def base_extend(self, S):
r"""
Returns the conic over ``S`` given by the same equation as ``self``.
EXAMPLES::
sage: c = Conic([1, 1, 1]); c
Projective Conic Curve over Rational Field defined by x^2 + y^2 + z^2
sage: c.has_rational_point()
False
sage: d = c.base_extend(QuadraticField(-1, 'i')); d
Projective Conic Curve over Number Field in i with defining polynomial x^2 + 1 with i = 1*I defined by x^2 + y^2 + z^2
sage: d.rational_point(algorithm = 'rnfisnorm')
(i : 1 : 0)
"""
if S in _Fields:
B = self.base_ring()
if B == S:
return self
if not S.has_coerce_map_from(B):
raise ValueError("No natural map from the base ring of self " \
"(= %s) to S (= %s)" % (self, S))
from .constructor import Conic
con = Conic([S(c) for c in self.coefficients()], \
self.variable_names())
if self._rational_point is not None:
pt = [S(c) for c in Sequence(self._rational_point)]
if not pt == [0,0,0]:
# The following line stores the point in the cache
# if (and only if) there is no point in the cache.
pt = con.point(pt)
return con
return ProjectivePlaneCurve.base_extend(self, S)
def point(self, v, check=True):
r"""
Constructs a point on ``self`` corresponding to the input ``v``.
If ``check`` is True, then checks if ``v`` defines a valid
point on ``self``.
If no rational point on ``self`` is known yet, then also caches the point
for use by ``self.rational_point()`` and ``self.parametrization()``.
EXAMPLES ::
sage: c = Conic([1, -1, 1])
sage: c.point([15, 17, 8])
(15/8 : 17/8 : 1)
sage: c.rational_point()
(15/8 : 17/8 : 1)
sage: d = Conic([1, -1, 1])
sage: d.rational_point()
(-1 : 1 : 0)
"""
if is_Vector(v):
v = Sequence(v)
p = ProjectivePlaneCurve.point(self, v, check=check)
if self._rational_point is None:
self._rational_point = p
return p
def point(self, v, check=True):
r"""
Constructs a point on ``self`` corresponding to the input ``v``.
If ``check`` is True, then checks if ``v`` defines a valid
point on ``self``.
If no rational point on ``self`` is known yet, then also caches the point
for use by ``self.rational_point()`` and ``self.parametrization()``.
EXAMPLES ::
sage: c = Conic([1, -1, 1])
sage: c.point([15, 17, 8])
(15/8 : 17/8 : 1)
sage: c.rational_point()
(15/8 : 17/8 : 1)
sage: d = Conic([1, -1, 1])
sage: d.rational_point()
(-1 : 1 : 0)
"""
if is_Vector(v):
v = Sequence(v)
p = ProjectivePlaneCurve.point(self, v, check=check)
if self._rational_point is None:
self._rational_point = p
return p
def base_extend(self, S):
r"""
Returns the conic over ``S`` given by the same equation as ``self``.
EXAMPLES::
sage: c = Conic([1, 1, 1]); c
Projective Conic Curve over Rational Field defined by x^2 + y^2 + z^2
sage: c.has_rational_point()
False
sage: d = c.base_extend(QuadraticField(-1, 'i')); d
Projective Conic Curve over Number Field in i with defining polynomial x^2 + 1 defined by x^2 + y^2 + z^2
sage: d.rational_point(algorithm = 'rnfisnorm')
(i : 1 : 0)
"""
if S in _Fields:
B = self.base_ring()
if B == S:
return self
if not S.has_coerce_map_from(B):
raise ValueError("No natural map from the base ring of self " \
"(= %s) to S (= %s)" % (self, S))
from .constructor import Conic
con = Conic([S(c) for c in self.coefficients()], \
self.variable_names())
if self._rational_point is not None:
pt = [S(c) for c in Sequence(self._rational_point)]
if not pt == [0,0,0]:
# The following line stores the point in the cache
# if (and only if) there is no point in the cache.
pt = con.point(pt)
return con
return ProjectivePlaneCurve.base_extend(self, S)
def __init__(self, A, f):
r"""
See ``Conic`` for full documentation.
EXAMPLES:
::
sage: c = Conic([1, 1, 1]); c
Projective Conic Curve over Rational Field defined by x^2 + y^2 + z^2
"""
ProjectivePlaneCurve.__init__(self, A, f)
self._coefficients = [f[(2,0,0)], f[(1,1,0)], f[(1,0,1)],
f[(0,2,0)], f[(0,1,1)], f[(0,0,2)]]
self._parametrization = None
self._diagonal_matrix = None
self._rational_point = None
def __init__(self, A, f):
r"""
See ``Conic`` for full documentation.
EXAMPLES:
::
sage: c = Conic([1, 1, 1]); c
Projective Conic Curve over Rational Field defined by x^2 + y^2 + z^2
"""
ProjectivePlaneCurve.__init__(self, A, f)
self._coefficients = [f[(2,0,0)], f[(1,1,0)], f[(1,0,1)],
f[(0,2,0)], f[(0,1,1)], f[(0,0,2)]]
self._parametrization = None
self._diagonal_matrix = None
self._rational_point = None
def hom(self, x, Y=None):
r"""
Return the scheme morphism from ``self`` to ``Y`` defined by ``x``.
Here ``x`` can be a matrix or a sequence of polynomials.
If ``Y`` is omitted, then a natural image is found if possible.
EXAMPLES:
Here are a few Morphisms given by matrices. In the first
example, ``Y`` is omitted, in the second example, ``Y`` is specified.
::
sage: c = Conic([-1, 1, 1])
sage: h = c.hom(Matrix([[1,1,0],[0,1,0],[0,0,1]])); h
Scheme morphism:
From: Projective Conic Curve over Rational Field defined by -x^2 + y^2 + z^2
To: Projective Conic Curve over Rational Field defined by -x^2 + 2*x*y + z^2
Defn: Defined on coordinates by sending (x : y : z) to
(x + y : y : z)
sage: h([-1, 1, 0])
(0 : 1 : 0)
sage: c = Conic([-1, 1, 1])
sage: d = Conic([4, 1, -1])
sage: c.hom(Matrix([[0, 0, 1/2], [0, 1, 0], [1, 0, 0]]), d)
Scheme morphism:
From: Projective Conic Curve over Rational Field defined by -x^2 + y^2 + z^2
To: Projective Conic Curve over Rational Field defined by 4*x^2 + y^2 - z^2
Defn: Defined on coordinates by sending (x : y : z) to
(1/2*z : y : x)
``ValueError`` is raised if the wrong codomain ``Y`` is specified:
::
sage: c = Conic([-1, 1, 1])
sage: c.hom(Matrix([[0, 0, 1/2], [0, 1, 0], [1, 0, 0]]), c)
Traceback (most recent call last):
...
ValueError: The matrix x (= [ 0 0 1/2]
[ 0 1 0]
[ 1 0 0]) does not define a map from self (= Projective Conic Curve over Rational Field defined by -x^2 + y^2 + z^2) to Y (= Projective Conic Curve over Rational Field defined by -x^2 + y^2 + z^2)
The identity map between two representations of the same conic:
::
sage: C = Conic([1,2,3,4,5,6])
sage: D = Conic([2,4,6,8,10,12])
sage: C.hom(identity_matrix(3), D)
Scheme morphism:
From: Projective Conic Curve over Rational Field defined by x^2 + 2*x*y + 4*y^2 + 3*x*z + 5*y*z + 6*z^2
To: Projective Conic Curve over Rational Field defined by 2*x^2 + 4*x*y + 8*y^2 + 6*x*z + 10*y*z + 12*z^2
Defn: Defined on coordinates by sending (x : y : z) to
(x : y : z)
An example not over the rational numbers:
::
sage: P.<t> = QQ[]
sage: C = Conic([1,0,0,t,0,1/t])
sage: D = Conic([1/t^2, 0, -2/t^2, t, 0, (t + 1)/t^2])
sage: T = Matrix([[t,0,1],[0,1,0],[0,0,1]])
sage: C.hom(T, D)
Scheme morphism:
From: Projective Conic Curve over Fraction Field of Univariate Polynomial Ring in t over Rational Field defined by x^2 + t*y^2 + 1/t*z^2
To: Projective Conic Curve over Fraction Field of Univariate Polynomial Ring in t over Rational Field defined by 1/t^2*x^2 + t*y^2 + (-2/t^2)*x*z + ((t + 1)/t^2)*z^2
Defn: Defined on coordinates by sending (x : y : z) to
(t*x + z : y : z)
"""
if is_Matrix(x):
from .constructor import Conic
y = x.inverse()
A = y.transpose()*self.matrix()*y
im = Conic(A)
if Y is None:
Y = im
elif not Y == im:
raise ValueError("The matrix x (= %s) does not define a " \
"map from self (= %s) to Y (= %s)" % \
(x, self, Y))
x = Sequence(x*vector(self.ambient_space().gens()))
return self.Hom(Y)(x, check = False)
return ProjectivePlaneCurve.hom(self, x, Y)
def hom(self, x, Y=None):
r"""
Return the scheme morphism from ``self`` to ``Y`` defined by ``x``.
Here ``x`` can be a matrix or a sequence of polynomials.
If ``Y`` is omitted, then a natural image is found if possible.
EXAMPLES:
Here are a few Morphisms given by matrices. In the first
example, ``Y`` is omitted, in the second example, ``Y`` is specified.
::
sage: c = Conic([-1, 1, 1])
sage: h = c.hom(Matrix([[1,1,0],[0,1,0],[0,0,1]])); h
Scheme morphism:
From: Projective Conic Curve over Rational Field defined by -x^2 + y^2 + z^2
To: Projective Conic Curve over Rational Field defined by -x^2 + 2*x*y + z^2
Defn: Defined on coordinates by sending (x : y : z) to
(x + y : y : z)
sage: h([-1, 1, 0])
(0 : 1 : 0)
sage: c = Conic([-1, 1, 1])
sage: d = Conic([4, 1, -1])
sage: c.hom(Matrix([[0, 0, 1/2], [0, 1, 0], [1, 0, 0]]), d)
Scheme morphism:
From: Projective Conic Curve over Rational Field defined by -x^2 + y^2 + z^2
To: Projective Conic Curve over Rational Field defined by 4*x^2 + y^2 - z^2
Defn: Defined on coordinates by sending (x : y : z) to
(1/2*z : y : x)
``ValueError`` is raised if the wrong codomain ``Y`` is specified:
::
sage: c = Conic([-1, 1, 1])
sage: c.hom(Matrix([[0, 0, 1/2], [0, 1, 0], [1, 0, 0]]), c)
Traceback (most recent call last):
...
ValueError: The matrix x (= [ 0 0 1/2]
[ 0 1 0]
[ 1 0 0]) does not define a map from self (= Projective Conic Curve over Rational Field defined by -x^2 + y^2 + z^2) to Y (= Projective Conic Curve over Rational Field defined by -x^2 + y^2 + z^2)
The identity map between two representations of the same conic:
::
sage: C = Conic([1,2,3,4,5,6])
sage: D = Conic([2,4,6,8,10,12])
sage: C.hom(identity_matrix(3), D)
Scheme morphism:
From: Projective Conic Curve over Rational Field defined by x^2 + 2*x*y + 4*y^2 + 3*x*z + 5*y*z + 6*z^2
To: Projective Conic Curve over Rational Field defined by 2*x^2 + 4*x*y + 8*y^2 + 6*x*z + 10*y*z + 12*z^2
Defn: Defined on coordinates by sending (x : y : z) to
(x : y : z)
An example not over the rational numbers:
::
sage: P.<t> = QQ[]
sage: C = Conic([1,0,0,t,0,1/t])
sage: D = Conic([1/t^2, 0, -2/t^2, t, 0, (t + 1)/t^2])
sage: T = Matrix([[t,0,1],[0,1,0],[0,0,1]])
sage: C.hom(T, D)
Scheme morphism:
From: Projective Conic Curve over Fraction Field of Univariate Polynomial Ring in t over Rational Field defined by x^2 + t*y^2 + 1/t*z^2
To: Projective Conic Curve over Fraction Field of Univariate Polynomial Ring in t over Rational Field defined by 1/t^2*x^2 + t*y^2 + (-2/t^2)*x*z + ((t + 1)/t^2)*z^2
Defn: Defined on coordinates by sending (x : y : z) to
(t*x + z : y : z)
"""
if is_Matrix(x):
from .constructor import Conic
y = x.inverse()
A = y.transpose()*self.matrix()*y
im = Conic(A)
if Y is None:
Y = im
elif not Y == im:
raise ValueError("The matrix x (= %s) does not define a " \
"map from self (= %s) to Y (= %s)" % \
(x, self, Y))
x = Sequence(x*vector(self.ambient_space().gens()))
return self.Hom(Y)(x, check = False)
return ProjectivePlaneCurve.hom(self, x, Y)