Curves in Manifolds#

Given a differentiable manifold \(M\), a differentiable curve in \(M\) is a differentiable mapping

\[\gamma: I \longrightarrow M,\]

where \(I\) is an interval of \(\RR\).

Differentiable curves are implemented by DifferentiableCurve.

AUTHORS:

  • Eric Gourgoulhon (2015): initial version

  • Travis Scrimshaw (2016): review tweaks

REFERENCES:

class sage.manifolds.differentiable.curve.DifferentiableCurve(parent, coord_expression=None, name=None, latex_name=None, is_isomorphism=False, is_identity=False)#

Bases: DiffMap

Curve in a differentiable manifold.

Given a differentiable manifold \(M\), a differentiable curve in \(M\) is a differentiable map

\[\gamma: I \longrightarrow M,\]

where \(I\) is an interval of \(\RR\).

INPUT:

  • parent – DifferentiableCurveSet the set of curves \(\mathrm{Hom}(I, M)\) to which the curve belongs

  • coord_expression – (default: None) dictionary (possibly empty) of the functions of the curve parameter \(t\) expressing the curve in various charts of \(M\), the keys of the dictionary being the charts and the values being lists or tuples of \(n\) symbolic expressions of \(t\), where \(n\) is the dimension of \(M\)

  • name – (default: None) string; symbol given to the curve

  • latex_name – (default: None) string; LaTeX symbol to denote the curve; if none is provided, name will be used

  • is_isomorphism – (default: False) determines whether the constructed object is a diffeomorphism; if set to True, then \(M\) must have dimension one

  • is_identity – (default: False) determines whether the constructed object is the identity map; if set to True, then \(M\) must be the interval \(I\)

EXAMPLES:

The lemniscate of Gerono in the 2-dimensional Euclidean plane:

sage: M = Manifold(2, 'M')
sage: X.<x,y> = M.chart()
sage: t = var('t')
sage: c = M.curve({X: [sin(t), sin(2*t)/2]}, (t, 0, 2*pi), name='c') ; c
Curve c in the 2-dimensional differentiable manifold M
sage: type(c)
<class 'sage.manifolds.differentiable.manifold_homset.DifferentiableCurveSet_with_category.element_class'>

Instead of declaring the parameter \(t\) as a symbolic variable by means of var('t'), it is equivalent to get it as the canonical coordinate of the real number line (see RealLine):

sage: R.<t> = manifolds.RealLine()
sage: c = M.curve({X: [sin(t), sin(2*t)/2]}, (t, 0, 2*pi), name='c') ; c
Curve c in the 2-dimensional differentiable manifold M

A graphical view of the curve is provided by the method plot():

sage: c.plot(aspect_ratio=1)                                                    # needs sage.plot
Graphics object consisting of 1 graphics primitive
../../../_images/curve-1.svg

Curves are considered as (manifold) morphisms from real intervals to differentiable manifolds:

sage: c.parent()
Set of Morphisms from Real interval (0, 2*pi) to 2-dimensional
 differentiable manifold M in Category of smooth manifolds over Real
 Field with 53 bits of precision
sage: I = R.open_interval(0, 2*pi)
sage: c.parent() is Hom(I, M)
True
sage: c.domain()
Real interval (0, 2*pi)
sage: c.domain() is I
True
sage: c.codomain()
2-dimensional differentiable manifold M

Accordingly, all methods of DiffMap are available for them. In particular, the method display() shows the coordinate representations in various charts of manifold M:

sage: c.display()
c: (0, 2*pi) → M
   t ↦ (x, y) = (sin(t), 1/2*sin(2*t))

Another map method is using the usual call syntax, which returns the image of a point in the curve’s domain:

sage: t0 = pi/2
sage: I(t0)
Point on the Real number line ℝ
sage: c(I(t0))
Point on the 2-dimensional differentiable manifold M
sage: c(I(t0)).coord(X)
(1, 0)

For curves, the value of the parameter, instead of the corresponding point in the real line manifold, can be passed directly:

sage: c(t0)
Point c(1/2*pi) on the 2-dimensional differentiable manifold M
sage: c(t0).coord(X)
(1, 0)
sage: c(t0) == c(I(t0))
True

Instead of a dictionary of coordinate expressions, the curve can be defined by a single coordinate expression in a given chart:

sage: c = M.curve([sin(t), sin(2*t)/2], (t, 0, 2*pi), chart=X, name='c') ; c
Curve c in the 2-dimensional differentiable manifold M
sage: c.display()
c: (0, 2*pi) → M
   t ↦ (x, y) = (sin(t), 1/2*sin(2*t))

Since X is the default chart on M, it can be omitted:

sage: c = M.curve([sin(t), sin(2*t)/2], (t, 0, 2*pi), name='c') ; c
Curve c in the 2-dimensional differentiable manifold M
sage: c.display()
c: (0, 2*pi) → M
   t ↦ (x, y) = (sin(t), 1/2*sin(2*t))

Note that a curve in \(M\) can also be created as a differentiable map \(I \to M\):

sage: c1 = I.diff_map(M, coord_functions={X: [sin(t), sin(2*t)/2]},
....:                 name='c') ; c1
Curve c in the 2-dimensional differentiable manifold M
sage: c1.parent() is c.parent()
True
sage: c1 == c
True

LaTeX symbols representing a curve:

sage: c = M.curve([sin(t), sin(2*t)/2], (t, 0, 2*pi))
sage: latex(c)
\text{Curve in the 2-dimensional differentiable manifold M}
sage: c = M.curve([sin(t), sin(2*t)/2], (t, 0, 2*pi), name='c')
sage: latex(c)
c
sage: c = M.curve([sin(t), sin(2*t)/2], (t, 0, 2*pi), name='c',
....:             latex_name=r'\gamma')
sage: latex(c)
\gamma

The curve’s tangent vector field (velocity vector):

sage: v = c.tangent_vector_field() ; v
Vector field c' along the Real interval (0, 2*pi) with values on the
 2-dimensional differentiable manifold M
sage: v.display()
c' = cos(t) ∂/∂x + (2*cos(t)^2 - 1) ∂/∂y

Plot of the curve and its tangent vector field:

sage: show(c.plot(thickness=2, aspect_ratio=1) +
....:      v.plot(chart=X, number_values=17, scale=0.5))
../../../_images/curve-2.svg

Value of the tangent vector field at \(t = \pi\):

sage: v.at(R(pi))
Tangent vector c' at Point on the 2-dimensional differentiable
 manifold M
sage: v.at(R(pi)) in M.tangent_space(c(R(pi)))
True
sage: v.at(R(pi)).display()
c' = -∂/∂x + ∂/∂y

Curves \(\RR \to \RR\) can be composed: the operator \(\circ\) is given by *:

sage: f = R.curve(t^2, (t,-oo,+oo))
sage: g = R.curve(cos(t), (t,-oo,+oo))
sage: s = g*f ; s
Differentiable map from the Real number line ℝ to itself
sage: s.display()
ℝ → ℝ
   t ↦ cos(t^2)
sage: s = f*g ; s
Differentiable map from the Real number line ℝ to itself
sage: s.display()
ℝ → ℝ
   t ↦ cos(t)^2

Curvature and torsion of a curve in a Riemannian manifold

Let us consider a helix \(C\) in the Euclidean space \(\mathbb{E}^3\) parametrized by its arc length \(s\):

sage: E.<x,y,z> = EuclideanSpace()
sage: R.<s> = manifolds.RealLine()
sage: C = E.curve((2*cos(s/3), 2*sin(s/3), sqrt(5)*s/3), (s, 0, 6*pi),
....:             name='C')

Its tangent vector field is:

sage: T = C.tangent_vector_field()
sage: T.display()
C' = -2/3*sin(1/3*s) e_x + 2/3*cos(1/3*s) e_y + 1/3*sqrt(5) e_z

Since \(C\) is parametrized by its arc length \(s\), \(T\) is a unit vector (with respect to the Euclidean metric of \(\mathbb{E}^3\)):

sage: norm(T)
Scalar field |C'| on the Real interval (0, 6*pi)
sage: norm(T).display()
|C'|: (0, 6*pi) → ℝ
   s ↦ 1

Vector fields along \(C\) are defined by the method vector_field() of the domain of \(C\) with the keyword argument dest_map set to \(C\). For instance the derivative vector \(T'=\mathrm{d}T/\mathrm{d}s\) is:

sage: I = C.domain(); I
Real interval (0, 6*pi)
sage: Tp = I.vector_field([diff(T[i], s) for i in E.irange()], dest_map=C,
....:                     name="T'")
sage: Tp.display()
T' = -2/9*cos(1/3*s) e_x - 2/9*sin(1/3*s) e_y

The norm of \(T'\) is the curvature of the helix:

sage: kappa = norm(Tp)
sage: kappa
Scalar field |T'| on the Real interval (0, 6*pi)
sage: kappa.expr()
2/9

The unit normal vector along \(C\) is:

sage: N = Tp / kappa
sage: N.display()
-cos(1/3*s) e_x - sin(1/3*s) e_y

while the binormal vector along \(C\) is \(B = T \times N\):

sage: B = T.cross_product(N)
sage: B
Vector field along the Real interval (0, 6*pi) with values on the
 Euclidean space E^3
sage: B.display()
1/3*sqrt(5)*sin(1/3*s) e_x - 1/3*sqrt(5)*cos(1/3*s) e_y + 2/3 e_z

The three vector fields \((T, N, B)\) form the Frenet-Serret frame along \(C\):

sage: FS = I.vector_frame(('T', 'N', 'B'), (T, N, B),
....:                     symbol_dual=('t', 'n', 'b'))
sage: FS
Vector frame ((0, 6*pi), (T,N,B)) with values on the Euclidean space E^3

The Frenet-Serret frame is orthonormal:

sage: matrix([[u.dot(v).expr() for v in FS] for u in FS])
[1 0 0]
[0 1 0]
[0 0 1]

The derivative vectors \(N'\) and \(B'\):

sage: Np = I.vector_field([diff(N[i], s) for i in E.irange()],
....:                     dest_map=C, name="N'")
sage: Np.display()
N' = 1/3*sin(1/3*s) e_x - 1/3*cos(1/3*s) e_y
sage: Bp = I.vector_field([diff(B[i], s) for i in E.irange()],
....:                     dest_map=C, name="B'")
sage: Bp.display()
B' = 1/9*sqrt(5)*cos(1/3*s) e_x + 1/9*sqrt(5)*sin(1/3*s) e_y

The Frenet-Serret formulas:

sage: for v in (Tp, Np, Bp):
....:     v.display(FS)
....:
T' = 2/9 N
N' = -2/9 T + 1/9*sqrt(5) B
B' = -1/9*sqrt(5) N

The torsion of \(C\) is obtained as the third component of \(N'\) in the Frenet-Serret frame:

sage: tau = Np[FS, 3]
sage: tau
1/9*sqrt(5)
coord_expr(chart=None)#

Return the coordinate functions expressing the curve in a given chart.

INPUT:

  • chart – (default: None) chart on the curve’s codomain; if None, the codomain’s default chart is assumed

OUTPUT:

  • symbolic expression representing the curve in the above chart

EXAMPLES:

Cartesian and polar expression of a curve in the Euclidean plane:

sage: M = Manifold(2, 'R^2', r'\RR^2')  # the Euclidean plane R^2
sage: c_xy.<x,y> = M.chart() # Cartesian coordinate on R^2
sage: U = M.open_subset('U', coord_def={c_xy: (y!=0, x<0)}) # the complement of the segment y=0 and x>0
sage: c_cart = c_xy.restrict(U) # Cartesian coordinates on U
sage: c_spher.<r,ph> = U.chart(r'r:(0,+oo) ph:(0,2*pi):\phi') # spherical coordinates on U

Links between spherical coordinates and Cartesian ones:

sage: ch_cart_spher = c_cart.transition_map(c_spher, [sqrt(x*x+y*y), atan2(y,x)])
sage: ch_cart_spher.set_inverse(r*cos(ph), r*sin(ph))
Check of the inverse coordinate transformation:
  x == x  *passed*
  y == y  *passed*
  r == r  *passed*
  ph == arctan2(r*sin(ph), r*cos(ph))  **failed**
NB: a failed report can reflect a mere lack of simplification.
sage: R.<t> = manifolds.RealLine()
sage: c = U.curve({c_spher: (1,t)}, (t, 0, 2*pi), name='c')
sage: c.coord_expr(c_spher)
(1, t)
sage: c.coord_expr(c_cart)
(cos(t), sin(t))

Since c_cart is the default chart on U, it can be omitted:

sage: c.coord_expr()
(cos(t), sin(t))

Cartesian expression of a cardioid:

sage: c = U.curve({c_spher: (2*(1+cos(t)), t)}, (t, 0, 2*pi), name='c')
sage: c.coord_expr(c_cart)
(2*cos(t)^2 + 2*cos(t), 2*(cos(t) + 1)*sin(t))
plot(chart=None, ambient_coords=None, mapping=None, prange=None, include_end_point=(True, True), end_point_offset=(0.001, 0.001), parameters=None, color='red', style='-', label_axes=True, thickness=1, plot_points=75, max_range=8, aspect_ratio='automatic', **kwds)#

Plot the current curve in a Cartesian graph based on the coordinates of some ambient chart.

The curve is drawn in terms of two (2D graphics) or three (3D graphics) coordinates of a given chart, called hereafter the ambient chart. The ambient chart’s domain must overlap with the curve’s codomain or with the codomain of the composite curve \(\Phi\circ c\), where \(c\) is the current curve and \(\Phi\) some manifold differential map (argument mapping below).

INPUT:

  • chart – (default: None) the ambient chart (see above); if None, the default chart of the codomain of the curve (or of the curve composed with \(\Phi\)) is used

  • 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) differentiable mapping \(\Phi\) (instance of DiffMap) providing the link between the curve and the ambient chart chart (cf. above); if None, the ambient chart is supposed to be defined on the codomain of the curve.

  • prange – (default: None) range of the curve parameter for the plot; if None, the entire parameter range declared during the curve construction is considered (with -Infinity replaced by -max_range and +Infinity by max_range)

  • include_end_point – (default: (True, True)) determines whether the end points of prange are included in the plot

  • end_point_offset – (default: (0.001, 0.001)) offsets from the end points when they are not included in the plot: if include_end_point[0] == False, the minimal value of the curve parameter used for the plot is prange[0] + end_point_offset[0], while if include_end_point[1] == False, the maximal value is prange[1] - end_point_offset[1].

  • max_range – (default: 8) numerical value substituted to +Infinity if the latter is the upper bound of the parameter range; similarly -max_range is the numerical valued substituted for -Infinity

  • parameters – (default: None) dictionary giving the numerical values of the parameters that may appear in the coordinate expression of the curve

  • color – (default: ‘red’) color of the drawn curve

  • style – (default: ‘-’) color of the drawn curve; NB: style is effective only for 2D plots

  • thickness – (default: 1) thickness of the drawn curve

  • plot_points – (default: 75) number of points to plot the curve

  • label_axes – (default: True) boolean determining whether the labels of the coordinate axes of chart shall be added to the graph; can be set to False if the graph is 3D and must be superposed with another graph.

  • aspect_ratio – (default: 'automatic') aspect ratio of the plot; the default value ('automatic') applies only for 2D plots; for 3D plots, the default value is 1 instead

OUTPUT:

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

EXAMPLES:

Plot of the lemniscate of Gerono:

sage: R2 = Manifold(2, 'R^2')
sage: X.<x,y> = R2.chart()
sage: R.<t> = manifolds.RealLine()
sage: c = R2.curve([sin(t), sin(2*t)/2], (t, 0, 2*pi), name='c')
sage: c.plot()  # 2D plot
Graphics object consisting of 1 graphics primitive
../../../_images/curve-3.svg

Plot for a subinterval of the curve’s domain:

sage: c.plot(prange=(0,pi))
Graphics object consisting of 1 graphics primitive
../../../_images/curve-4.svg

Plot with various options:

sage: c.plot(color='green', style=':', thickness=3, aspect_ratio=1)
Graphics object consisting of 1 graphics primitive
../../../_images/curve-5.svg

Cardioid defined in terms of polar coordinates and plotted with respect to Cartesian coordinates, as an example of use of the optional argument chart:

sage: E.<r,ph> = EuclideanSpace(coordinates='polar')
sage: c = E.curve((1 + cos(ph), ph), (ph, 0, 2*pi))
sage: c.plot(chart=E.cartesian_coordinates(), aspect_ratio=1)
Graphics object consisting of 1 graphics primitive
../../../_images/curve-6.svg

Plot via a mapping to another manifold: loxodrome of a sphere viewed in \(\RR^3\):

sage: S2 = Manifold(2, 'S^2')
sage: U = S2.open_subset('U')
sage: XS.<th,ph> = U.chart(r'th:(0,pi):\theta ph:(0,2*pi):\phi')
sage: R3 = Manifold(3, 'R^3')
sage: X3.<x,y,z> = R3.chart()
sage: F = S2.diff_map(R3, {(XS, X3): [sin(th)*cos(ph),
....:                      sin(th)*sin(ph), cos(th)]}, name='F')
sage: F.display()
F: S^2 → R^3
on U: (th, ph) ↦ (x, y, z) = (cos(ph)*sin(th), sin(ph)*sin(th), cos(th))
sage: c = S2.curve([2*atan(exp(-t/10)), t], (t, -oo, +oo), name='c')
sage: graph_c = c.plot(mapping=F, max_range=40,
....:                  plot_points=200, thickness=2, label_axes=False)  # 3D plot
sage: graph_S2 = XS.plot(X3, mapping=F, number_values=11, color='black') # plot of the sphere
sage: show(graph_c + graph_S2) # the loxodrome + the sphere
../../../_images/curve-7.svg

Example of use of the argument parameters: we define a curve with some symbolic parameters a and b:

sage: a, b = var('a b')
sage: c = R2.curve([a*cos(t) + b, a*sin(t)], (t, 0, 2*pi), name='c')

To make a plot, we set specific values for a and b by means of the Python dictionary parameters:

sage: c.plot(parameters={a: 2, b: -3}, aspect_ratio=1)
Graphics object consisting of 1 graphics primitive
../../../_images/curve-8.svg
tangent_vector_field(name=None, latex_name=None)#

Return the tangent vector field to the curve (velocity vector).

INPUT:

  • name – (default: None) string; symbol given to the tangent vector field; if none is provided, the primed curve symbol (if any) will be used

  • latex_name – (default: None) string; LaTeX symbol to denote the tangent vector field; if None then (i) if name is None as well, the primed curve LaTeX symbol (if any) will be used or (ii) if name is not None, name will be used

OUTPUT:

  • the tangent vector field, as an instance of VectorField

EXAMPLES:

Tangent vector field to a circle curve in \(\RR^2\):

sage: M = Manifold(2, 'R^2')
sage: X.<x,y> = M.chart()
sage: R.<t> = manifolds.RealLine()
sage: c = M.curve([cos(t), sin(t)], (t, 0, 2*pi), name='c')
sage: v = c.tangent_vector_field() ; v
Vector field c' along the Real interval (0, 2*pi) with values on
 the 2-dimensional differentiable manifold R^2
sage: v.display()
c' = -sin(t) ∂/∂x + cos(t) ∂/∂y
sage: latex(v)
{c'}
sage: v.parent()
Free module X((0, 2*pi),c) of vector fields along the Real interval
 (0, 2*pi) mapped into the 2-dimensional differentiable manifold R^2

Value of the tangent vector field for some specific value of the curve parameter (\(t = \pi\)):

sage: R(pi) in c.domain()  # pi in (0, 2*pi)
True
sage: vp = v.at(R(pi)) ; vp
Tangent vector c' at Point on the 2-dimensional differentiable
 manifold R^2
sage: vp.parent() is M.tangent_space(c(R(pi)))
True
sage: vp.display()
c' = -∂/∂y

Tangent vector field to a curve in a non-parallelizable manifold (the 2-sphere \(S^2\)): first, we introduce the 2-sphere:

sage: M = Manifold(2, 'M') # the 2-dimensional sphere S^2
sage: U = M.open_subset('U') # complement of the North pole
sage: c_xy.<x,y> = U.chart() # stereographic coordinates from the North pole
sage: V = M.open_subset('V') # complement of the South pole
sage: c_uv.<u,v> = V.chart() # stereographic coordinates from the South pole
sage: M.declare_union(U,V)   # S^2 is the union of U and V
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: W = U.intersection(V)
sage: A = W.open_subset('A', coord_def={c_xy.restrict(W): (y!=0, x<0)})
sage: c_spher.<th,ph> = A.chart(r'th:(0,pi):\theta ph:(0,2*pi):\phi') # spherical coordinates
sage: spher_to_xy = c_spher.transition_map(c_xy.restrict(A),
....:           (sin(th)*cos(ph)/(1-cos(th)), sin(th)*sin(ph)/(1-cos(th))) )
sage: spher_to_xy.set_inverse(2*atan(1/sqrt(x^2+y^2)), atan2(y, x), check=False)

Then we define a curve (a loxodrome) by its expression in terms of spherical coordinates and evaluate the tangent vector field:

sage: R.<t> = manifolds.RealLine()
sage: c = M.curve({c_spher: [2*atan(exp(-t/10)), t]}, (t, -oo, +oo),
....:              name='c') ; c
Curve c in the 2-dimensional differentiable manifold M
sage: vc = c.tangent_vector_field() ; vc
Vector field c' along the Real number line ℝ with values on
 the 2-dimensional differentiable manifold M
sage: vc.parent()
Module X(ℝ,c) of vector fields along the Real number line ℝ
 mapped into the 2-dimensional differentiable manifold M
sage: vc.display(c_spher.frame().along(c.restrict(R,A)))
c' = -1/5*e^(1/10*t)/(e^(1/5*t) + 1) ∂/∂th + ∂/∂ph