def affine_patch(self, i):
r"""
Return the `i^{th}` affine patch of this projective space.
This is an ambient affine space `\mathbb{A}^n_R,` where
`R` is the base ring of self, whose "projective embedding"
map is `1` in the `i^{th}` factor.
INPUT:
- ``i`` -- integer between 0 and dimension of self, inclusive.
OUTPUT:
- An ambient affine space with fixed projective_embedding map.
EXAMPLES::
sage: PP = ProjectiveSpace(5) / QQ
sage: AA = PP.affine_patch(2)
sage: AA
Affine Space of dimension 5 over Rational Field
sage: AA.projective_embedding()
Scheme morphism:
From: Affine Space of dimension 5 over Rational Field
To: Projective Space of dimension 5 over Rational Field
Defn: Defined on coordinates by sending (x0, x1, x2, x3, x4) to
(x0 : x1 : 1 : x2 : x3 : x4)
sage: AA.projective_embedding(0)
Scheme morphism:
From: Affine Space of dimension 5 over Rational Field
To: Projective Space of dimension 5 over Rational Field
Defn: Defined on coordinates by sending (x0, x1, x2, x3, x4) to
(1 : x0 : x1 : x2 : x3 : x4)
"""
i = int(i) # implicit type checking
n = self.dimension_relative()
if i < 0 or i > n:
raise ValueError("Argument i (= %s) must be between 0 and %s." %
(i, n))
try:
return self.__affine_patches[i]
except AttributeError:
self.__affine_patches = {}
except KeyError:
pass
from sage.schemes.affine.affine_space import AffineSpace
AA = AffineSpace(n, self.base_ring(), names='x')
AA._default_embedding_index = i
phi = AA.projective_embedding(i, self)
self.__affine_patches[i] = AA
return AA
def affine_patch(self, i):
r"""
Return the `i^{th}` affine patch of this projective space.
This is an ambient affine space `\mathbb{A}^n_R,` where
`R` is the base ring of self, whose "projective embedding"
map is `1` in the `i^{th}` factor.
INPUT:
- ``i`` -- integer between 0 and dimension of self, inclusive.
OUTPUT:
- An ambient affine space with fixed projective_embedding map.
EXAMPLES::
sage: PP = ProjectiveSpace(5) / QQ
sage: AA = PP.affine_patch(2)
sage: AA
Affine Space of dimension 5 over Rational Field
sage: AA.projective_embedding()
Scheme morphism:
From: Affine Space of dimension 5 over Rational Field
To: Projective Space of dimension 5 over Rational Field
Defn: Defined on coordinates by sending (x0, x1, x2, x3, x4) to
(x0 : x1 : 1 : x2 : x3 : x4)
sage: AA.projective_embedding(0)
Scheme morphism:
From: Affine Space of dimension 5 over Rational Field
To: Projective Space of dimension 5 over Rational Field
Defn: Defined on coordinates by sending (x0, x1, x2, x3, x4) to
(1 : x0 : x1 : x2 : x3 : x4)
"""
i = int(i) # implicit type checking
n = self.dimension_relative()
if i < 0 or i > n:
raise ValueError("Argument i (= %s) must be between 0 and %s."%(i, n))
try:
return self.__affine_patches[i]
except AttributeError:
self.__affine_patches = {}
except KeyError:
pass
from sage.schemes.affine.affine_space import AffineSpace
AA = AffineSpace(n, self.base_ring(), names='x')
AA._default_embedding_index = i
phi = AA.projective_embedding(i, self)
self.__affine_patches[i] = AA
return AA
def affine_patch(self, i, AA=None):
r"""
Return the `i^{th}` affine patch of this projective space.
This is an ambient affine space `\mathbb{A}^n_R,` where
`R` is the base ring of self, whose "projective embedding"
map is `1` in the `i^{th}` factor.
INPUT:
- ``i`` -- integer between 0 and dimension of self, inclusive.
- ``AA`` -- (default: None) ambient affine space, this is constructed
if it is not given.
OUTPUT:
- An ambient affine space with fixed projective_embedding map.
EXAMPLES::
sage: PP = ProjectiveSpace(5) / QQ
sage: AA = PP.affine_patch(2)
sage: AA
Affine Space of dimension 5 over Rational Field
sage: AA.projective_embedding()
Scheme morphism:
From: Affine Space of dimension 5 over Rational Field
To: Projective Space of dimension 5 over Rational Field
Defn: Defined on coordinates by sending (x0, x1, x2, x3, x4) to
(x0 : x1 : 1 : x2 : x3 : x4)
sage: AA.projective_embedding(0)
Scheme morphism:
From: Affine Space of dimension 5 over Rational Field
To: Projective Space of dimension 5 over Rational Field
Defn: Defined on coordinates by sending (x0, x1, x2, x3, x4) to
(1 : x0 : x1 : x2 : x3 : x4)
::
sage: P.<x,y> = ProjectiveSpace(QQ,1)
sage: P.affine_patch(0).projective_embedding(0).codomain() == P
True
"""
i = int(i) # implicit type checking
n = self.dimension_relative()
if i < 0 or i > n:
raise ValueError("Argument i (= %s) must be between 0 and %s." %
(i, n))
try:
A = self.__affine_patches[i]
#assume that if you've passed in a new affine space you want to override
#the existing patch
if AA is None or A == AA:
return (A)
except AttributeError:
self.__affine_patches = {}
except KeyError:
pass
#if no ith patch exists, we may still be here with AA==None
if AA == None:
from sage.schemes.affine.affine_space import AffineSpace
AA = AffineSpace(n, self.base_ring(), names='x')
elif AA.dimension_relative() != n:
raise ValueError("Affine Space must be of the dimension %s" % (n))
AA._default_embedding_index = i
phi = AA.projective_embedding(i, self)
self.__affine_patches[i] = AA
return AA
def affine_patch(self, i, AA = None):
r"""
Return the `i^{th}` affine patch of this projective space.
This is an ambient affine space `\mathbb{A}^n_R,` where
`R` is the base ring of self, whose "projective embedding"
map is `1` in the `i^{th}` factor.
INPUT:
- ``i`` -- integer between 0 and dimension of self, inclusive.
- ``AA`` -- (default: None) ambient affine space, this is constructed
if it is not given.
OUTPUT:
- An ambient affine space with fixed projective_embedding map.
EXAMPLES::
sage: PP = ProjectiveSpace(5) / QQ
sage: AA = PP.affine_patch(2)
sage: AA
Affine Space of dimension 5 over Rational Field
sage: AA.projective_embedding()
Scheme morphism:
From: Affine Space of dimension 5 over Rational Field
To: Projective Space of dimension 5 over Rational Field
Defn: Defined on coordinates by sending (x0, x1, x2, x3, x4) to
(x0 : x1 : 1 : x2 : x3 : x4)
sage: AA.projective_embedding(0)
Scheme morphism:
From: Affine Space of dimension 5 over Rational Field
To: Projective Space of dimension 5 over Rational Field
Defn: Defined on coordinates by sending (x0, x1, x2, x3, x4) to
(1 : x0 : x1 : x2 : x3 : x4)
::
sage: P.<x,y> = ProjectiveSpace(QQ,1)
sage: P.affine_patch(0).projective_embedding(0).codomain() == P
True
"""
i = int(i) # implicit type checking
n = self.dimension_relative()
if i < 0 or i > n:
raise ValueError("Argument i (= %s) must be between 0 and %s."%(i, n))
try:
A = self.__affine_patches[i]
#assume that if you've passed in a new affine space you want to override
#the existing patch
if AA is None or A == AA:
return(A)
except AttributeError:
self.__affine_patches = {}
except KeyError:
pass
#if no ith patch exists, we may still be here with AA==None
if AA == None:
from sage.schemes.affine.affine_space import AffineSpace
AA = AffineSpace(n, self.base_ring(), names = 'x')
elif AA.dimension_relative() != n:
raise ValueError("Affine Space must be of the dimension %s"%(n))
AA._default_embedding_index = i
phi = AA.projective_embedding(i, self)
self.__affine_patches[i] = AA
return AA