Curvature and Curved Space
(2008-11-27)
[Geodesic] Curvature of a Planar Curve
Along a smooth curve in the
Euclidean plane, the
curvilinear abscisssa s of a point M
can be defined
(up to a choice of origin and a chaoice of sign)
by the differential relation:
(ds)2 = (dx)2 + (dy)2
This is just the Pythagorean theorem applied to infinitesimal quantities
(since, at a large enough magnification, any smooth curve looks perfectly straight).
The tangent at point M is oriented along the direction of the
unit vector T :
| T = |
dM |
= |
|
dx/ds |
|
= |
|
cos j |
|
|
 |
| ds |
dy/ds |
sin j |
The angle
j between the x-axis and T
is the inclination of the curve at M.
The derivative of
j (with respect to s)
is the [geodesic] curvature :
kg
=
1/r
=
dj / ds
In this, the signed quantitity r
= ds / dj
is called the geodesic radius of curvature.
Its absolute value is the radius of curvature
(often denoted R).
Changing the orientation of the plane changes the signs of
dj,
kg
and r.
Changing the orientation of the curve changes the signs of
ds,
kg
and r.
When M
is given as an explicit function of the parameter t instead of s,
the above curvature can be expressed in terms of
v = M' = dM/dt :
| kg |
= |
1 |
= |
dj |
= |
det ( v, v' ) |
= |
x' y'' - y'x'' |
|
|
|
|
| r |
ds |
||v|| 3 |
[ (x' ) 2 + (y' ) 2 ] 3/2 |
|
|
The subscript "g" (for "geodesic") is usually dropped in an introductory
context concerned with planar curves only.
However, we retain it here to avoid a conflict of notations
when we distinguish the
curvature of a spatial curve
(k, which is always nonnegative, by definition)
from the geodesic curvature of a curve
drawn on a curved surface (of which the flat plane is a
special case).
To prove the above relation, we introduce
the geodesic normal vector
g which is obtained by rotating T
one quarter of a turn counterclockwise (again, the qualifier "geodesic"
is rarely used for the planar case but we shall
soon generalize to curves drawn on other surfaces).
The definitions of
j and
kg yield:
| dT |
= |
dj |
|
|
-sin j |
|
=
kg g |
|
|
| ds |
ds |
cos j |
If the parameters t and s correspond to the same
orientation of the curve, then the speed v = ds/dt
is positive and we have v = v T. Therefore:
v' =
(dv/dt) T + v [ (ds/dt) (dT/ds) ]
=
(dv/dt) T +
v2 kg g
Since
v and T are collinear, we obtain
v ´ v' =
v2 kg
( v T ´ g )
The third component of that vectorial equation yields the advertised result.
(2008-11-30) Curvature and Torsion of a 3-dimensional Curve
The Frenet-Serret trihedron
(T,N,B) and formulas (1832, 1847, 1851).
In three dimensions, the curvilinear abcissa s
along a curve G is defined via:
(ds)2 = (dx)2 + (dy)2 + (dz)2
= (dM)2
So, the tangent vector T = dM / ds
is a unit vector (T2 = 1).
Therefore, unless its left side vanishes, the following relation
defines
both a unit vector N perpendicular to
T and a
positive number
k, called curvature
of G at M.
dT / ds =
k N
N is called the principal normal
and B = T´N
is the binormal.
The direct trihedron (T,N,B)
is the Frenet or Frenet-Serret trihedron.
The derivatives of the three vectors in a moving orthonormal trihedron
are antisymmetric linear combinations of themselves
(this is what gives rise to the three components of the
rotation vector in rigid kinematics).
For the Frenet trihedron, the above defining relation yields two
of the three coefficients involved in the derivatives with respect to s
(one is the curvature, the other one is zero).
The third component (t)
appearing in the following formulas is dubbed torsion.
Frenet Formulas :
| |
dT / ds | = |
| k N | |
| dN / ds | = |
- k T | + |
t B |
|
| dB / ds | = |
| - t N | |
|
Equivalently, the rotation vector with respect to s
is equal to
t T + k B
For a straight line,
the curvature k is zero.
N and B are undefined, so is
t.
The Frenet-Serret formulas were obtained independently by
Jean Frenet
(1816-1900) and by
Joseph
Serret (1819-1985; X1838) respectively in 1847 and 1851.
The Frenet-Serret trihedron was first introduced in 1832 by
the Piedmontese [or Sardinian?] political refugee
Gasparo
Mario Pagani (1796-1855) who was a professor in Begium, at the Universities of
Louvain (1826-1832, 1835-1854)
and Liège (1832-1835).
Curves of Constant Curvature and Torsion :
Putting, for convenience,
k = (cos q) / a
and
t = (sin q) / a
we obtain:
d2 N / ds2
=
-k dT/ds
+
t dB/ds
=
- ( k 2 + t 2 ) N
=
- N / a 2
The unit vector N is thus an harmonic function of s.
With the proper choice of base vectors, we have:
| N |
= |
|
-cos s/a
- sin s/a
0 |
|
= |
a |
|
dT |
|
|
| cos q |
ds |
Integrating the rightmost equation to solve for T, we obtain:
| T |
= |
|
- cos q sin s/a
cos q cos s/a
sin q |
|
= |
dM |
|
| ds |
Adding a nonzero vectorial constant of integration would yield something that fails to
be of unit length
(except, possibly, at isolated values of s).
Another integration gives the equation of the curve,
up to an irrelevant translation:
| M |
= |
|
a cos q cos s/a
a cos q sin s/a
s sin q |
|
-
This is the equation of an helix, parametrized by s.
| |
Michel-Ange Lancret |
Lancret's theorem states that a curve is a
generalized helix
if and only if its torsion to curvature ratio is a constant
(positive for a right-handed helix, negative for a left-handed one).
This result was stated in 1802 by
Michel-Ange Lancret
(1774-1807; X1794)
and first proved in 1845 by
Jean-Claude Barré de Saint Venant (1797-1866; X1813).
(2008-11-30) Curve drawn on a surface
The Darboux-Ribaucour trihedron
(T,g,k).
The Darboux-Ribaucour trihedron includes the unit tangent T
to the curve G
and the unit normal k to the surface
S (respectively determining the orientation
of the curve and that of the surface).
In the picture at right, the dotted circle is in the plane orthogonal
to T and oriented by it.
The third vector
g = k ´ T
is called the geodesic normal to
G on S.
The fundamental angle
q
which goes around the axis of T from
N (the principal normal to the curve)
to k can be introduced via the relations:
k =
cos q N
+
sin q B
g =
sin q N
-
cos q B
The following definitions involve the plain
curvature and torsion of the curve...
Normal Curvature
|
kk =
k cos q
|
|
Geodesic Curvature
|
kg =
k sin q
|
|
Geodesic Torsion
|
tg =
t + dq/ds
|
|
Those definitions are tailored for the following formulas
(which state that
the rotation vector of the trihedron
with respect to s is
tg T
- kk g
+ kg k ).
Darboux Formulas :
| |
dT / ds | = |
| |
kg g |
+ kk k |
|
| dg / ds | = |
- kg T |
|
+ tg k |
| dk / ds | = |
- kk T |
- tg g |
|
|
Those formulas could also be presented as definitions of the normal
curvature, geodesic curvature and geodesic torsion, in which case the
previous expressions are theorems.
The first formula is simply
dT/ds = k N
with N = cos q k + sin q g.
The other two equations are obtained by differentiating
with respect to s the above expressions for
g or k in terms of
q, N and B.
For example:
dg / ds =
sin q dN/ds
-
cos q dB/ds
+
dq/ds (
cos q N
+
sin q B )
The advertised result follows from
the Frenet formulas for dN/ds
and dB/ds.
Gaston
Darboux (1842-1917; X1861)
|
Albert
Ribaucour (1845-1893; X1865)
(2008-12-09) Lines of curvature and geodesic lines
(2008-12-09) Meusnier's theorem(s) for
lines drawn on a surface.
The osculating circles of all lines with the same tangent form a sphere!
Jean-Baptiste
Marie Charles Meusnier de la Place (1754-1793)
(2008-11-30) Gaussian Curvature of a Surface
(Intrinsic Curvature)
The product of the two principal curvatures
at a point of a surface.
Pierre
Bonnet (1819-1892; X1838)
(2009-07-22) Holonomic Angle around a Curve on a Surface
A parallel-transported vector may be rotated (Levi-Civita, 1917)
Around a given loop drawn on a surface,
the parallel-transport of all vectors (tangent to the surface)
rotates them through the same angle.
This angle is called the holonomic angle
of the loop; its value in radians is simply the integral of the
Gaussian curvature over the curved surface
bordered by the loop.
Levi-Civita's
Concept of the Parallel Transport of Vectors on a Surface by
Thayer Watkins
(2003-11-15) Total Curvature of a 3-dimensional Loop
Statements related to the Fary-Milnor Theorem (1949, 1950).
The integral of the curvature of a closed 3-dimensional curve is no less
than 2p.
This minimum is achieved for any simple convex planar curve.
The integral of the
signed curvature (geodesic curvature) of any smooth planar loop
is 2p times an integer called the "turning number"
of the curve (which is, loosely speaking, the number of times the extremity of
its tangent vector goes counterclockwise around the origin).
The turning number is either +1 or -1 for a simple loop
(i.e., a closed oriented curve which does not intersect itself).
If that loop is convex, the geodesic curvature has always the same sign, so the absolute
value of its integral (2p) is indeed the integral
of its absolute value 1/R, as advertised.
For a knotted curve, the integral of the curvature is no less
than 4p.
This statement is the
Fary-Milnor theorem
which was proved independently in 1949 and 1950, respectively, by
István
Fáry (1922-1984) and
John Milnor
(1931-).
It's natural to ask whether the integral of any combination of
curvature and torsion can remain invariant by homotopy among 3D loops, in the
same way the turning number does for 2D loops.
Let's use the 2D case as a hint...
K =
( v ´ dv/dt )
/ ||v|| 3
where v = dM / dt
The integral of K´ds
over the whole curve G
is a vector of length 2p,
whenever G happens to be
a simple closed planar curve...
(2009-07-22) Linearly Independent Curvature Components
In n dimensions,
the Riemann curvature tensor is a tensor of rank 4
whose n4 covariant coordinates
obey the following relations:
R abcd =
- R bacd =
- R abdc =
R cdab
R abcd +
R adcb +
R acdb = 0
Thus, it has only
n 2 ( n 2 - 1 ) / 12
linearly independent components:
0, 1, 6, 20, 50, 105, 196, 336, 540, 825, 1210, ...
(A002415)
The fact that this sequence starts with 0 for
n = 1 indicates that a manifold of dimension 1
has no intrinsic curvature...
The number of scalars (i.e., tensors of rank zero)
which can be constructed from the Riemann tensor
is just 1
when n = 2.
Otherwise, it is equal to
n (n-1) (n-2) (n+3) / 12
[which is 0 for n = 1].
The whole sequence is:
0, 1, 3, 14, 40, 90, 175, 308, 504, 780, 1155, ...
(A050297)
For n > 2 , this differs from the previous
sequence by ½ n (n-1)
That numerical evidence suggests that the
curvature information which cannot be specified by scalars
corresponds to a single antisymmetrical tensor
of rank 2 which is not defined
at all for 2-dimensional surfaces...
Riemann
Curvature Tensor (Wikipedia)
|
Riemann
Tensor (MathWorld)
Appendix 7:
Independent
Components of the Curvature Tensor
by Kevin S. Brown
|
