There is nothing in the World except empty curved space.
Matter, charge, electromagnetism, and other fields are only manifestations of the curvature of space. John Archibald Wheeler
(1911-2008)
(2008-11-27) [Geodesic] Curvature of a Planar Curve
Longitudinal
curvature is a signed quantity.
With the common conventions, a curve with positive curvature veers to the left when
we stand on the plane facing forward in the direction of progression.
This sign depends on which way we travel along the curve and which way we orient the plane
(standing up normally or doing a headstand, which switches left and right).
Let's quantify this:
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 choice of sign)
by the following differential relation,
which can be construed as the Pythagorean theorem
applied to infinitesimal quantities
(since, at a large enough magnification, any smooth curve looks perfectly straight).
(ds)2 = (dx)2 + (dy)2
The tangent at 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) which is a signed quantity,
as noted above.
To prove the above relation, we introduce
the geodesic normal vectorg which is obtained by rotating T
one quarter of a turn counterclockwise
(by convention,
that's positive).
The above definitions of
j and
kg yield:
dT
=
dj
-sin j
=
kgg
ds
ds
cos j
Again, the qualifier "geodesic"
is rarely used for the planar case but we shall
soon generalize to curves drawn on other surfaces.
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 kgg
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 3 dimensions, the curvilinear abcissa s
along a curve G is defined via:
(ds)2 = (dx)2 + (dy)2 + (dz)2
= (dM)2
The tangent vector T = dM / ds
is a unit vector (T2 = 1).
So, 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.
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 specify two
of the three coefficients involved in the derivatives with respect to s
(one is equal to the curvature, the other one vanishes).
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 0.
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 itself had actually been introduced in 1832 by
the Piedmontese [or Sardinian?] political refugee
Gasparo
Mario Pagani (1796-1855) who was a professor of mathematics in Belgium,
at the Universities of Louvain (1826-1832, 1835-1854)
and Liège (1832-1835).
Curves of Constant Curvature and Torsion :
We must rule out the case of constant zero curvature, which trivially
implies that the curve is straight (in which case the torsion is undefined).
Otherwise, the following relation does define a positive constant:
a = ( k 2 +
t 2)-½
With this notation, we have:
d2 N / ds2
=
-k dT/ds
+
t dB/ds
=
- ( k 2 + t 2 ) N
=
- N / a 2
This shows that the unit vector N is an harmonic function of s.
Therefore, with the proper choice of base vectors, we have:
N
=
-cos s/a - sin s/a 0
=
1
dT
k
ds
For the sake of future convenience, we may introduce a constant angle
q, uniquely defined by its sine and cosine
(whose squares add up to 1):
cos q = a k
and
sin q = a t
With this new notation, the above reads:
-cos s/a - sin s/a 0
=
a
dT
cos q
ds
Integrating this equation, 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.
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
Let's introduce the rotation vector of the
trihedron with respect to s as:
tgT
- kkg
+ kgk
That's just another way to state the following traditional formulas:
Darboux Formulas :
dT / ds
=
kgg
+ kkk
dg / ds
=
- kgT
+ tgk
dk / ds
=
- kkT
- tgg
The three new quantities so introduced
can be expressed in terms of the curve's own
curvature k
and torsion t, namely:
Normal Curvature
kk =
k cos q
Geodesic Curvature
kg =
k sin q
Geodesic Torsion
tg =
t + dq/ds
Proof :
Those expressions of
kk and
kg are easily established by
deriving the first
Darboux formula from the firstFrenet formula:
dT/ds = k N
using N = cos q k + sin q g.
tg comes from a (tougher)
derivation of either remaining Darboux formula:
To obtain the second Darboux formula, we may differentiate
(with respect to s ) the above expression of
g (in terms of
q, N and B)
and substitute in the result the values given by
the Frenet formulas for dN/ds
and dB/ds :
dg / ds =
sin q dN/ds
-
cos q dB/ds
+
dq/ds (
cos q N
+
sin q B )
=
sin q
(- k T +
t B )
-
cos q (- t N )
+
dq/ds (
cos q N
+
sin q B )
=
(- k sin q ) T
+ ( t +
dq/ds )
( cos q N
+
sin q B )
=
- kgT
+
tgk
As a mere check, let's also obtain the last formula
by differentiating k :
dk / ds =
cos q dN/ds
+
sin q dB/ds
+
dq/ds (
cos q B
-
sin q N )
=
cos q
(- k T +
t B )
+
sin q (- t N )
+
dq/ds (
cos q B
-
sin q N )
=
(- k cos q ) T
+ ( t +
dq/ds )
( cos q B
-
sin q N )
=
- kkT
-
tgg
(2012-03-17) The Two Fundamental Local Quadratic Forms
Expressing the normal curvature of a curve of given tangent at point M.
At a given point M of a parametrized surface M(u,v),
all the partial derivatives of first and second order are vectorial quantities that can
be defined via the following differential relations:
dM = M'u du + M'v dv
d2M = M''uu (du)2
+ 2 M''uv du dv
+ M''vv (dv)2
The 1st and 2ndfundamental quadratic forms are traditionally
denoted:
F1 (du,dv)
= (dM)2
= E (du)2 + 2 F du dv + G (dv)2
F2 (du,dv) = k . d2M
= L (du)2 + 2 M du dv + N (dv)2
Clearly, the six scalar quantities
E,F,G and L,M,N are given by:
E = || M'u ||2
F = M'u . M'v
G = || M'v ||2
L = k . M''uu
M = k . M''uv
N = k . M''vv
Some authors use
P,Q,R instead of E,F,G. Others use e,f,g instead of L,M,N.
Normal curvature kk :
If dM = T ds, then
d2M = (dT/ds) (ds)2 + T d2s
and it follows (by projection on k
of the first Darboux formula) that:
k . d2M = k . (dT/ds) (ds)2
+ 0 . d2s =
kk (dM)2
kkis the ratio of the two fundamental forms :
kk
=
k . d2M
=
F2 (du,dv)
(dM)2
F1 (du,dv)
Unless it is constant, the normal curvature
kk takes on two
distinct extreme values
k1 and
k2
for two perpendicular directions
(called principal directions of curvature )
each of which corresponding to a solution in (du,dv)
of the following equation
(obtained, modulo an irrelevant factor,
by differentiating the above with respect to the ratio of du and dv).
General characterization of the two principal directions of curvature :
(EM-FL) (du)2 + (EN-GL) du dv + (FN-GM) (dv)2
= 0
In the general case, the above is a quadratic equation in x = du/dv
with two distinct solutions x1 and x2 corresponding,
as advertised, to directions that are easily checked to be perpendicular because
of a vanishing dot product.
Hint :
(EM-FL) [ E x1 x2 + F (x1+x2) + G ]
= E (FN-GM) - F (EN-GL) + G (EM-FL) = 0
Principal curvatures
k1 and
k2
(extremes of kk )
Extreme normal curvatures are solutions of
k2 - 2 H k + K = 0
(2008-12-09) Lines of Curvature of a Surface (1776)
Everywhere tangent to a principal direction of curvature.
The concept was introduced by the founder of Polytechnique,
Gaspard Monge
(1746-1818) in 1776. It was investigated in depth by his student
Charles Dupin (1784-1846; X1801) in 1813.
For a surface of revolution, the two sets of lines of curvature are the
meridians and the parallels.
Rodrigues's Formula :
In a parametrized surface, a curve M(u(t),v(t))
parametrized with t is a line of curvature if and only if
there is a scaling factor k(t)
[which turns out to be the relevant principal curvature]
such that:
N(u(t),v(t))' = k(t) M(u(t),v(t))'
Proof :
Triply Orthogonal System of Surfaces :
Such a system is formed by three families of surfaces, each depending on
a single continuous parameter, if they verify the following condition:
At any point where three surface of the system intersect
(one from each of the three single-parameter families) their
three tangent planes are pairwise othogonal.
The theorem of Dupin says that the intersection
of two surfaces from such a system is a line of curvature of both surfaces.
(2008-12-09) Geodesics (geodesic lines)
The geodesics are curves of zero geodesic curvature.
The path of least length between two points on a surface is a geodesic.
Tannery's Pear
Pictured at left is the lower half (z ≤ 0)
of a degree-4 algebraic surface (due to
Jules Tannery) of equation:
8 a2 (x2 + y2 )
=
(a2 - z2 ) z2
A convenient parametrization is:
x =
(a / Ö32) sin u cos v
y =
(a / Ö32) sin u sin v
z =
a sin u/2
This surface has the very remarkable property that every geodesic is an
algebraic closed curve that crosses itself once.
In particular, the double point of all meridians is the conical point
(all other geodesics go around the axis of symmetry twice ).
(2008-12-09) Meusnier's theorem(s) for lines drawn on a surface (1776)
The osculating circles of all lines with the same tangent form a sphere!
Jean-Baptiste Meusnier (1754-1793)
annouced this result in 1776. He only published it formally in 1785.
In modern terms, this states that tangent lines have the same normal curvature.
The Gaussian curvature at a point P
appears in the Taylor series
expansion of the curvilinear hypothenuse h(s) of a small isoceles right triangle
with two sides of
length s on perpendicular geodesics intersecting at P.
h(s) = Ö2
[ 1 -
K s2 / 12 + O(s3 ) ]
Likewise, the perimeter of a small circle of radius r centered on P is:
2 p s
[ 1 -
K s2 / 6 + O(s3 ) ]
One way to check or memorize that formula is to consider the special case of a sphere of
radius R (with K = 1/R2 ) where the
exact circumference is:
2 p R sin ( s/R ) =
2 p s [ 1 -
K s2 / 3! +
K2 s4 / 5! + ... ]
(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 the integral of the
Gaussian curvature over the curved surface
bordered by the loop.
(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...
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's equal to:
n (n-1) (n-2) (n+3) / 12
[which is 0 for n = 1].
The whole sequence is:
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...
(2008-11-27) [Geodesic] Curvature of a Planar Curve
