Coordinate Charts#

The class Chart implements coordinate charts on a topological manifold over a topological field \(K\). The subclass RealChart is devoted to the case \(K=\RR\), for which the concept of coordinate range is meaningful. Moreover, RealChart is endowed with some plotting capabilities (cf. method plot()).

Transition maps between charts are implemented via the class CoordChange.

AUTHORS:

Eric Gourgoulhon, Michal Bejger (2013-2015) : initial version
Travis Scrimshaw (2015): review tweaks
Eric Gourgoulhon (2019): periodic coordinates, add calculus_method()

REFERENCES:

Chap. 2 of [Lee2011]
Chap. 1 of [Lee2013]

class sage.manifolds.chart.Chart(domain, coordinates, calc_method=None, periods=None, coord_restrictions=None)#

Bases: UniqueRepresentation, SageObject

Chart on a topological manifold.

Given a topological manifold \(M\) of dimension \(n\) over a topological field \(K\), a chart on \(M\) is a pair \((U, \varphi)\), where \(U\) is an open subset of \(M\) and \(\varphi : U \rightarrow V \subset K^n\) is a homeomorphism from \(U\) to an open subset \(V\) of \(K^n\).

The components \((x^1, \ldots, x^n)\) of \(\varphi\), defined by \(\varphi(p) = (x^1(p), \ldots, x^n(p)) \in K^n\) for any point \(p \in U\), are called the coordinates of the chart \((U, \varphi)\).

INPUT:

domain – open subset \(U\) on which the chart is defined (must be an instance of TopologicalManifold)
coordinates – (default: ‘’ (empty string)) single string defining the coordinate symbols, with ' ' (whitespace) as a separator; each item has at most three fields, separated by a colon (:):
1. the coordinate symbol (a letter or a few letters)
2. (optional) the period of the coordinate if the coordinate is periodic; the period field must be written as period=T, where T is the period (see examples below)
3. (optional) the LaTeX spelling of the coordinate; if not provided the coordinate symbol given in the first field will be used
The order of fields 2 and 3 does not matter and each of them can be omitted. If it contains any LaTeX expression, the string coordinates must be declared with the prefix ‘r’ (for “raw”) to allow for a proper treatment of LaTeX’s backslash character (see examples below). If no period and no LaTeX spelling are to be set for any coordinate, the argument coordinates can be omitted when the shortcut operator <,> is used to declare the chart (see examples below).
calc_method – (default: None) string defining the calculus method for computations involving coordinates of the chart; must be one of
- 'SR': Sage’s default symbolic engine (Symbolic Ring)
- 'sympy': SymPy
- None: the default of CalculusMethod will be used
names – (default: None) unused argument, except if coordinates is not provided; it must then be a tuple containing the coordinate symbols (this is guaranteed if the shortcut operator <,> is used)
coord_restrictions: Additional restrictions on the coordinates. A restriction can be any symbolic equality or inequality involving the coordinates, such as x > y or x^2 + y^2 != 0. The items of the list (or set or frozenset) coord_restrictions are combined with the and operator; if some restrictions are to be combined with the or operator instead, they have to be passed as a tuple in some single item of the list (or set or frozenset) coord_restrictions. For example:
```
coord_restrictions=[x > y, (x != 0, y != 0), z^2 < x]
```
means (x > y) and ((x != 0) or (y != 0)) and (z^2 < x). If the list coord_restrictions contains only one item, this item can be passed as such, i.e. writing x > y instead of the single element list [x > y]. If the chart variables have not been declared as variables yet, coord_restrictions must be lambda-quoted.

EXAMPLES:

A chart on a complex 2-dimensional topological manifold:

sage: M = Manifold(2, 'M', field='complex', structure='topological')
sage: X = M.chart('x y'); X
Chart (M, (x, y))
sage: latex(X)
\left(M,(x, y)\right)
sage: type(X)
<class 'sage.manifolds.chart.Chart'>

To manipulate the coordinates \((x,y)\) as global variables, one has to set:

sage: x,y = X[:]

However, a shortcut is to use the declarator <x,y> in the left-hand side of the chart declaration (there is then no need to pass the string 'x y' to chart()):

sage: M = Manifold(2, 'M', field='complex', structure='topological')
sage: X.<x,y> = M.chart(); X
Chart (M, (x, y))

The coordinates are then immediately accessible:

sage: y
y
sage: x is X[0] and y is X[1]
True

Note that x and y declared in <x,y> are mere Python variable names and do not have to coincide with the coordinate symbols; for instance, one may write:

sage: M = Manifold(2, 'M', field='complex', structure='topological')
sage: X.<x1,y1> = M.chart('x y'); X
Chart (M, (x, y))

Then y is not known as a global Python variable and the coordinate \(y\) is accessible only through the global variable y1:

sage: y1
y
sage: latex(y1)
y
sage: y1 is X[1]
True

However, having the name of the Python variable coincide with the coordinate symbol is quite convenient; so it is recommended to declare:

sage: M = Manifold(2, 'M', field='complex', structure='topological')
sage: X.<x,y> = M.chart()

In the above example, the chart X covers entirely the manifold M:

sage: X.domain()
Complex 2-dimensional topological manifold M

Of course, one may declare a chart only on an open subset of M:

sage: U = M.open_subset('U')
sage: Y.<z1, z2> = U.chart(r'z1:\zeta_1 z2:\zeta_2'); Y
Chart (U, (z1, z2))
sage: Y.domain()
Open subset U of the Complex 2-dimensional topological manifold M

In the above declaration, we have also specified some LaTeX writing of the coordinates different from the text one:

sage: latex(z1)
{\zeta_1}

Note the prefix r in front of the string r'z1:\zeta_1 z2:\zeta_2'; it makes sure that the backslash character is treated as an ordinary character, to be passed to the LaTeX interpreter.

Periodic coordinates are declared through the keyword period= in the coordinate field:

sage: N = Manifold(2, 'N', field='complex', structure='topological')
sage: XN.<Z1,Z2> = N.chart('Z1:period=1+2*I Z2')
sage: XN.periods()
(2*I + 1, None)

Coordinates are Sage symbolic variables (see sage.symbolic.expression):

sage: type(z1)
<class 'sage.symbolic.expression.Expression'>

In addition to the Python variable name provided in the operator <.,.>, the coordinates are accessible by their indices:

sage: Y[0], Y[1]
(z1, z2)

The index range is that declared during the creation of the manifold. By default, it starts at 0, but this can be changed via the parameter start_index:

sage: M1 = Manifold(2, 'M_1', field='complex', structure='topological',
....:               start_index=1)
sage: Z.<u,v> = M1.chart()
sage: Z[1], Z[2]
(u, v)

The full set of coordinates is obtained by means of the slice operator [:]:

sage: Y[:]
(z1, z2)

Some partial sets of coordinates:

sage: Y[:1]
(z1,)
sage: Y[1:]
(z2,)

Each constructed chart is automatically added to the manifold’s user atlas:

sage: M.atlas()
[Chart (M, (x, y)), Chart (U, (z1, z2))]

and to the atlas of the chart’s domain:

sage: U.atlas()
[Chart (U, (z1, z2))]

Manifold subsets have a default chart, which, unless changed via the method set_default_chart(), is the first defined chart on the subset (or on a open subset of it):

sage: M.default_chart()
Chart (M, (x, y))
sage: U.default_chart()
Chart (U, (z1, z2))

The default charts are not privileged charts on the manifold, but rather charts whose name can be skipped in the argument list of functions having an optional chart= argument.

The chart map \(\varphi\) acting on a point is obtained by passing it as an input to the map:

sage: p = M.point((1+i, 2), chart=X); p
Point on the Complex 2-dimensional topological manifold M
sage: X(p)
(I + 1, 2)
sage: X(p) == p.coord(X)
True

Setting additional coordinate restrictions:

sage: M = Manifold(2, 'M', field='complex', structure='topological')
sage: X.<x,y> = M.chart(coord_restrictions=lambda x,y: abs(x) > 1)
sage: X.valid_coordinates(2+i, 1)
True
sage: X.valid_coordinates(i, 1)
False

See also

CalculusMethod for a complete documentation.

OUTPUT:

an instance of CalculusMethod

EXAMPLES:

The default calculus method relies on Sage’s Symbolic Ring:

sage: M = Manifold(2, 'M', structure='topological')
sage: X.<x,y> = M.chart()
sage: X.calculus_method()
Available calculus methods (* = current):
 - SR (default) (*)
 - sympy

Accordingly the method expr() of a function f defined on the chart X returns a Sage symbolic expression:

sage: f = X.function(x^2 + cos(y)*sin(x))
sage: f.expr()
x^2 + cos(y)*sin(x)
sage: type(f.expr())
<class 'sage.symbolic.expression.Expression'>
sage: parent(f.expr())
Symbolic Ring
sage: f.display()
(x, y) ↦ x^2 + cos(y)*sin(x)

Changing to SymPy:

sage: X.calculus_method().set('sympy')
sage: f.expr()
x**2 + sin(x)*cos(y)
sage: type(f.expr())
<class 'sympy.core.add.Add'>
sage: parent(f.expr())
<class 'sympy.core.add.Add'>
sage: f.display()
(x, y) ↦ x**2 + sin(x)*cos(y)

Back to the Symbolic Ring:

sage: X.calculus_method().set('SR')
sage: f.display()
(x, y) ↦ x^2 + cos(y)*sin(x)

codomain()#

Return the codomain of self as a set.

EXAMPLES:

sage: M = Manifold(2, 'M', field='complex', structure='topological')
sage: X.<x,y> = M.chart()
sage: X.codomain()
Vector space of dimension 2 over Complex Field with 53 bits of precision

domain()#

Return the open subset on which the chart is defined.

EXAMPLES:

sage: M = Manifold(2, 'M', structure='topological')
sage: X.<x,y> = M.chart()
sage: X.domain()
2-dimensional topological manifold M
sage: U = M.open_subset('U')
sage: Y.<u,v> = U.chart()
sage: Y.domain()
Open subset U of the 2-dimensional topological manifold M

function(expression, calc_method=None, expansion_symbol=None, order=None)#

Define a coordinate function to the base field.

If the current chart belongs to the atlas of a \(n\)-dimensional manifold over a topological field \(K\), a coordinate function is a map

\[\begin{split}\begin{array}{cccc} f:& V\subset K^n & \longrightarrow & K \\ & (x^1,\ldots, x^n) & \longmapsto & f(x^1,\ldots, x^n), \end{array}\end{split}\]

where \(V\) is the chart codomain and \((x^1, \ldots, x^n)\) are the chart coordinates.

INPUT:

expression – a symbolic expression involving the chart coordinates, to represent \(f(x^1,\ldots, x^n)\)
calc_method – string (default: None): the calculus method with respect to which the internal expression of the function must be initialized from expression; one of
- 'SR': Sage’s default symbolic engine (Symbolic Ring)
- 'sympy': SymPy
- None: the chart current calculus method is assumed
expansion_symbol – (default: None) symbolic variable (the “small parameter”) with respect to which the coordinate expression is expanded in power series (around the zero value of this variable)
order – integer (default: None); the order of the expansion if expansion_symbol is not None; the order is defined as the degree of the polynomial representing the truncated power series in expansion_symbol.

Warning

The value of order is \(n-1\), where \(n\) is the order of the big \(O\) in the power series expansion

OUTPUT:

instance of ChartFunction representing the coordinate function \(f\)

EXAMPLES:

A symbolic coordinate function:

sage: M = Manifold(2, 'M', structure='topological')
sage: X.<x,y> = M.chart()
sage: f = X.function(sin(x*y))
sage: f
sin(x*y)
sage: type(f)
<class 'sage.manifolds.chart_func.ChartFunctionRing_with_category.element_class'>
sage: f.display()
(x, y) ↦ sin(x*y)
sage: f(2,3)
sin(6)

Using SymPy for the internal representation of the function (dictionary _express):

sage: g = X.function(x^2 + x*cos(y), calc_method='sympy')
sage: g._express
{'sympy': x**2 + x*cos(y)}

On the contrary, for f, only the SR part has been initialized:

sage: f._express
{'SR': sin(x*y)}

See ChartFunction for more examples.

function_ring()#

Return the ring of coordinate functions on self.

EXAMPLES:

sage: M = Manifold(2, 'M', structure='topological')
sage: X.<x,y> = M.chart()
sage: X.function_ring()
Ring of chart functions on Chart (M, (x, y))

manifold()#

Return the manifold on which the chart is defined.

EXAMPLES:

sage: M = Manifold(2, 'M', structure='topological')
sage: U = M.open_subset('U')
sage: X.<x,y> = U.chart()
sage: X.manifold()
2-dimensional topological manifold M
sage: X.domain()
Open subset U of the 2-dimensional topological manifold M

multifunction(*expressions)#

Define a coordinate function to some Cartesian power of the base field.

If \(n\) and \(m\) are two positive integers and \((U, \varphi)\) is a chart on a topological manifold \(M\) of dimension \(n\) over a topological field \(K\), a multi-coordinate function associated to \((U,\varphi)\) is a map

\[\begin{split}\begin{array}{llcl} f:& V \subset K^n & \longrightarrow & K^m \\ & (x^1, \ldots, x^n) & \longmapsto & (f_1(x^1, \ldots, x^n), \ldots, f_m(x^1, \ldots, x^n)), \end{array}\end{split}\]

where \(V\) is the codomain of \(\varphi\). In other words, \(f\) is a \(K^m\)-valued function of the coordinates associated to the chart \((U, \varphi)\).

See MultiCoordFunction for a complete documentation.

INPUT:

expressions – list (or tuple) of \(m\) elements to construct the coordinate functions \(f_i\) (\(1\leq i \leq m\)); for symbolic coordinate functions, this must be symbolic expressions involving the chart coordinates, while for numerical coordinate functions, this must be data file names

OUTPUT:

a MultiCoordFunction representing \(f\)

EXAMPLES:

Function of two coordinates with values in \(\RR^3\):

sage: M = Manifold(2, 'M', structure='topological')
sage: X.<x,y> = M.chart()
sage: f = X.multifunction(x+y, sin(x*y), x^2 + 3*y); f
Coordinate functions (x + y, sin(x*y), x^2 + 3*y) on the Chart (M, (x, y))
sage: f(2,3)
(5, sin(6), 13)

one_function()#

Return the constant function of the coordinates equal to one.

If the current chart belongs to the atlas of a \(n\)-dimensional manifold over a topological field \(K\), the “one” coordinate function is the map

\[\begin{split}\begin{array}{cccc} f:& V\subset K^n & \longrightarrow & K \\ & (x^1,\ldots, x^n) & \longmapsto & 1, \end{array}\end{split}\]

where \(V\) is the chart codomain.

See class ChartFunction for a complete documentation.

OUTPUT:

a ChartFunction representing the one coordinate function \(f\)

EXAMPLES:

sage: M = Manifold(2, 'M', structure='topological')
sage: X.<x,y> = M.chart()
sage: X.one_function()
1
sage: X.one_function().display()
(x, y) ↦ 1
sage: type(X.one_function())
<class 'sage.manifolds.chart_func.ChartFunctionRing_with_category.element_class'>

The result is cached:

sage: X.one_function() is X.one_function()
True

One function on a p-adic manifold:

sage: # needs sage.rings.padics
sage: M = Manifold(2, 'M', structure='topological', field=Qp(5)); M
2-dimensional topological manifold M over the 5-adic Field with
 capped relative precision 20
sage: X.<x,y> = M.chart()
sage: X.one_function()
1 + O(5^20)
sage: X.one_function().display()
(x, y) ↦ 1 + O(5^20)

periods()#

Return the coordinate periods.

OUTPUT:

a tuple containing the period of each coordinate, with the value None if the coordinate is not periodic

EXAMPLES:

A chart without any periodic coordinate:

sage: M = Manifold(2, 'M', structure='topological')
sage: X.<x,y> = M.chart()
sage: X.periods()
(None, None)

Charts with a periodic coordinate:

sage: Y.<u,v> = M.chart("u v:(0,2*pi):periodic")
sage: Y.periods()
(None, 2*pi)
sage: Z.<a,b> = M.chart(r"a:period=sqrt(2):\alpha b:\beta")
sage: Z.periods()
(sqrt(2), None)

Complex manifold with a periodic coordinate:

sage: M = Manifold(2, 'M', field='complex', structure='topological',
....:              start_index=1)
sage: X.<x,y> = M.chart("x y:period=1+I")
sage: X.periods()
(None, I + 1)

preimage(codomain_subset, name=None, latex_name=None)#

Return the preimage (pullback) of codomain_subset under self.

It is the subset of the domain of self formed by the points whose coordinate vectors lie in codomain_subset.

INPUT:

codomain_subset - an instance of ConvexSet_base or another object with a __contains__ method that accepts coordinate vectors
name – string; name (symbol) given to the subset
latex_name – (default: None) string; LaTeX symbol to denote the subset; if none are provided, it is set to name

OUTPUT:

either a TopologicalManifold or a ManifoldSubsetPullback

EXAMPLES:

sage: M = Manifold(2, 'R^2', structure='topological')
sage: c_cart.<x,y> = M.chart() # Cartesian coordinates on R^2

Pulling back a polytope under a chart:

sage: # needs sage.geometry.polyhedron
sage: P = Polyhedron(vertices=[[0, 0], [1, 2], [2, 1]]); P
A 2-dimensional polyhedron in ZZ^2 defined as the convex hull of 3 vertices
sage: McP = c_cart.preimage(P); McP
Subset x_y_inv_P of the 2-dimensional topological manifold R^2
sage: M((1, 2)) in McP
True
sage: M((2, 0)) in McP
False

Pulling back the interior of a polytope under a chart:

sage: # needs sage.geometry.polyhedron
sage: int_P = P.interior(); int_P
Relative interior of
 a 2-dimensional polyhedron in ZZ^2 defined as the convex hull of 3 vertices
sage: McInt_P = c_cart.preimage(int_P, name='McInt_P'); McInt_P
Open subset McInt_P of the 2-dimensional topological manifold R^2
sage: M((0, 0)) in McInt_P
False
sage: M((1, 1)) in McInt_P
True

Pulling back a point lattice:

sage: W = span([[1, 0], [3, 5]], ZZ); W
Free module of degree 2 and rank 2 over Integer Ring
Echelon basis matrix:
[1 0]
[0 5]
sage: McW = c_cart.pullback(W, name='McW'); McW
Subset McW of the 2-dimensional topological manifold R^2
sage: M((4, 5)) in McW
True
sage: M((4, 4)) in McW
False

Pulling back a real vector subspaces:

sage: V = span([[1, 2]], RR); V
Vector space of degree 2 and dimension 1 over Real Field with 53 bits of precision
Basis matrix:
[1.00000000000000 2.00000000000000]
sage: McV = c_cart.pullback(V, name='McV'); McV
Subset McV of the 2-dimensional topological manifold R^2
sage: M((2, 4)) in McV
True
sage: M((1, 0)) in McV
False

Pulling back a finite set of points:

sage: F = Family([vector(QQ, [1, 2], immutable=True),
....:             vector(QQ, [2, 3], immutable=True)])
sage: McF = c_cart.pullback(F, name='McF'); McF
Subset McF of the 2-dimensional topological manifold R^2
sage: M((2, 3)) in McF
True
sage: M((0, 0)) in McF
False

Pulling back the integers:

sage: R = manifolds.RealLine(); R
Real number line ℝ
sage: McZ = R.canonical_chart().pullback(ZZ, name='ℤ'); McZ
Subset ℤ of the Real number line ℝ
sage: R((3/2,)) in McZ
False
sage: R((-2,)) in McZ
True

pullback(codomain_subset, name=None, latex_name=None)#

Return the preimage (pullback) of codomain_subset under self.

It is the subset of the domain of self formed by the points whose coordinate vectors lie in codomain_subset.

INPUT:

codomain_subset - an instance of ConvexSet_base or another object with a __contains__ method that accepts coordinate vectors
name – string; name (symbol) given to the subset
latex_name – (default: None) string; LaTeX symbol to denote the subset; if none are provided, it is set to name

OUTPUT:

either a TopologicalManifold or a ManifoldSubsetPullback

EXAMPLES:

sage: M = Manifold(2, 'R^2', structure='topological')
sage: c_cart.<x,y> = M.chart() # Cartesian coordinates on R^2

Pulling back a polytope under a chart:

sage: # needs sage.geometry.polyhedron
sage: P = Polyhedron(vertices=[[0, 0], [1, 2], [2, 1]]); P
A 2-dimensional polyhedron in ZZ^2 defined as the convex hull of 3 vertices
sage: McP = c_cart.preimage(P); McP
Subset x_y_inv_P of the 2-dimensional topological manifold R^2
sage: M((1, 2)) in McP
True
sage: M((2, 0)) in McP
False

Pulling back the interior of a polytope under a chart:

sage: # needs sage.geometry.polyhedron
sage: int_P = P.interior(); int_P
Relative interior of
 a 2-dimensional polyhedron in ZZ^2 defined as the convex hull of 3 vertices
sage: McInt_P = c_cart.preimage(int_P, name='McInt_P'); McInt_P
Open subset McInt_P of the 2-dimensional topological manifold R^2
sage: M((0, 0)) in McInt_P
False
sage: M((1, 1)) in McInt_P
True

Pulling back a point lattice:

sage: W = span([[1, 0], [3, 5]], ZZ); W
Free module of degree 2 and rank 2 over Integer Ring
Echelon basis matrix:
[1 0]
[0 5]
sage: McW = c_cart.pullback(W, name='McW'); McW
Subset McW of the 2-dimensional topological manifold R^2
sage: M((4, 5)) in McW
True
sage: M((4, 4)) in McW
False

Pulling back a real vector subspaces:

sage: V = span([[1, 2]], RR); V
Vector space of degree 2 and dimension 1 over Real Field with 53 bits of precision
Basis matrix:
[1.00000000000000 2.00000000000000]
sage: McV = c_cart.pullback(V, name='McV'); McV
Subset McV of the 2-dimensional topological manifold R^2
sage: M((2, 4)) in McV
True
sage: M((1, 0)) in McV
False

Pulling back a finite set of points:

sage: F = Family([vector(QQ, [1, 2], immutable=True),
....:             vector(QQ, [2, 3], immutable=True)])
sage: McF = c_cart.pullback(F, name='McF'); McF
Subset McF of the 2-dimensional topological manifold R^2
sage: M((2, 3)) in McF
True
sage: M((0, 0)) in McF
False

Pulling back the integers:

sage: R = manifolds.RealLine(); R
Real number line ℝ
sage: McZ = R.canonical_chart().pullback(ZZ, name='ℤ'); McZ
Subset ℤ of the Real number line ℝ
sage: R((3/2,)) in McZ
False
sage: R((-2,)) in McZ
True

restrict(subset, restrictions=None)#

Return the restriction of self to some open subset of its domain.

If the current chart is \((U,\varphi)\), a restriction (or subchart) is a chart \((V,\psi)\) such that \(V\subset U\) and \(\psi = \varphi |_V\).

If such subchart has not been defined yet, it is constructed here.

The coordinates of the subchart bare the same names as the coordinates of the current chart.

INPUT:

subset – open subset \(V\) of the chart domain \(U\) (must be an instance of TopologicalManifold)
restrictions – (default: None) list of coordinate restrictions defining the subset \(V\)

restrictions = [x > y, (x != 0, y != 0), z^2 < x]

OUTPUT:

chart \((V, \psi)\) as a Chart

EXAMPLES:

Coordinates on the unit open ball of \(\CC^2\) as a subchart of the global coordinates of \(\CC^2\):

sage: M = Manifold(2, 'C^2', field='complex', structure='topological')
sage: X.<z1, z2> = M.chart()
sage: B = M.open_subset('B')
sage: X_B = X.restrict(B, abs(z1)^2 + abs(z2)^2 < 1); X_B
Chart (B, (z1, z2))

transition_map(other, transformations, intersection_name=None, restrictions1=None, restrictions2=None)#

Construct the transition map between the current chart, \((U, \varphi)\) say, and another one, \((V, \psi)\) say.

If \(n\) is the manifold’s dimension, the transition map is the map

\[\psi\circ\varphi^{-1}: \varphi(U\cap V) \subset K^n \rightarrow \psi(U\cap V) \subset K^n,\]

where \(K\) is the manifold’s base field. In other words, the transition map expresses the coordinates \((y^1, \ldots, y^n)\) of \((V, \psi)\) in terms of the coordinates \((x^1, \ldots, x^n)\) of \((U, \varphi)\) on the open subset where the two charts intersect, i.e. on \(U \cap V\).

INPUT:

other – the chart \((V, \psi)\)
transformations – tuple (or list) \((Y_1, \ldots, Y_n)\), where \(Y_i\) is the symbolic expression of the coordinate \(y^i\) in terms of the coordinates \((x^1, \ldots, x^n)\)
intersection_name – (default: None) name to be given to the subset \(U \cap V\) if the latter differs from \(U\) or \(V\)
restrictions1 – (default: None) list of conditions on the coordinates of the current chart that define \(U \cap V\) if the latter differs from \(U\)
restrictions2 – (default: None) list of conditions on the coordinates of the chart \((V,\psi)\) that define \(U \cap V\) if the latter differs from \(V\)

restrictions = [x > y, (x != 0, y != 0), z^2 < x]

OUTPUT:

the transition map \(\psi \circ \varphi^{-1}\) defined on \(U \cap V\) as a CoordChange

EXAMPLES:

Transition map between two stereographic charts on the circle \(S^1\):

sage: M = Manifold(1, 'S^1', structure='topological')
sage: U = M.open_subset('U') # Complement of the North pole
sage: cU.<x> = U.chart() # Stereographic chart from the North pole
sage: V = M.open_subset('V') # Complement of the South pole
sage: cV.<y> = V.chart() # Stereographic chart from the South pole
sage: M.declare_union(U,V)   # S^1 is the union of U and V
sage: trans = cU.transition_map(cV, 1/x, intersection_name='W',
....:                           restrictions1= x!=0, restrictions2 = y!=0)
sage: trans
Change of coordinates from Chart (W, (x,)) to Chart (W, (y,))
sage: trans.display()
y = 1/x

The subset \(W\), intersection of \(U\) and \(V\), has been created by transition_map():

sage: F = M.subset_family(); F
Set {S^1, U, V, W} of open subsets of the 1-dimensional topological manifold S^1
sage: W = F['W']
sage: W is U.intersection(V)
True
sage: M.atlas()
[Chart (U, (x,)), Chart (V, (y,)), Chart (W, (x,)), Chart (W, (y,))]

Transition map between the spherical chart and the Cartesian one on \(\RR^2\):

sage: M = Manifold(2, 'R^2', structure='topological')
sage: c_cart.<x,y> = M.chart()
sage: U = M.open_subset('U') # the complement of the half line {y=0, x >= 0}
sage: c_spher.<r,phi> = U.chart(r'r:(0,+oo) phi:(0,2*pi):\phi')
sage: trans = c_spher.transition_map(c_cart, (r*cos(phi), r*sin(phi)),
....:                                restrictions2=(y!=0, x<0))
sage: trans
Change of coordinates from Chart (U, (r, phi)) to Chart (U, (x, y))
sage: trans.display()
x = r*cos(phi)
y = r*sin(phi)

In this case, no new subset has been created since \(U \cap M = U\):

sage: M.subset_family()
Set {R^2, U} of open subsets of the 2-dimensional topological manifold R^2

but a new chart has been created: \((U, (x, y))\):

sage: M.atlas()
[Chart (R^2, (x, y)), Chart (U, (r, phi)), Chart (U, (x, y))]

valid_coordinates(*coordinates, **kwds)#

Check whether a tuple of coordinates can be the coordinates of a point in the chart domain.

INPUT:

*coordinates – coordinate values
**kwds – options:
- parameters=None, dictionary to set numerical values to some parameters (see example below)

OUTPUT:

True if the coordinate values are admissible in the chart image, False otherwise

EXAMPLES:

sage: M = Manifold(2, 'M', field='complex', structure='topological')
sage: X.<x,y> = M.chart(coord_restrictions=lambda x,y: [abs(x)<1, y!=0])
sage: X.valid_coordinates(0, i)
True
sage: X.valid_coordinates(i, 1)
False
sage: X.valid_coordinates(i/2, 1)
True
sage: X.valid_coordinates(i/2, 0)
False
sage: X.valid_coordinates(2, 0)
False

Example of use with the keyword parameters to set a specific value to a parameter appearing in the coordinate restrictions:

sage: var('a')  # the parameter is a symbolic variable
a
sage: Y.<u,v> = M.chart(coord_restrictions=lambda u,v: abs(v)<a)
sage: Y.valid_coordinates(1, i, parameters={a: 2})  # setting a=2
True
sage: Y.valid_coordinates(1, 2*i, parameters={a: 2})
False

zero_function()#

Return the zero function of the coordinates.

If the current chart belongs to the atlas of a \(n\)-dimensional manifold over a topological field \(K\), the zero coordinate function is the map

\[\begin{split}\begin{array}{cccc} f:& V\subset K^n & \longrightarrow & K \\ & (x^1,\ldots, x^n) & \longmapsto & 0, \end{array}\end{split}\]

where \(V\) is the chart codomain.

See class ChartFunction for a complete documentation.

OUTPUT:

a ChartFunction representing the zero coordinate function \(f\)

EXAMPLES:

sage: M = Manifold(2, 'M', structure='topological')
sage: X.<x,y> = M.chart()
sage: X.zero_function()
0
sage: X.zero_function().display()
(x, y) ↦ 0
sage: type(X.zero_function())
<class 'sage.manifolds.chart_func.ChartFunctionRing_with_category.element_class'>

The result is cached:

sage: X.zero_function() is X.zero_function()
True

Zero function on a p-adic manifold:

sage: # needs sage.rings.padics
sage: M = Manifold(2, 'M', structure='topological', field=Qp(5)); M
2-dimensional topological manifold M over the 5-adic Field with
 capped relative precision 20
sage: X.<x,y> = M.chart()
sage: X.zero_function()
0
sage: X.zero_function().display()
(x, y) ↦ 0

class sage.manifolds.chart.CoordChange(chart1, chart2, *transformations)#

Bases: SageObject

Transition map between two charts of a topological manifold.

Giving two coordinate charts \((U, \varphi)\) and \((V, \psi)\) on a topological manifold \(M\) of dimension \(n\) over a topological field \(K\), the transition map from \((U, \varphi)\) to \((V, \psi)\) is the map

\[\psi\circ\varphi^{-1}: \varphi(U\cap V) \subset K^n \rightarrow \psi(U\cap V) \subset K^n.\]

In other words, the transition map \(\psi \circ \varphi^{-1}\) expresses the coordinates \((y^1, \ldots, y^n)\) of \((V, \psi)\) in terms of the coordinates \((x^1, \ldots, x^n)\) of \((U, \varphi)\) on the open subset where the two charts intersect, i.e. on \(U \cap V\).

INPUT:

chart1 – chart \((U, \varphi)\)
chart2 – chart \((V, \psi)\)
transformations – tuple (or list) \((Y_1, \ldots, Y_2)\), where \(Y_i\) is the symbolic expression of the coordinate \(y^i\) in terms of the coordinates \((x^1, \ldots, x^n)\)

EXAMPLES:

Transition map on a 2-dimensional topological manifold:

sage: M = Manifold(2, 'M', structure='topological')
sage: X.<x,y> = M.chart()
sage: Y.<u,v> = M.chart()
sage: X_to_Y = X.transition_map(Y, [x+y, x-y])
sage: X_to_Y
Change of coordinates from Chart (M, (x, y)) to Chart (M, (u, v))
sage: type(X_to_Y)
<class 'sage.manifolds.chart.CoordChange'>
sage: X_to_Y.display()
u = x + y
v = x - y

disp()#

Display of the coordinate transformation.

The output is either text-formatted (console mode) or LaTeX-formatted (notebook mode).

EXAMPLES:

From spherical coordinates to Cartesian ones in the plane:

sage: M = Manifold(2, 'R^2', structure='topological')
sage: U = M.open_subset('U') # the complement of the half line {y=0, x>= 0}
sage: c_cart.<x,y> = U.chart()
sage: c_spher.<r,ph> = U.chart(r'r:(0,+oo) ph:(0,2*pi):\phi')
sage: spher_to_cart = c_spher.transition_map(c_cart, [r*cos(ph), r*sin(ph)])
sage: spher_to_cart.display()
x = r*cos(ph)
y = r*sin(ph)
sage: latex(spher_to_cart.display())
\left\{\begin{array}{lcl} x & = & r \cos\left({\phi}\right) \\
 y & = & r \sin\left({\phi}\right) \end{array}\right.

A shortcut is disp():

sage: spher_to_cart.disp()
x = r*cos(ph)
y = r*sin(ph)

display()#

Display of the coordinate transformation.

The output is either text-formatted (console mode) or LaTeX-formatted (notebook mode).

EXAMPLES:

From spherical coordinates to Cartesian ones in the plane:

sage: M = Manifold(2, 'R^2', structure='topological')
sage: U = M.open_subset('U') # the complement of the half line {y=0, x>= 0}
sage: c_cart.<x,y> = U.chart()
sage: c_spher.<r,ph> = U.chart(r'r:(0,+oo) ph:(0,2*pi):\phi')
sage: spher_to_cart = c_spher.transition_map(c_cart, [r*cos(ph), r*sin(ph)])
sage: spher_to_cart.display()
x = r*cos(ph)
y = r*sin(ph)
sage: latex(spher_to_cart.display())
\left\{\begin{array}{lcl} x & = & r \cos\left({\phi}\right) \\
 y & = & r \sin\left({\phi}\right) \end{array}\right.

A shortcut is disp():

sage: spher_to_cart.disp()
x = r*cos(ph)
y = r*sin(ph)

inverse()#

Return the inverse coordinate transformation.

If the inverse is not already known, it is computed here. If the computation fails, the inverse can be set by hand via the method set_inverse().

OUTPUT:

an instance of CoordChange representing the inverse of the current coordinate transformation

EXAMPLES:

Inverse of a coordinate transformation corresponding to a rotation in the Cartesian plane:

sage: M = Manifold(2, 'M', structure='topological')
sage: c_xy.<x,y> = M.chart()
sage: c_uv.<u,v> = M.chart()
sage: phi = var('phi', domain='real')
sage: xy_to_uv = c_xy.transition_map(c_uv,
....:                                [cos(phi)*x + sin(phi)*y,
....:                                 -sin(phi)*x + cos(phi)*y])
sage: M.coord_changes()
{(Chart (M, (x, y)),
  Chart (M, (u, v))): Change of coordinates from Chart (M, (x, y)) to Chart (M, (u, v))}
sage: uv_to_xy = xy_to_uv.inverse(); uv_to_xy
Change of coordinates from Chart (M, (u, v)) to Chart (M, (x, y))
sage: uv_to_xy.display()
x = u*cos(phi) - v*sin(phi)
y = v*cos(phi) + u*sin(phi)
sage: M.coord_changes()  # random (dictionary output)
{(Chart (M, (u, v)),
  Chart (M, (x, y))): Change of coordinates from Chart (M, (u, v)) to Chart (M, (x, y)),
 (Chart (M, (x, y)),
  Chart (M, (u, v))): Change of coordinates from Chart (M, (x, y)) to Chart (M, (u, v))}

The result is cached:

sage: xy_to_uv.inverse() is uv_to_xy
True

We have as well:

sage: uv_to_xy.inverse() is xy_to_uv
True

restrict(dom1, dom2=None)#

Restriction to subsets.

INPUT:

dom1 – open subset of the domain of chart1
dom2 – (default: None) open subset of the domain of chart2; if None, dom1 is assumed

OUTPUT:

the transition map between the charts restricted to the specified subsets

EXAMPLES:

sage: M = Manifold(2, 'M', structure='topological')
sage: X.<x,y> = M.chart()
sage: Y.<u,v> = M.chart()
sage: X_to_Y = X.transition_map(Y, [x+y, x-y])
sage: U = M.open_subset('U', coord_def={X: x>0, Y: u+v>0})
sage: X_to_Y_U = X_to_Y.restrict(U); X_to_Y_U
Change of coordinates from Chart (U, (x, y)) to Chart (U, (u, v))
sage: X_to_Y_U.display()
u = x + y
v = x - y

The result is cached:

sage: X_to_Y.restrict(U) is X_to_Y_U
True

set_inverse(*transformations, **kwds)#

Sets the inverse of the coordinate transformation.

This is useful when the automatic computation via inverse() fails.

INPUT:

transformations – the inverse transformations expressed as a list of the expressions of the “old” coordinates in terms of the “new” ones
kwds – optional arguments; valid keywords are
- check (default: True) – boolean determining whether the provided transformations are checked to be indeed the inverse coordinate transformations
- verbose (default: False) – boolean determining whether some details of the check are printed out; if False, no output is printed if the check is passed (see example below)

EXAMPLES:

From spherical coordinates to Cartesian ones in the plane:

sage: M = Manifold(2, 'R^2', structure='topological')
sage: U = M.open_subset('U') # complement of the half line {y=0, x>= 0}
sage: c_cart.<x,y> = U.chart()
sage: c_spher.<r,ph> = U.chart(r'r:(0,+oo) ph:(0,2*pi):\phi')
sage: spher_to_cart = c_spher.transition_map(c_cart,
....:                                        [r*cos(ph), r*sin(ph)])
sage: spher_to_cart.set_inverse(sqrt(x^2+y^2), atan2(y,x))
Check of the inverse coordinate transformation:
  r == r  *passed*
  ph == arctan2(r*sin(ph), r*cos(ph))  **failed**
  x == x  *passed*
  y == y  *passed*
NB: a failed report can reflect a mere lack of simplification.

As indicated, the failure for ph is due to a lack of simplification of the arctan2 term, not to any error in the provided inverse formulas.

We have now:

sage: spher_to_cart.inverse()
Change of coordinates from Chart (U, (x, y)) to Chart (U, (r, ph))
sage: spher_to_cart.inverse().display()
r = sqrt(x^2 + y^2)
ph = arctan2(y, x)
sage: M.coord_changes()  # random (dictionary output)
{(Chart (U, (r, ph)),
  Chart (U, (x, y))): Change of coordinates from Chart (U, (r, ph))
   to Chart (U, (x, y)),
 (Chart (U, (x, y)),
  Chart (U, (r, ph))): Change of coordinates from Chart (U, (x, y))
   to Chart (U, (r, ph))}

One can suppress the check of the provided formulas by means of the optional argument check=False:

sage: spher_to_cart.set_inverse(sqrt(x^2+y^2), atan2(y,x),
....:                           check=False)

However, it is not recommended to do so, the check being (obviously) useful to avoid some mistake. For instance, if the term sqrt(x^2+y^2) contains a typo (x^3 instead of x^2), we get:

sage: spher_to_cart.set_inverse(sqrt(x^3+y^2), atan2(y,x))
Check of the inverse coordinate transformation:
  r == sqrt(r*cos(ph)^3 + sin(ph)^2)*r  **failed**
  ph == arctan2(r*sin(ph), r*cos(ph))  **failed**
  x == sqrt(x^3 + y^2)*x/sqrt(x^2 + y^2)  **failed**
  y == sqrt(x^3 + y^2)*y/sqrt(x^2 + y^2)  **failed**
NB: a failed report can reflect a mere lack of simplification.

If the check is passed, no output is printed out:

sage: M = Manifold(2, 'M')
sage: X1.<x,y> = M.chart()
sage: X2.<u,v> = M.chart()
sage: X1_to_X2 = X1.transition_map(X2, [x+y, x-y])
sage: X1_to_X2.set_inverse((u+v)/2, (u-v)/2)

unless the option verbose is set to True:

sage: X1_to_X2.set_inverse((u+v)/2, (u-v)/2, verbose=True)
Check of the inverse coordinate transformation:
  x == x  *passed*
  y == y  *passed*
  u == u  *passed*
  v == v  *passed*

class sage.manifolds.chart.RealChart(domain, coordinates, calc_method=None, bounds=None, periods=None, coord_restrictions=None)#

Bases: Chart

Chart on a topological manifold over \(\RR\).

Given a topological manifold \(M\) of dimension \(n\) over \(\RR\), a chart on \(M\) is a pair \((U,\varphi)\), where \(U\) is an open subset of \(M\) and \(\varphi : U \to V \subset \RR^n\) is a homeomorphism from \(U\) to an open subset \(V\) of \(\RR^n\).

The components \((x^1, \ldots, x^n)\) of \(\varphi\), defined by \(\varphi(p) = (x^1(p), \ldots, x^n(p))\in \RR^n\) for any point \(p \in U\), are called the coordinates of the chart \((U, \varphi)\).

INPUT:

domain – open subset \(U\) on which the chart is defined
coordinates – (default: ‘’ (empty string)) single string defining the coordinate symbols, with ' ' (whitespace) as a separator; each item has at most four fields, separated by a colon (:):
1. the coordinate symbol (a letter or a few letters)
2. (optional) the interval \(I\) defining the coordinate range: if not provided, the coordinate is assumed to span all \(\RR\); otherwise \(I\) must be provided in the form (a,b) (or equivalently ]a,b[); the bounds a and b can be +/-Infinity, Inf, infinity, inf or oo; for singular coordinates, non-open intervals such as [a,b] and (a,b] (or equivalently ]a,b]) are allowed; note that the interval declaration must not contain any whitespace
3. (optional) indicator of the periodic character of the coordinate, either as period=T, where T is the period, or as the keyword periodic (the value of the period is then deduced from the interval \(I\) declared in field 2; see examples below)
4. (optional) the LaTeX spelling of the coordinate; if not provided the coordinate symbol given in the first field will be used
The order of fields 2 to 4 does not matter and each of them can be omitted. If it contains any LaTeX expression, the string coordinates must be declared with the prefix ‘r’ (for “raw”) to allow for a proper treatment of LaTeX’s backslash character (see examples below). If interval range, no period and no LaTeX spelling are to be set for any coordinate, the argument coordinates can be omitted when the shortcut operator <,> is used to declare the chart (see examples below).
calc_method – (default: None) string defining the calculus method for computations involving coordinates of the chart; must be one of
- 'SR': Sage’s default symbolic engine (Symbolic Ring)
- 'sympy': SymPy
- None: the default of CalculusMethod will be used
names – (default: None) unused argument, except if coordinates is not provided; it must then be a tuple containing the coordinate symbols (this is guaranteed if the shortcut operator <,> is used)
coord_restrictions: Additional restrictions on the coordinates. A restriction can be any symbolic equality or inequality involving the coordinates, such as x > y or x^2 + y^2 != 0. The items of the list (or set or frozenset) coord_restrictions are combined with the and operator; if some restrictions are to be combined with the or operator instead, they have to be passed as a tuple in some single item of the list (or set or frozenset) coord_restrictions. For example:
```
coord_restrictions=[x > y, (x != 0, y != 0), z^2 < x]
```
means (x > y) and ((x != 0) or (y != 0)) and (z^2 < x). If the list coord_restrictions contains only one item, this item can be passed as such, i.e. writing x > y instead of the single element list [x > y]. If the chart variables have not been declared as variables yet, coord_restrictions must be lambda-quoted.

EXAMPLES:

Cartesian coordinates on \(\RR^3\):

sage: M = Manifold(3, 'R^3', r'\RR^3', structure='topological',
....:              start_index=1)
sage: c_cart = M.chart('x y z'); c_cart
Chart (R^3, (x, y, z))
sage: type(c_cart)
<class 'sage.manifolds.chart.RealChart'>

To have the coordinates accessible as global variables, one has to set:

sage: (x,y,z) = c_cart[:]

However, a shortcut is to use the declarator <x,y,z> in the left-hand side of the chart declaration (there is then no need to pass the string 'x y z' to chart()):

sage: M = Manifold(3, 'R^3', r'\RR^3', structure='topological',
....:              start_index=1)
sage: c_cart.<x,y,z> = M.chart(); c_cart
Chart (R^3, (x, y, z))

The coordinates are then immediately accessible:

sage: y
y
sage: y is c_cart[2]
True

Note that x, y, z declared in <x,y,z> are mere Python variable names and do not have to coincide with the coordinate symbols; for instance, one may write:

sage: M = Manifold(3, 'R^3', r'\RR^3', structure='topological',
....:              start_index=1)
sage: c_cart.<x1,y1,z1> = M.chart('x y z'); c_cart
Chart (R^3, (x, y, z))

Then y is not known as a global variable and the coordinate \(y\) is accessible only through the global variable y1:

sage: y1
y
sage: y1 is c_cart[2]
True

However, having the name of the Python variable coincide with the coordinate symbol is quite convenient; so it is recommended to declare:

sage: forget()   # for doctests only
sage: M = Manifold(3, 'R^3', r'\RR^3', structure='topological', start_index=1)
sage: c_cart.<x,y,z> = M.chart()

Spherical coordinates on the subset \(U\) of \(\RR^3\) that is the complement of the half-plane \(\{y=0, x \geq 0\}\):

sage: U = M.open_subset('U')
sage: c_spher.<r,th,ph> = U.chart(r'r:(0,+oo) th:(0,pi):\theta ph:(0,2*pi):\phi')
sage: c_spher
Chart (U, (r, th, ph))

Note the prefix ‘r’ for the string defining the coordinates in the arguments of chart.

Coordinates are Sage symbolic variables (see sage.symbolic.expression):

sage: type(th)
<class 'sage.symbolic.expression.Expression'>
sage: latex(th)
{\theta}
sage: assumptions(th)
[th is real, th > 0, th < pi]

Coordinate are also accessible by their indices:

sage: x1 = c_spher[1]; x2 = c_spher[2]; x3 = c_spher[3]
sage: [x1, x2, x3]
[r, th, ph]
sage: (x1, x2, x3) == (r, th, ph)
True

The full set of coordinates is obtained by means of the slice [:]:

sage: c_cart[:]
(x, y, z)
sage: c_spher[:]
(r, th, ph)

Let us check that the declared coordinate ranges have been taken into account:

sage: c_cart.coord_range()
x: (-oo, +oo); y: (-oo, +oo); z: (-oo, +oo)
sage: c_spher.coord_range()
r: (0, +oo); th: (0, pi); ph: (0, 2*pi)
sage: bool(th>0 and th<pi)
True
sage: assumptions()  # list all current symbolic assumptions
[x is real, y is real, z is real, r is real, r > 0, th is real,
 th > 0, th < pi, ph is real, ph > 0, ph < 2*pi]

The coordinate ranges are used for simplifications:

sage: simplify(abs(r)) # r has been declared to lie in the interval (0,+oo)
r
sage: simplify(abs(x)) # no positive range has been declared for x
abs(x)

A coordinate can be declared periodic by adding the keyword periodic to its range:

sage: V = M.open_subset('V')
sage: c_spher1.<r,th,ph1> = \
....: V.chart(r'r:(0,+oo) th:(0,pi):\theta ph1:(0,2*pi):periodic:\phi_1')
sage: c_spher1.periods()
(None, None, 2*pi)
sage: c_spher1.coord_range()
r: (0, +oo); th: (0, pi); ph1: [0, 2*pi] (periodic)

It is equivalent to give the period as period=2*pi, skipping the coordinate range:

sage: c_spher2.<r,th,ph2> = \
....: V.chart(r'r:(0,+oo) th:(0,pi):\theta ph2:period=2*pi:\phi_2')
sage: c_spher2.periods()
(None, None, 2*pi)
sage: c_spher2.coord_range()
r: (0, +oo); th: (0, pi); ph2: [0, 2*pi] (periodic)

Each constructed chart is automatically added to the manifold’s user atlas:

sage: M.atlas()
[Chart (R^3, (x, y, z)), Chart (U, (r, th, ph)),
 Chart (V, (r, th, ph1)), Chart (V, (r, th, ph2))]

and to the atlas of its domain:

sage: U.atlas()
[Chart (U, (r, th, ph))]

Manifold subsets have a default chart, which, unless changed via the method set_default_chart(), is the first defined chart on the subset (or on a open subset of it):

sage: M.default_chart()
Chart (R^3, (x, y, z))
sage: U.default_chart()
Chart (U, (r, th, ph))

The default charts are not privileged charts on the manifold, but rather charts whose name can be skipped in the argument list of functions having an optional chart= argument.

The chart map \(\varphi\) acting on a point is obtained by means of the call operator, i.e. the operator ():

sage: p = M.point((1,0,-2)); p
Point on the 3-dimensional topological manifold R^3
sage: c_cart(p)
(1, 0, -2)
sage: c_cart(p) == p.coord(c_cart)
True
sage: q = M.point((2,pi/2,pi/3), chart=c_spher) # point defined by its spherical coordinates
sage: c_spher(q)
(2, 1/2*pi, 1/3*pi)
sage: c_spher(q) == q.coord(c_spher)
True
sage: a = U.point((1,pi/2,pi)) # the default coordinates on U are the spherical ones
sage: c_spher(a)
(1, 1/2*pi, pi)
sage: c_spher(a) == a.coord(c_spher)
True

Cartesian coordinates on \(U\) as an example of chart construction with coordinate restrictions: since \(U\) is the complement of the half-plane \(\{y = 0, x \geq 0\}\), we must have \(y \neq 0\) or \(x < 0\) on U. Accordingly, we set:

sage: c_cartU.<x,y,z> = U.chart(coord_restrictions=lambda x,y,z: (y!=0, x<0))
sage: U.atlas()
[Chart (U, (r, th, ph)), Chart (U, (x, y, z))]
sage: M.atlas()
[Chart (R^3, (x, y, z)), Chart (U, (r, th, ph)),
 Chart (V, (r, th, ph1)), Chart (V, (r, th, ph2)),
 Chart (U, (x, y, z))]
sage: c_cartU.valid_coordinates(-1,0,2)
True
sage: c_cartU.valid_coordinates(1,0,2)
False
sage: c_cart.valid_coordinates(1,0,2)
True

Note that, as an example, the following would have meant \(y \neq 0\) and \(x < 0\):

c_cartU.<x,y,z> = U.chart(coord_restrictions=lambda x,y,z: [y!=0, x<0])

Chart grids can be drawn in 2D or 3D graphics thanks to the method plot().

add_restrictions(restrictions)#

Add some restrictions on the coordinates.

This is deprecated; provide the restrictions at the time of creating the chart.

INPUT:

restrictions – list of restrictions on the coordinates, in addition to the ranges declared by the intervals specified in the chart constructor

restrictions = [x > y, (x != 0, y != 0), z^2 < x]

EXAMPLES:

Cartesian coordinates on the open unit disc in \(\RR^2\):

sage: M = Manifold(2, 'M', structure='topological') # the open unit disc
sage: X.<x,y> = M.chart()
sage: X.add_restrictions(x^2+y^2<1)
doctest:warning...
DeprecationWarning: Chart.add_restrictions is deprecated; provide the
restrictions at the time of creating the chart
See https://github.com/sagemath/sage/issues/32102 for details.
sage: X.valid_coordinates(0,2)
False
sage: X.valid_coordinates(0,1/3)
True

The restrictions are transmitted to subcharts:

sage: A = M.open_subset('A') # annulus 1/2 < r < 1
sage: X_A = X.restrict(A, x^2+y^2 > 1/4)
sage: X_A._restrictions
[x^2 + y^2 < 1, x^2 + y^2 > (1/4)]
sage: X_A.valid_coordinates(0,1/3)
False
sage: X_A.valid_coordinates(2/3,1/3)
True

If appropriate, the restrictions are transformed into bounds on the coordinate ranges:

sage: U = M.open_subset('U')
sage: X_U = X.restrict(U)
sage: X_U.coord_range()
x: (-oo, +oo); y: (-oo, +oo)
sage: X_U.add_restrictions([x<0, y>1/2])
sage: X_U.coord_range()
x: (-oo, 0); y: (1/2, +oo)

codomain()#

Return the codomain of self as a set.

EXAMPLES:

sage: M = Manifold(2, 'R^2', structure='topological')
sage: U = M.open_subset('U') # the complement of the half line {y=0, x >= 0}
sage: c_spher.<r,phi> = U.chart(r'r:(0,+oo) phi:(0,2*pi):\phi')
sage: c_spher.codomain()
The Cartesian product of ((0, +oo), (0, 2*pi))

sage: M = Manifold(3, 'R^3', r'\RR^3', structure='topological', start_index=1)
sage: c_cart.<x,y,z> = M.chart()
sage: c_cart.codomain()
Vector space of dimension 3 over Real Field with 53 bits of precision

In the current implementation, the codomain of periodic coordinates are represented by a fundamental domain:

sage: V = M.open_subset('V')
sage: c_spher1.<r,th,ph1> = \
....: V.chart(r'r:(0,+oo) th:(0,pi):\theta ph1:(0,2*pi):periodic:\phi_1')
sage: c_spher1.codomain()
The Cartesian product of ((0, +oo), (0, pi), [0, 2*pi))

coord_bounds(i=None)#

Return the lower and upper bounds of the range of a coordinate.

For a nicely formatted output, use coord_range() instead.

INPUT:

i – (default: None) index of the coordinate; if None, the bounds of all the coordinates are returned

OUTPUT:

the coordinate bounds as the tuple ((xmin, min_included), (xmax, max_included)) where
- xmin is the coordinate lower bound
- min_included is a boolean, indicating whether the coordinate can take the value xmin, i.e. xmin is a strict lower bound iff min_included is False
- xmin is the coordinate upper bound
- max_included is a boolean, indicating whether the coordinate can take the value xmax, i.e. xmax is a strict upper bound iff max_included is False

EXAMPLES:

Some coordinate bounds on a 2-dimensional manifold:

sage: forget()  # for doctests only
sage: M = Manifold(2, 'M', structure='topological')
sage: c_xy.<x,y> = M.chart('x y:[0,1)')
sage: c_xy.coord_bounds(0)  # x in (-oo,+oo) (the default)
((-Infinity, False), (+Infinity, False))
sage: c_xy.coord_bounds(1)  # y in [0,1)
((0, True), (1, False))
sage: c_xy.coord_bounds()
(((-Infinity, False), (+Infinity, False)), ((0, True), (1, False)))
sage: c_xy.coord_bounds() == (c_xy.coord_bounds(0), c_xy.coord_bounds(1))
True

The coordinate bounds can also be recovered via the method coord_range():

sage: c_xy.coord_range()
x: (-oo, +oo); y: [0, 1)
sage: c_xy.coord_range(y)
y: [0, 1)

or via Sage’s function sage.symbolic.assumptions.assumptions():

sage: assumptions(x)
[x is real]
sage: assumptions(y)
[y is real, y >= 0, y < 1]

coord_range(xx=None)#

Display the range of a coordinate (or all coordinates), as an interval.

INPUT:

xx – (default: None) symbolic expression corresponding to a coordinate of the current chart; if None, the ranges of all coordinates are displayed

EXAMPLES:

Ranges of coordinates on a 2-dimensional manifold:

sage: M = Manifold(2, 'M', structure='topological')
sage: X.<x,y> = M.chart()
sage: X.coord_range()
x: (-oo, +oo); y: (-oo, +oo)
sage: X.coord_range(x)
x: (-oo, +oo)
sage: U = M.open_subset('U', coord_def={X: [x>1, y<pi]})
sage: XU = X.restrict(U)  # restriction of chart X to U
sage: XU.coord_range()
x: (1, +oo); y: (-oo, pi)
sage: XU.coord_range(x)
x: (1, +oo)
sage: XU.coord_range(y)
y: (-oo, pi)

The output is LaTeX-formatted for the notebook:

sage: latex(XU.coord_range(y))
y :\ \left( -\infty, \pi \right)

plot(chart=None, ambient_coords=None, mapping=None, fixed_coords=None, ranges=None, number_values=None, steps=None, parameters=None, max_range=8, color='red', style='-', thickness=1, plot_points=75, label_axes=True, **kwds)#

Plot self as a grid in a Cartesian graph based on the coordinates of some ambient chart.

The grid is formed by curves along which a chart coordinate varies, the other coordinates being kept fixed. It is drawn in terms of two (2D graphics) or three (3D graphics) coordinates of another chart, called hereafter the ambient chart.

The ambient chart is related to the current chart either by a transition map if both charts are defined on the same manifold, or by the coordinate expression of some continuous map (typically an immersion). In the latter case, the two charts may be defined on two different manifolds.

INPUT:

chart – (default: None) the ambient chart (see above); if None, the ambient chart is set to the current chart
ambient_coords – (default: None) tuple containing the 2 or 3 coordinates of the ambient chart in terms of which the plot is performed; if None, all the coordinates of the ambient chart are considered
mapping – (default: None) ContinuousMap; continuous manifold map providing the link between the current chart and the ambient chart (cf. above); if None, both charts are supposed to be defined on the same manifold and related by some transition map (see transition_map())
fixed_coords – (default: None) dictionary with keys the chart coordinates that are not drawn and with values the fixed value of these coordinates; if None, all the coordinates of the current chart are drawn
ranges – (default: None) dictionary with keys the coordinates to be drawn and values tuples (x_min, x_max) specifying the coordinate range for the plot; if None, the entire coordinate range declared during the chart construction is considered (with -Infinity replaced by -max_range and +Infinity by max_range)
number_values – (default: None) either an integer or a dictionary with keys the coordinates to be drawn and values the number of constant values of the coordinate to be considered; if number_values is a single integer, it represents the number of constant values for all coordinates; if number_values is None, it is set to 9 for a 2D plot and to 5 for a 3D plot
steps – (default: None) dictionary with keys the coordinates to be drawn and values the step between each constant value of the coordinate; if None, the step is computed from the coordinate range (specified in ranges) and number_values. On the contrary if the step is provided for some coordinate, the corresponding number of constant values is deduced from it and the coordinate range.
parameters – (default: None) dictionary giving the numerical values of the parameters that may appear in the relation between the two coordinate systems
max_range – (default: 8) numerical value substituted to +Infinity if the latter is the upper bound of the range of a coordinate for which the plot is performed over the entire coordinate range (i.e. for which no specific plot range has been set in ranges); similarly -max_range is the numerical valued substituted for -Infinity
color – (default: 'red') either a single color or a dictionary of colors, with keys the coordinates to be drawn, representing the colors of the lines along which the coordinate varies, the other being kept constant; if color is a single color, it is used for all coordinate lines
style – (default: '-') either a single line style or a dictionary of line styles, with keys the coordinates to be drawn, representing the style of the lines along which the coordinate varies, the other being kept constant; if style is a single style, it is used for all coordinate lines; NB: style is effective only for 2D plots
thickness – (default: 1) either a single line thickness or a dictionary of line thicknesses, with keys the coordinates to be drawn, representing the thickness of the lines along which the coordinate varies, the other being kept constant; if thickness is a single value, it is used for all coordinate lines
plot_points – (default: 75) either a single number of points or a dictionary of integers, with keys the coordinates to be drawn, representing the number of points to plot the lines along which the coordinate varies, the other being kept constant; if plot_points is a single integer, it is used for all coordinate lines
label_axes – (default: True) boolean determining whether the labels of the ambient coordinate axes shall be added to the graph; can be set to False if the graph is 3D and must be superposed with another graph

OUTPUT:

a graphic object, either a Graphics for a 2D plot (i.e. based on 2 coordinates of the ambient chart) or a Graphics3d for a 3D plot (i.e. based on 3 coordinates of the ambient chart)

EXAMPLES:

A 2-dimensional chart plotted in terms of itself results in a rectangular grid:

sage: R2 = Manifold(2, 'R^2', structure='topological')  # the Euclidean plane
sage: c_cart.<x,y> = R2.chart()  # Cartesian coordinates
sage: g = c_cart.plot(); g  # equivalent to c_cart.plot(c_cart)             # needs sage.plot
Graphics object consisting of 18 graphics primitives

Grid of polar coordinates in terms of Cartesian coordinates in the Euclidean plane:

sage: U = R2.open_subset('U', coord_def={c_cart: (y!=0, x<0)})  # the complement of the segment y=0 and x>0
sage: c_pol.<r,ph> = U.chart(r'r:(0,+oo) ph:(0,2*pi):\phi')  # polar coordinates on U
sage: pol_to_cart = c_pol.transition_map(c_cart, [r*cos(ph), r*sin(ph)])
sage: g = c_pol.plot(c_cart); g                                             # needs sage.plot
Graphics object consisting of 18 graphics primitives

Call with non-default values:

sage: g = c_pol.plot(c_cart, ranges={ph:(pi/4,pi)},                         # needs sage.plot
....:                number_values={r:7, ph:17},
....:                color={r:'red', ph:'green'},
....:                style={r:'-', ph:'--'})

A single coordinate line can be drawn:

sage: g = c_pol.plot(c_cart,    # draw a circle of radius r=2               # needs sage.plot
....:                fixed_coords={r: 2})

sage: g = c_pol.plot(c_cart,    # draw a segment at phi=pi/4                # needs sage.plot
....:                fixed_coords={ph: pi/4})

An example with the ambient chart lying in an another manifold (the plot is then performed via some manifold map passed as the argument mapping): 3D plot of the stereographic charts on the 2-sphere:

sage: S2 = Manifold(2, 'S^2', structure='topological')  # the 2-sphere
sage: U = S2.open_subset('U'); V = S2.open_subset('V')  # complement of the North and South pole, respectively
sage: S2.declare_union(U,V)
sage: c_xy.<x,y> = U.chart()  # stereographic coordinates from the North pole
sage: c_uv.<u,v> = V.chart()  # stereographic coordinates from the South pole
sage: xy_to_uv = c_xy.transition_map(c_uv, (x/(x^2+y^2), y/(x^2+y^2)),
....:                 intersection_name='W', restrictions1= x^2+y^2!=0,
....:                 restrictions2= u^2+v^2!=0)
sage: uv_to_xy = xy_to_uv.inverse()
sage: R3 = Manifold(3, 'R^3', structure='topological')  # the Euclidean space R^3
sage: c_cart.<X,Y,Z> = R3.chart()  # Cartesian coordinates on R^3
sage: Phi = S2.continuous_map(R3, {(c_xy, c_cart): [2*x/(1+x^2+y^2),
....:                          2*y/(1+x^2+y^2), (x^2+y^2-1)/(1+x^2+y^2)],
....:                          (c_uv, c_cart): [2*u/(1+u^2+v^2),
....:                          2*v/(1+u^2+v^2), (1-u^2-v^2)/(1+u^2+v^2)]},
....:                         name='Phi', latex_name=r'\Phi')  # Embedding of S^2 in R^3
sage: g = c_xy.plot(c_cart, mapping=Phi); g                                 # needs sage.plot
Graphics3d Object

NB: to get a better coverage of the whole sphere, one should increase the coordinate sampling via the argument number_values or the argument steps (only the default value, number_values = 5, is used here, which is pretty low).

The same plot without the (X,Y,Z) axes labels:

sage: g = c_xy.plot(c_cart, mapping=Phi, label_axes=False)                  # needs sage.plot

The North and South stereographic charts on the same plot:

sage: g2 = c_uv.plot(c_cart, mapping=Phi, color='green')                    # needs sage.plot
sage: g + g2                                                                # needs sage.plot
Graphics3d Object

South stereographic chart drawn in terms of the North one (we split the plot in four parts to avoid the singularity at \((u,v)=(0,0)\)):

sage: # long time, needs sage.plot
sage: W = U.intersection(V) # the subset common to both charts
sage: c_uvW = c_uv.restrict(W) # chart (W,(u,v))
sage: gSN1 = c_uvW.plot(c_xy, ranges={u:[-6.,-0.02], v:[-6.,-0.02]})
sage: gSN2 = c_uvW.plot(c_xy, ranges={u:[-6.,-0.02], v:[0.02,6.]})
sage: gSN3 = c_uvW.plot(c_xy, ranges={u:[0.02,6.], v:[-6.,-0.02]})
sage: gSN4 = c_uvW.plot(c_xy, ranges={u:[0.02,6.], v:[0.02,6.]})
sage: show(gSN1+gSN2+gSN3+gSN4, xmin=-1.5, xmax=1.5, ymin=-1.5, ymax=1.5)

The coordinate line \(u = 1\) (red) and the coordinate line \(v = 1\) (green) on the same plot:

sage: # long time, needs sage.plot
sage: gu1 = c_uvW.plot(c_xy, fixed_coords={u: 1}, max_range=20,
....:                  plot_points=300)
sage: gv1 = c_uvW.plot(c_xy, fixed_coords={v: 1}, max_range=20,
....:                  plot_points=300, color='green')
sage: gu1 + gv1
Graphics object consisting of 2 graphics primitives

Note that we have set max_range=20 to have a wider range for the coordinates \(u\) and \(v\), i.e. to have \([-20, 20]\) instead of the default \([-8, 8]\).

A 3-dimensional chart plotted in terms of itself results in a 3D rectangular grid:

sage: # long time, needs sage.plot
sage: g = c_cart.plot()  # equivalent to c_cart.plot(c_cart)
sage: g
Graphics3d Object

A 4-dimensional chart plotted in terms of itself (the plot is performed for at most 3 coordinates, which must be specified via the argument ambient_coords):

sage: # needs sage.plot
sage: M = Manifold(4, 'M', structure='topological')
sage: X.<t,x,y,z> = M.chart()
sage: g = X.plot(ambient_coords=(t,x,y))  # the coordinate z is not depicted  # long time
sage: g                                                                       # long time
Graphics3d Object

sage: # needs sage.plot
sage: g = X.plot(ambient_coords=(t,y))  # the coordinates x and z are not depicted
sage: g
Graphics object consisting of 18 graphics primitives

Note that the default values of some arguments of the method plot are stored in the dictionary plot.options:

sage: X.plot.options  # random (dictionary output)
{'color': 'red', 'label_axes': True, 'max_range': 8,
 'plot_points': 75, 'style': '-', 'thickness': 1}

so that they can be adjusted by the user:

sage: X.plot.options['color'] = 'blue'

From now on, all chart plots will use blue as the default color. To restore the original default options, it suffices to type:

sage: X.plot.reset()

restrict(subset, restrictions=None)#

Return the restriction of the chart to some open subset of its domain.

If the current chart is \((U, \varphi)\), a restriction (or subchart) is a chart \((V, \psi)\) such that \(V \subset U\) and \(\psi = \varphi|_V\).

If such subchart has not been defined yet, it is constructed here.

The coordinates of the subchart bare the same names as the coordinates of the current chart.

INPUT:

subset – open subset \(V\) of the chart domain \(U\) (must be an instance of TopologicalManifold)
restrictions – (default: None) list of coordinate restrictions defining the subset \(V\)

restrictions = [x > y, (x != 0, y != 0), z^2 < x]

OUTPUT:

the chart \((V, \psi)\) as a RealChart

EXAMPLES:

Cartesian coordinates on the unit open disc in \(\RR^2\) as a subchart of the global Cartesian coordinates:

sage: M = Manifold(2, 'R^2', structure='topological')
sage: c_cart.<x,y> = M.chart() # Cartesian coordinates on R^2
sage: D = M.open_subset('D') # the unit open disc
sage: c_cart_D = c_cart.restrict(D, x^2+y^2<1)
sage: p = M.point((1/2, 0))
sage: p in D
True
sage: q = M.point((1, 2))
sage: q in D
False

Cartesian coordinates on the annulus \(1 < \sqrt{x^2 + y^2} < 2\):

sage: A = M.open_subset('A')
sage: c_cart_A = c_cart.restrict(A, [x^2+y^2>1, x^2+y^2<4])
sage: p in A, q in A
(False, False)
sage: a = M.point((3/2,0))
sage: a in A
True

valid_coordinates(*coordinates, **kwds)#

Check whether a tuple of coordinates can be the coordinates of a point in the chart domain.

INPUT:

*coordinates – coordinate values
**kwds – options:
- tolerance=0, to set the absolute tolerance in the test of coordinate ranges
- parameters=None, to set some numerical values to parameters

OUTPUT:

True if the coordinate values are admissible in the chart range and False otherwise

EXAMPLES:

Cartesian coordinates on a square interior:

sage: forget()  # for doctest only
sage: M = Manifold(2, 'M', structure='topological')  # the square interior
sage: X.<x,y> = M.chart('x:(-2,2) y:(-2,2)')
sage: X.valid_coordinates(0,1)
True
sage: X.valid_coordinates(-3/2,5/4)
True
sage: X.valid_coordinates(0,3)
False

The unit open disk inside the square:

sage: D = M.open_subset('D', coord_def={X: x^2+y^2<1})
sage: XD = X.restrict(D)
sage: XD.valid_coordinates(0,1)
False
sage: XD.valid_coordinates(-3/2,5/4)
False
sage: XD.valid_coordinates(-1/2,1/2)
True
sage: XD.valid_coordinates(0,0)
True

Another open subset of the square, defined by \(x^2+y^2<1\) or (\(x>0\) and \(|y|<1\)):

sage: B = M.open_subset('B',
....:                   coord_def={X: (x^2+y^2<1,
....:                                  [x>0, abs(y)<1])})
sage: XB = X.restrict(B)
sage: XB.valid_coordinates(-1/2, 0)
True
sage: XB.valid_coordinates(-1/2, 3/2)
False
sage: XB.valid_coordinates(3/2, 1/2)
True

valid_coordinates_numerical(*coordinates)#

Check whether a tuple of float coordinates can be the coordinates of a point in the chart domain.

This version is optimized for float numbers, and cannot accept parameters nor tolerance. The chart restriction must also be specified in CNF (i.e. a list of tuples).

INPUT:

*coordinates – coordinate values

OUTPUT:

True if the coordinate values are admissible in the chart range and False otherwise

EXAMPLES:

Cartesian coordinates on a square interior:

sage: forget()  # for doctest only
sage: M = Manifold(2, 'M', structure='topological')  # the square interior
sage: X.<x,y> = M.chart('x:(-2,2) y:(-2,2)')
sage: X.valid_coordinates_numerical(0,1)
True
sage: X.valid_coordinates_numerical(-3/2,5/4)
True
sage: X.valid_coordinates_numerical(0,3)
False

The unit open disk inside the square:

sage: D = M.open_subset('D', coord_def={X: x^2+y^2<1})
sage: XD = X.restrict(D)
sage: XD.valid_coordinates_numerical(0,1)
False
sage: XD.valid_coordinates_numerical(-3/2,5/4)
False
sage: XD.valid_coordinates_numerical(-1/2,1/2)
True
sage: XD.valid_coordinates_numerical(0,0)
True

Another open subset of the square, defined by \(x^2 + y^2 < 1\) or (\(x > 0\) and \(|y| < 1\)):

sage: B = M.open_subset('B',coord_def={X: [(x^2+y^2<1, x>0),
....:                   (x^2+y^2<1,  abs(y)<1)]})
sage: XB = X.restrict(B)
sage: XB.valid_coordinates_numerical(-1/2, 0)
True
sage: XB.valid_coordinates_numerical(-1/2, 3/2)
False
sage: XB.valid_coordinates_numerical(3/2, 1/2)
True