets
base_place
discriminant_points
closest_discriminant_point
branch_points
path_to_place
monodromy_group
monodromy_path
a_cycles
b_cycles
c_cycles
show_paths
RiemannSurfacePath_from_cycle
RiemannSurfacePath_from_complex_path
_path_to_discriminant_place
_path_to_regular_
"""
@property
def base_point(self):
return self.complex_path_factory.base_point
@property
def base_sheets(self):
return self._base_sheets
@property
def base_place(self):
return RegularPlace(self, self.base_point, self.base_sheets[0])
@property
def branch_points(self):
return self.monodromy_group()[0]
@property
def discriminant_points(self):
return self.complex_path_factory.discriminant_points
@property
def skeleton(self):
if not self._skeleton:
mon = (self.base_point, self.base_sheets) + \
self.monodromy_group()
self._skeleton = Skeleton(self.riemann_surface, mon)
return self._skeleton
def __init__(self, riemann_surface, base_point=None, base_sheets=None,
kappa=3./5.):
r"""Initialize the Riemann surface path factory.
The user can manually supply a base point in the complex x-plane as
well as an ordering of the base sheets at the base point. The base
point must be a regular point of the curve.
A "relaxation parameter" `kappa` is used when determining the radii of
the "bounding circles" around each discriminant point of the curve.
Paths in the complex plane inside of these bounding circles need to be
constructed with care since the y-roots of the curve are (possibly)
beginning to converge to one another.
Parameters
----------
RS : RiemannSurface
The Riemann surface.
kappa : double
A relaxation factor between 0 and 1.
base_point : complex
An optional user-defined base x-point for the mondromy group
and Riemann surface.
base_sheets : complex, list
An optional user-defined ordered list of y-roots above the
base point.
Returns
-------
None
"""
self.riemann_surface = riemann_surface
self.complex_path_factory = ComplexPathFactory(
riemann_surface.f, base_point=base_point, kappa=kappa)
# now that the compelx path factory is built we have a base point
# (either provided or chosen). we now need to determine the base
# sheets.
#
# if not provided then compute some. otherwise, check they are roots
if not base_sheets:
x,y = self.riemann_surface.f.parent().gens()
p = self.riemann_surface.f(self.base_point, y).univariate_polynomial()
roots = p.roots(CDF, multiplicities=False)
base_sheets = numpy.array(roots, dtype=numpy.complex)
else:
f = self.riemann_surface.f
for sheet in base_sheets:
value = f(self.base_point, sheet)
if abs(value) > 1e-15:
raise ValueError('Base sheets %s do not lie above base '
'point %s'%(base_sheets, self.base_point))
self._base_sheets = base_sheets
# skeleton is computed when first used
self._skeleton = None
def __repr__(self):
s = 'Path Factory for %s'%(self.riemann_surface)
return s
def show_paths(self, *args, **kwds):
r"""Plots all of the monodromy paths of the curve.
Parameters
----------
args : list
See :func:`ComplexPathFactory.show_paths`
kwds : dict
See :func:`ComplexPathFactory.show_paths`
Returns
-------
plt : Sage plot
A plot of the monodromy paths.
"""
plt = self.complex_path_factory.show_paths(*args, **kwds)
return plt
def closest_discriminant_point(self, x, **kwds):
r"""Returns the discriminant point closest to `x`.
See Also
--------
:func:`ComplexPathFactory.closest_discriminant_point`
"""
b = self.complex_path_factory.closest_discriminant_point(x)
return b
def path_to_place(self, P):
r"""Returns a path to a specified place `P` on the Riemann surface.
Parameters
----------
P : complex tuple
A place on the Riemann surface.
See Also
--------
:func:`PathFactory._path_to_infinite_place`
:func:`PathFactory._path_to_discriminant_place`
:func:`PathFactory._path_to_regular_place`
"""
if P.is_infinite():
return self._path_to_infinite_place(P)
elif P.is_discriminant():
return self._path_to_discriminant_place(P)
else:
return self._path_to_regular_place(P)
def _path_to_infinite_place(self, P):
r"""Returns a path to a place at an infintiy of the surface.
A place at infinity is one where the `x`-projection of the place is the
point `x = \infty` of the complex Riemann sphere. An infinite place is
a type of discirminant place.
Parameters
----------
P : Place
The target infinite place.
Returns
-------
gamma : RiemannSurfacePath
A path from the base place to the place at infinity.
"""
# determine a place Q from where we can reach the target place P. first,
# pick an appropriate x-point over which a Q is chosen
x0 = self.base_point
if numpy.real(x0) < 0:
xa = 5*x0 # move away from the origin
else:
xa = -5 # arbitrary choice away from the origin
# next, determine an appropriate y-part from where we can reach the
# place at infinity
p = P.puiseux_series
center, coefficient, ramification_index = p.xdata
ta = CC(xa/coefficient).nth_root(abs(ramification_index))
ta = ta if ramification_index > 0 else 1/ta
p.extend_to_t(ta)
ya = complex(p.eval_y(ta))
# construct the place Q and compute the path going from P0 to Q
Q = self.riemann_surface(xa,ya)
gamma_P0_to_Q = self.path_to_place(Q)
# construct the path going from Q to P
xend = complex(gamma_P0_to_Q.get_x(1.0))
yend = array(gamma_P0_to_Q.get_y(1.0), dtype=complex)
gamma_x = ComplexRay(xend)
gamma_Q_to_P = RiemannSurfacePathPuiseux(
self.riemann_surface, gamma_x, yend)
gamma = gamma_P0_to_Q + gamma_Q_to_P
return gamma
def _path_to_discriminant_place(self, P):
r"""Returns a path to a discriminant place on the surface.
A "discriminant" place :math:`P` is a place on the Riemann
surface where Puiseux series are required to determine the x-
and y-projections of the place.
Parameters
----------
P : Place
A place on the Riemann surface whose x-projection is a discriminant
point of the curve.
Returns
-------
gamma : RiemannSurfacePath
A path from the base place to the discriminant place `P`.
"""
# compute a valid y-value at x=a=b-R, where b is the
# discriminant point center of the series and R is the radius of
# bounding circle around b, such that analytically continuing
# from that (x,y) to x=b will reach the designated place
p = P.puiseux_series
center, coefficient, ramification_index = p.xdata
R = self.complex_path_factory.radius(center)
a = center - R
t = (-R/coefficient)**(1.0/ramification_index)
# p.coerce_to_numerical()
p.extend_to_t(t)
y = p.eval_y(t)
# construct a path going from the base place to this regular
# place to the left of the target discriminant point
P1 = self.riemann_surface(a,y)
gamma1 = self._path_to_regular_place(P1)
# construct the RiemannSurfacePath going from this regular place to the
# discriminant point.
xend = complex(gamma1.get_x(1.0))
yend = array(gamma1.get_y(1.0), dtype=complex)
gamma_x = ComplexLine(complex(a), complex(center))
segment = RiemannSurfacePathPuiseux(
self.riemann_surface, gamma_x, yend)
gamma = gamma1 + segment
return gamma
def _path_to_regular_place(self, P):
r"""Returns a path to a regular place on the surface.
A "regular" place :math:`P` is a place on the Riemann surface
where the x-projection of the place is not a discriminant point
of the curve.
Parameters
----------
P : Place
A place on the Riemann surface whose x-projection is a regular
point of the curve.
Returns
-------
gamma : RiemannSurfacePath
A path from the base place to the regular place `P`.
"""
# construct the reverse path from the place to the base x-point in
# order to determine which sheet the place is on
P0 = self.base_place
gamma_x = self.complex_path_factory.path(P0.x, P.x)
gamma = self.RiemannSurfacePath_from_complex_path(gamma_x)
# determine the index of the sheet (sheet_index) of the base place
# continues to the y-component of the target
base_sheets = self.base_sheets
end_sheets = array(gamma.get_y(1.0), dtype=complex)
end_diffs = abs(end_sheets - complex(P.y))
if min(end_diffs) > 1.0e-8:
raise ValueError('Error in constructing Abel path: end of regular '
'path does not match with target place.')
sheet_index = numpy.argmin(end_diffs)
# if this sheet index is zero (the base sheet) then simply return the
# constructed path gamma. otherwise, construct the branch switching
# path
if sheet_index != 0:
ypath = self.skeleton.ypath_from_base_to_sheet(sheet_index)
gamma_swap = self.RiemannSurfacePath_from_cycle(ypath)
ordered_base_sheets = numpy.array(gamma_swap.get_y(1.0), dtype=complex)
# take the new ordering of branch points and construct the
# path to the target place. there is an inherent check that
# the y-values above the base point match
gamma_rest = self.RiemannSurfacePath_from_complex_path(
gamma_x, y0=ordered_base_sheets)
gamma = gamma_swap + gamma_rest
# sanity check
yend = gamma.get_y(1.0)[0]
if numpy.abs(yend - numpy.complex(P.y)) > 1.0e-8:
raise ValueError('Error in constructing Abel path.')
return gamma
def _path_near_discriminant_point(self, P):
"""Returns a path to a place `P` where the x-coordinate is a
discriminant point on the surface.
"""
raise NotImplementedError()
@cached_method
def monodromy_group(self):
"""Returns the monodromy group of the algebraic curve defining the
Riemann surface.
Returns
-------
list : list
A list containing the base point, base sheets, branch points, and
corresponding monodromy group elements.
"""
x0 = self.base_point
y0 = self.base_sheets
# compute the monodromy element for each discriminant point. if
# it's the identity permutation, don't add to monodromy group
branch_points = []
permutations = []
for bi in self.discriminant_points:
gamma = self.monodromy_path(bi)
# compute the end fibre and the corresponding permutation
yend = array(gamma.get_y(1.0), dtype=complex)
phi = matching_permutation(y0, yend)
# add the point to the monodromy group if the permutation is
# not the identity.
if not phi.is_identity():
branch_points.append(bi)
permutations.append(phi)
# compute the monodromy element of the point at infinity
gamma_x = self.complex_path_factory.monodromy_path_infinity()
gamma = self.RiemannSurfacePath_from_complex_path(gamma_x, x0, y0)
yend = array(gamma.get_y(1.0), dtype=complex)
phi_oo = matching_permutation(y0, yend)
# sanity check: the product of the finite branch point permutations
# should be equal to the inverse of the permutation at infinity
phi_prod = reduce(lambda phi1,phi2: phi2*phi1, permutations)
phi_prod_all = phi_prod * phi_oo
if not phi_prod_all.is_identity():
raise ValueError('Contradictory permutation at infinity.')
# add infinity if it's a branch point
if not phi_oo.is_identity():
branch_points.append(infinity)
permutations.append(phi_oo)
return branch_points, permutations
def monodromy_path(self, bi, nrots=1):
"""Returns the monodromy path around the discriminant point `bi`.
Parameters
----------
bi : complex
A discriminant point of the curve.
nrots : optional, integer
The number of times to rotate around `bi`. (Default 1).
Returns
-------
RiemannSurfacePath
The path encircling the branch point `bi`.
"""
gammax = self.complex_path_factory.monodromy_path(bi, nrots=nrots)
gamma = self.RiemannSurfacePath_from_complex_path(gammax)
return gamma
def a_cycles(self):
"""Returns the a-cycles on the Riemann surface.
Returns
-------
list, RiemannSurfacePath
A list of the a-cycles of the Riemann surface as
:class:`RiemannSurfacePath` objects.
"""
cycles = self.skeleton.a_cycles()
paths = map(self.RiemannSurfacePath_from_cycle, cycles)
return paths
def b_cycles(self):
"""Returns the b-cycles on the Riemann surface.
Returns
-------
list, RiemannSurfacePath
A list of the b-cycles of the Riemann surface as
:class:`RiemannSurfacePath` objects.
"""
cycles = self.skeleton.b_cycles()
paths = map(self.RiemannSurfacePath_from_cycle, cycles)
return paths
def c_cycles(self):
"""Returns the c-cycles of the Riemann surface and the linear
combination matrix defining the a- and b-cycles from the c-cycles.
The a- and b- cycles of the Riemann surface are formed from linear
combinations of the c-cycles. These linear combinations are obtained
from the :py::meth:`linear_combinations` method.
.. note::
It may be computationally more efficient to integrate over
the (necessary) c-cycles and take linear combinations of the
results than to integrate over the a- and b-cycles
separately. Sometimes the column rank of the linear
combination matrix (that is, the number of c-cycles used to
construct a- and b-cycles) is lower than the size of the
homology group and sometimes the c-cycles are simpler and
shorter than the homology cycles.
Returns
-------
list, RiemannSurfacePath
A list of the c-cycles of the Riemann surface as
:class:`RiemannSurfacePath` objects.
"""
# prune the linear combinations matrix of the columns that don't
# contribute to a c-cycles
cycles, linear_combinations = self.skeleton.c_cycles()
ncols = linear_combinations.shape[1]
indices = [j for j in range(ncols)
if not (linear_combinations[:,j] == 0).all()]
c_cycles = [cycles[i] for i in indices]
linear_combinations = linear_combinations[:,indices]
paths = map(self.RiemannSurfacePath_from_cycle, c_cycles)
return paths, linear_combinations
def RiemannSurfacePath_from_cycle(self, cycle):
"""Constructs a :class:`RiemannSurfacePath` object from a list of
x-path data.
Parameters
----------
ypath : list
A list fo tuples defining a y-path. The output of
`self.a_cycles()`, `self.b_cycles()`, etc.
"""
segments = []
for bi, nrots in cycle:
gammax_i = self.complex_path_factory.monodromy_path(bi, nrots=nrots)
segments.extend(gammax_i.segments)
gammax = ComplexPath(segments)
gamma = self.RiemannSurfacePath_from_complex_path(gammax)
return gamma
def RiemannSurfacePath_from_complex_path(self, complex_path, x0=None, y0=None):
r"""Constructs a :class:`RiemannSurfacePath` object from x-path data.
Parameters
----------
complex_path : ComplexPath
A complex path.
x0 : complex (default `self.base_point`)
The starting x-point of the path.
y0 : complex list (default `self.base_sheets`)
The starting ordering of the y-sheets.
Returns
-------
RiemannSurfacePath
A path on the Riemann surface with the prescribed x-path.
"""
if x0 is None:
x0 = self.base_point
if y0 is None:
y0 = self.base_sheets
# coerce and assert that x0,y0 lies on the path and curve
x0 = complex(x0)
y0 = array(y0, dtype=complex)
if abs(x0 - complex_path(0)) > 1e-7:
raise ValueError('The point %s is not at the start of the '
'ComplexPath %s'%(x0, complex_path))
f = self.riemann_surface.f
curve_error = [abs(complex(f(x0,y0k))) for y0k in y0]
if max(curve_error) > 1e-7:
raise ValueError('The fibre %s above %s does not lie on the '
'curve %s'%(y0.tolist(), x0, f))
# build a list of path segments from each tuple of xdata. build a line
# segment or arc depending on xdata input
x0_segment = x0
y0_segment = y0
segments = []
for segment_x in complex_path.segments:
# for each segment determine if we're far enough away to use Smale
# alpha theory paths or if we have to use Puiseux. we add a
# relazation factor to the radius to account for monodromy paths
xend = segment_x(1.0)
b = self.complex_path_factory.closest_discriminant_point(xend)
R = self.complex_path_factory.radius(b)
if abs(xend - b) > 0.9*R:
segment = RiemannSurfacePathSmale(
self.riemann_surface, segment_x, y0_segment)
else:
segment = RiemannSurfacePathPuiseux(
self.riemann_surface, segment_x, y0_segment)
# determine the starting place of the next segment
x0_segment = segment.get_x(1.0)
y0_segment = segment.get_y(1.0)
segments.append(segment)
# build the entire path from the path segments
gamma = RiemannSurfacePath(self.riemann_surface, complex_path, y0,
segments)
return gamma
def test_path_to_discriminant_point(self):
#
f = self.f1
CPF = ComplexPathFactory(f, -2, kappa=1.0)
path = CPF.path_to_discriminant_point(0)
self.assertEqual(path, ComplexLine(-2, -1))
#
f = self.f2
CPF = ComplexPathFactory(f, -1, kappa=1.0)
path = CPF.path_to_discriminant_point(-1.j)
z = -sqrt(2.0) / 2.0 - (1 - sqrt(2.0) / 2.0) * 1.j
w = path(1.0)
self.assertAlmostEqual(z, w)
path = CPF.path_to_discriminant_point(1.j)
z = z.conjugate()
w = path(1.0)
self.assertAlmostEqual(z, w)
# distance between 4th roots of unity = sqrt(2)
# kappa = 0.5 ==> radius = sqrt(2)/4.0
f = self.f3
CPF = ComplexPathFactory(f, -2, kappa=0.5)
path = CPF.path_to_discriminant_point(-1.j)
alpha = 1.0 - sqrt(10.0) / 20.0
z = (2 * alpha - 2) - alpha * 1.0j
w = path(1.0)
self.assertAlmostEqual(z, w)
path = CPF.path_to_discriminant_point(1.j)
z = z.conjugate()
w = path(1.0)
self.assertAlmostEqual(z, w)
def test_monodromy_path_infinity(self):
f = self.f3
CPF = ComplexPathFactory(f, -3, kappa=0.5)
gamma = CPF.monodromy_path_infinity()
self.assertAlmostEqual(gamma(0), CPF.base_point)
self.assertAlmostEqual(gamma(1.0), CPF.base_point)