def test_gamma0(self):
# no-op
s = (0, 0, 1)
f, alpha, beta = _transform(self.f1, s)
self.assertEqual(self.f1, f)
f, alpha, beta = _transform(self.f2, s)
self.assertEqual(self.f2, f)
f, alpha, beta = _transform(self.f3, s)
self.assertEqual(self.f3, f)
def test_gamma0(self):
# no-op
s = (0,0,1)
f,alpha,beta = _transform(self.f1, s)
self.assertEqual(self.f1, f)
f,alpha,beta = _transform(self.f2, s)
self.assertEqual(self.f2, f)
f,alpha,beta = _transform(self.f3, s)
self.assertEqual(self.f3, f)
def recenter_curve(g, singular_point):
r"""Returns a curve centered at a given singular point.
Given a singular point :math:`(x : y : z) = (\alpha : \beta : \gamma)`on a
Riemann surface :func:`recenter_curve` returns an affine curve :math:`h =
h(u,v)` such that the singularity occurs at :math:`u = 0` where
* :math:`u,v = x,y` if :math:`\gamma = 1`
* :math:`u,v = x,z` if :math:`\gamma = 0`
:func:`recenter_curve` is written in such a way to preserve the base ring
of the original curve in the case when it's a polynomial ring. For example,
if :math:`g \in R[c][x,y]` then `h \in R[c][u,v]`.
See Also
--------
abelfunctions.singularities._transform : recenters a given curve at the
singular point such that the singularity occurs at :math:`u = u0`
"""
# recenter the curve and adjoint polynomial at the singular point: find
# the affine plane u,v such that the singularity occurs at u=0
gsing,u0,v0 = _transform(g,singular_point)
R = gsing.parent()
u,v = R.gens()
h = gsing(u+u0,v)
return h
def recenter_curve(g, singular_point):
r"""Returns a curve centered at a given singular point.
Given a singular point :math:`(x : y : z) = (\alpha : \beta : \gamma)`on a
Riemann surface :func:`recenter_curve` returns an affine curve :math:`h =
h(u,v)` such that the singularity occurs at :math:`u = 0` where
* :math:`u,v = x,y` if :math:`\gamma = 1`
* :math:`u,v = x,z` if :math:`\gamma = 0`
:func:`recenter_curve` is written in such a way to preserve the base ring
of the original curve in the case when it's a polynomial ring. For example,
if :math:`g \in R[c][x,y]` then `h \in R[c][u,v]`.
See Also
--------
abelfunctions.singularities._transform : recenters a given curve at the
singular point such that the singularity occurs at :math:`u = u0`
"""
# recenter the curve and adjoint polynomial at the singular point: find
# the affine plane u,v such that the singularity occurs at u=0
gsing,u0,v0 = _transform(g,singular_point)
R = gsing.parent()
u,v = R.gens()
h = gsing(u+u0,v)
return h
def test_beta0(self):
s = (1, 0, 0)
f, _, _ = _transform(self.f1, s)
u, v = f.parent().gens()
g = u**2 - u**2 * v + u**2 * v**2 - 2 * u * v + 1
self.assertEqual(f, g)
def test_alpha0(self):
s = (0, 1, 0)
f, _, _ = _transform(self.f1, s)
u, v = f.parent().gens()
g = u**2 - u * v + v**2 - 2 * u**2 * v + u**4
self.assertEqual(f, g)
def test_beta0(self):
s = (1,0,0)
f,_,_ = _transform(self.f1, s)
u,v = f.parent().gens()
g = u**2 - u**2*v + u**2*v**2 - 2*u*v + 1
self.assertEqual(f, g)
def test_alpha0(self):
s = (0,1,0)
f,_,_ = _transform(self.f1, s)
u,v = f.parent().gens()
g = u**2 - u*v + v**2 - 2*u**2*v + u**4
self.assertEqual(f, g)