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Final Answers
© 2000-2018   Gérard P. Michon, Ph.D.

Curvature  &  Torsion

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

Related articles on this site:

Related Links (Outside this Site)

The History of Curvature   |   Curved Space and the Metric
Curvature, Intrinsic and Extrinsic   |   Did Archimedes Know Gauss-Bonnet?
Courbes et surfaces, TD à Paris XII  (exercises, in French)

Carl Friedrich Gauss  1777-1855
Jean Frenet  1816-1900
Joseph Serret  1819-1885
Pierre Bonnet  1819-1892
Bernhard Riemann  1826-1866
Elwin Christoffel  1829-1900
Camille Jordan  1839-1922
Gaston Darboux  1842-1917
Albert Ribaucour  1845-1893
François Cosserat  1852-1914
Gregorio Ricci-Curbastro  1853-1925
Luigi Bianchi  1856-1928
Eugène Cosserat  1866-1931
Alphonse Demoulin  1869-1947
Elie Cartan  1869-1951
Emile Cotton  1872-1950
Tullio Levi-Civita  1873-1941
Attilio Palatini  1889-1949

Wikipedia :   Frenet-Serret formulas   |   Darboux frame   |   Einstein-Cartan theory   |   Palatini variation

"Synthetic Theory of Ricci Curvature" (IHES video)  Cédric Villani   1 2 3 4 5 6 7


Curvature and Curved Space

 Curvilinear abscissa of M 
 and tangent vector at M (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     =     bracket
  dx/ds   bracket
    =     bracket
  cos j   bracket
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  djkg  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''    
Vinculum Vinculum Vinculum Vinculum
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 vector  g  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    bracket
 -sin j  bracket
    =    kg g
Vinculum Vinculum
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 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.   QED

(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     =     bracket
 -cos s/a 
- sin s/a
    =     1    dT
Vinculum Vinculum
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
    =     a    dT
Vinculum Vinculum
cos q ds

Integrating this equation, we obtain:

T     =     bracket
 - cos q  sin s/a
  cos q  cos s/a
sin q
    =     dM

Adding a nonzero vectorial constant of integration would yield something that fails to be of unit length  (except, possibly, at  isolated values  of  s).

 Right-Handed Helix

Another integration gives the equation of the curve, up to an irrelevant translation:

M     =     bracket
 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 
 (1774-1807) X1794
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).

 Curve drawn on 
 a surface (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:

tg T  -  kk g  +  kg k

That's just another way to state the following traditional formulas:

Darboux Formulas :
  dT / ds=     kg  g   +  kk  k  
dg / ds= - kg  T   +  tg  k
dk / ds= - kk  T - tg  g

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  first  Frenet 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  qN  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 qT   +   ( t + dq/ds )  ( cos q   N  +  sin q  B )
=   - kg  T   +   tg  k    QED

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 qT   +   ( t + dq/ds )  ( cos q  B  -  sin q   N )
=   - kk  T   -   tg  g

Gaston Darboux (1842-1917; X1861)   |   Albert Ribaucour (1845-1893; X1865)

(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 2nd  fundamental 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

kk  is the ratio of the two fundamental forms :
kk    =   k . d2M     =     F2 (du,dv)
Vinculum Vinculum
(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 k)

Extreme normal curvatures are solutions of     k2 - 2 H k + K  =  0

Mean Curvature   H
    k1 + k2     =     GL + EN - 2 FM
Vinculum Vinculum
2 2 ( EG - F2 )    
Gaussian Curvature   K
    k1 k2      =    LN - M2    
EG - F2

 Charles Dupin  Gaspard Monge (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  PolytechniqueGaspard 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 :

 Come back later, we're
 still working on this one...

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.

Line of Curvature  &  Triply Orthogonal System of Surfaces   (French & Multilingual)
Charles Dupin (1784-1873; X1801)

(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.

 A bold geodesic line drawn  
 on the surface of Tannery's Pear.
   Jules Tannery 
Jules Tannery

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 ).

Let's establish that:

 Come back later, we're
 still working on this one...

Wikipedia :   Geodesics   |   Spacetime geodesics

(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. 

 Come back later, we're
 still working on this one...

Jean-Baptiste Marie Charles Meusnier de la Place  (1754-1793)

 Sophie Germain 
 (1776-1831)  (2016-01-10)   Mean Curvature   (1776, 1831)
Half-sum of the two local  principal curvatures  on a surface.

The concept was brought to prominence by  Jean-Baptiste Meusnier  in 1776 and by  Sophie Germain  in 1831.

The  mean curvature  depends on the local  fundamental quadratic forms:

Mean Curvature   H
   H   =    k1 + k2     =     GL + EN - 2 FM
Vinculum Vinculum
2 2 ( EG - F2 )    

Minimal Surfaces  (Lagrange, 1762):

surface of least area  has zero mean curvature everywhere.  Thus, it verifies the following second-order  partial differential equation.

G L  +  E N   =   2 F M

 Come back later, we're
 still working on this one...

Wikipedia :   Mean curvature   |   Young-Laplace equation

(2008-11-30)   Gaussian Curvature of a Surface  (Intrinsic Curvature)
The product of the two  principal curvatures  at a point on a surface.

The  intrinsic curvature  depends on the local  fundamental quadratic forms:

Gaussian Curvature  K  (1824)
    K   =   k1 k2   =    LN - M2    
EG - F2

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/R)  where the  exact  circumference is:

2 p R  sin ( s/R )   =   2 p s   [ 1  -  K s2 / 3!  +  K2 s4 / 5!  +  ... ]

Sectional curvature :

 Come back later, we're
 still working on this one...

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 the integral of the  Gaussian curvature  over the curved surface bordered by the loop.

 Come back later, we're
 still working on this one...

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...

 Come back later, we're
 still working on this one...

(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

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