def __init__(self,
parent,
coord_expression=None,
name=None,
latex_name=None,
is_isomorphism=False,
is_identity=False):
r"""
Construct a curve.
TESTS::
sage: M = Manifold(2, 'M')
sage: X.<x,y> = M.chart()
sage: R.<t> = RealLine()
sage: I = R.open_interval(0, 2*pi)
sage: c = Hom(I,M)({X: (cos(t), sin(2*t))}, name='c') ; c
Curve c in the 2-dimensional differentiable manifold M
sage: TestSuite(c).run()
The identity of interval ``I``::
sage: c = Hom(I,I)({}, is_identity=True) ; c
Identity map Id_(0, 2*pi) of the Real interval (0, 2*pi)
sage: TestSuite(c).run()
"""
domain = parent.domain()
codomain = parent.codomain()
if coord_expression is None:
coord_functions = None
else:
if not isinstance(coord_expression, dict):
raise TypeError(
"{} is not a dictionary".format(coord_expression))
param_chart = domain.canonical_chart()
codom_atlas = codomain.atlas()
n = codomain.manifold().dim()
coord_functions = {}
for chart, expr in coord_expression.items():
if isinstance(chart, tuple):
# a pair of charts is passed:
coord_functions[chart] = expr
else:
coord_functions[(param_chart, chart)] = expr
DiffMap.__init__(self,
parent,
coord_functions=coord_functions,
name=name,
latex_name=latex_name,
is_isomorphism=is_isomorphism,
is_identity=is_identity)
def __init__(self, parent, coord_expression=None, name=None,
latex_name=None, is_isomorphism=False, is_identity=False):
r"""
Construct a curve.
TESTS::
sage: M = Manifold(2, 'M')
sage: X.<x,y> = M.chart()
sage: R.<t> = RealLine()
sage: I = R.open_interval(0, 2*pi)
sage: c = Hom(I,M)({X: (cos(t), sin(2*t))}, name='c') ; c
Curve c in the 2-dimensional differentiable manifold M
sage: TestSuite(c).run()
The identity of interval ``I``::
sage: c = Hom(I,I)({}, is_identity=True) ; c
Identity map Id_(0, 2*pi) of the Real interval (0, 2*pi)
sage: TestSuite(c).run()
"""
domain = parent.domain()
codomain = parent.codomain()
if coord_expression is None:
coord_functions = None
else:
if not isinstance(coord_expression, dict):
raise TypeError("{} is not a dictionary".format(
coord_expression))
param_chart = domain.canonical_chart()
codom_atlas = codomain.atlas()
n = codomain.manifold().dim()
coord_functions = {}
for chart, expr in coord_expression.items():
if isinstance(chart, tuple):
# a pair of charts is passed:
coord_functions[chart] = expr
else:
coord_functions[(param_chart, chart)] = expr
DiffMap.__init__(self, parent, coord_functions=coord_functions,
name=name, latex_name=latex_name,
is_isomorphism=is_isomorphism,
is_identity=is_identity)
def _repr_(self):
r"""
Return a string representation of ``self``.
TESTS::
sage: M = Manifold(2, 'M')
sage: X.<x,y> = M.chart()
sage: R.<t> = RealLine()
sage: M.curve([cos(t), sin(2*t)], (t, 0, 2*pi))
Curve in the 2-dimensional differentiable manifold M
sage: M.curve([cos(t), sin(2*t)], (t, 0, 2*pi), name='c')
Curve c in the 2-dimensional differentiable manifold M
"""
if self._codomain._dim == 1:
return DiffMap._repr_(self)
description = "Curve "
if self._name is not None:
description += self._name + " "
description += "in the {}".format(self._codomain)
return description
def _repr_(self):
r"""
Return a string representation of ``self``.
TESTS::
sage: M = Manifold(2, 'M')
sage: X.<x,y> = M.chart()
sage: R.<t> = RealLine()
sage: M.curve([cos(t), sin(2*t)], (t, 0, 2*pi))
Curve in the 2-dimensional differentiable manifold M
sage: M.curve([cos(t), sin(2*t)], (t, 0, 2*pi), name='c')
Curve c in the 2-dimensional differentiable manifold M
"""
if self._codomain._dim == 1:
return DiffMap._repr_(self)
description = "Curve "
if self._name is not None:
description += self._name + " "
description += "in the {}".format(self._codomain)
return description
def __call__(self, t, simplify=True):
r"""
Image for a given value of the curve parameter.
This is a redefinition of :meth:`sage.categories.map.Map.__call__`
to allow for the direct call with some value of the parameter
(numerical value or symbolic expression) instead of the element
(ManifoldPoint) of the domain corresponding to that value.
EXAMPLES:
Points on circle in the Euclidean plane::
sage: M = Manifold(2, 'M')
sage: X.<x,y> = M.chart()
sage: R.<t> = RealLine()
sage: c = M.curve([cos(t), sin(t)], (t, 0, 2*pi), name='c')
sage: c(0)
Point c(0) on the 2-dimensional differentiable manifold M
sage: c(0) in M
True
sage: c(0).coord(X)
(1, 0)
sage: c(pi/4).coord(X)
(1/2*sqrt(2), 1/2*sqrt(2))
sage: c(t)
Point c(t) on the 2-dimensional differentiable manifold M
sage: c(t).coord(X)
(cos(t), sin(t))
"""
# Case of a point in the domain:
if isinstance(t, ManifoldPoint):
return DiffMap.__call__(self, t)
# Case of a value of the canonical coordinate in the domain:
codom = self._codomain
canon_chart = self._domain._canon_chart
canon_coord = canon_chart._xx[0]
if (canon_chart, codom._def_chart) in self._coord_expression:
chart_pair = (canon_chart, codom._def_chart)
else:
chart_pair = next(iter(self._coord_expression.keys()))
# a chart is picked at random
coord_functions = self._coord_expression[chart_pair]._functions
n = codom._dim
dict_subs = {canon_coord: t}
coords = [coord_functions[i].expr().substitute(dict_subs)
for i in range(n)]
if simplify:
coords = [chart_pair[0].simplify(coords[i]) for i in range(n)]
if self._name is not None:
name = "{}({})".format(self._name, t)
else:
name = None
if self._latex_name is not None:
latex_name = r"{}\left({}\right)".format(self._latex_name, latex(t))
else:
latex_name = None
return codom.element_class(codom, coords=coords, chart=chart_pair[1],
name=name, latex_name=latex_name,
check_coords=False)
def __call__(self, t, simplify=True):
r"""
Image for a given value of the curve parameter.
This is a redefinition of :meth:`sage.categories.map.Map.__call__`
to allow for the direct call with some value of the parameter
(numerical value or symbolic expression) instead of the element
(ManifoldPoint) of the domain corresponding to that value.
EXAMPLES:
Points on circle in the Euclidean plane::
sage: M = Manifold(2, 'M')
sage: X.<x,y> = M.chart()
sage: R.<t> = RealLine()
sage: c = M.curve([cos(t), sin(t)], (t, 0, 2*pi), name='c')
sage: c(0)
Point c(0) on the 2-dimensional differentiable manifold M
sage: c(0) in M
True
sage: c(0).coord(X)
(1, 0)
sage: c(pi/4).coord(X)
(1/2*sqrt(2), 1/2*sqrt(2))
sage: c(t)
Point c(t) on the 2-dimensional differentiable manifold M
sage: c(t).coord(X)
(cos(t), sin(t))
"""
# Case of a point in the domain:
if isinstance(t, ManifoldPoint):
return DiffMap.__call__(self, t)
# Case of a value of the canonical coordinate in the domain:
codom = self._codomain
canon_chart = self._domain._canon_chart
canon_coord = canon_chart._xx[0]
if (canon_chart, codom._def_chart) in self._coord_expression:
chart_pair = (canon_chart, codom._def_chart)
else:
chart_pair = self._coord_expression.keys()[0] # a chart is picked
# at random
coord_functions = self._coord_expression[chart_pair]._functions
n = codom._dim
dict_subs = {canon_coord: t}
coords = [coord_functions[i].expr().substitute(dict_subs)
for i in range(n)]
if simplify:
coords = [simplify_chain_real(coords[i]) for i in range(n)]
if self._name is not None:
name = "{}({})".format(self._name, t)
else:
name = None
if self._latex_name is not None:
latex_name = r"{}\left({}\right)".format(self._latex_name, latex(t))
else:
latex_name = None
return codom.element_class(codom, coords=coords, chart=chart_pair[1],
name=name, latex_name=latex_name,
check_coords=False)