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Curvature of Surfaces in 3-Space by Michael Garman  &  Jessica Bonnie.
The History of Curvature  |  Curved Space and the Metric
Curvature, Intrinsic and Extrinsic  |  Did Archimedes Know Gauss-Bonnet?
Courbes et surfaces,  Paris XII  (in French).

(2008-11-27)  
Longitudinalcurvature is a signed  quantity.

With the common conventions,  a curve with positive curvature veers to the left whenwe 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 smoothcurve 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 followingdifferential relation,which can be construed as the Pythagorean theoremapplied 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
	ds			dy/ds				sin

The angle   between the  x-axis  and T is the inclination  of the curve at M. The derivative of   (with respect to  s) is the  [geodesic] curvature :

_g  =  1/  =  d / ds

In this,  the signed quantitity    =  ds / d  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 d, _g and  .
Changing the orientation of the curve  changes the signs of ds, _g and  .

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 :

g   =   1   =   d   =   det (v,v' )     =     x' y'' y'x''     ds ||v|| 3 [ (x') 2 + (y') 2] 3/2

The subscript  "g"   (for "geodesic")  is usually dropped in an introductorycontext concerned with planar  curves only. However, we retain it here to avoid a conflict of notationswhen we distinguish the curvature of a spatial curve (, which is always nonnegative, by definition) from the geodesic curvature  of a curve drawn on a curved surface  (of which the flat plane is aspecial case)  which is a signed quantity,as noted above.

To prove the above relation, we introducethe geodesic normal vector g  which is obtained by rotating T one quarter of a turn counterclockwise (by convention,that's positive). Theabove definitions of   and _g  yield:

dT	=	d			-sin		=   gg
ds		ds			cos

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_gg

Sincev  and T  are collinear, we obtain  vv'  = v²_g( vTg ).  The third component of that vectorial equation yields the advertised result. 

(2008-11-30)  
The Frenet-Serret trihedron (T,N,B)  and formulas (1832, 1847, 1851).

In 3 dimensions, the curvilinear abcissa  s along a curve    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 relationdefines both  a unit vector N  perpendicular to T  and a positive  number , called curvature of  .

dT / ds   =   N

N  is called the principal normal and B = TN  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 trihedronare antisymmetric linear combinations of themselves (this is what gives rise to the three components of therotation vector in rigid kinematics). For the Frenet trihedron, the above defining relation specify twoof the three coefficients involved in the derivatives with respect to  s (one is equal to the curvature, the other one vanishes). The third component  () appearing in the following formulas is dubbed torsion.

Frenet Formulas :

dT / ds =    N dN / ds =  T   +    B   dB / ds =  N

Equivalently, the rotation vector with respect to  s is equal to   T +  B

For a straight  line,the curvature    is 0. N andB are undefined, so is  .

Curves of Constant  Curvature and Torsion :

We must rule out the case of constant zero curvature,  which triviallyimplies that the curve is straight  (in which case the torsion is undefined). Otherwise, the following relation does define a positive constant:

a   =   ( ² + ²)^½

With this notation,  we have:

d²N / ds²  =  - dT/ds +  dB/ds  =  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
								ds

For the sake of future convenience,  we may introduce a constant angle ,  uniquely defined by its sine and cosine (whose squares add up to 1):

cos   =  a       and      sin   =  a

With this new notation,  the above reads:

	-cos s/a  - sin s/a 0		=	a		dT
				cos		ds

Integrating this equation, we obtain:

T	=		- cos   sin s/a   cos   cos s/a sin		=	dM
						ds

Adding a nonzero vectorial constant of integration would yield something that fails tobe 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   cos s/a   a  cos   sin s/a  s  sin

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). Thisresult was stated in 1802 byMichel-Ange Lancret(1774-1807; X1794) and first proved in 1845 by Jean-Claude Barré de Saint Venant (1797-1866; X1813).

(2008-11-30)  
The Darboux-Ribaucour trihedron  (T,g,k).

The Darboux-Ribaucour trihedron includes the unit tangent T to the curve   and the unit normal k  to the surface   (respectively determining the orientationof the curve and that of the surface). In the picture at right, the dotted circle is in the plane orthogonalto T  and oriented by it.

The third vector  g   =  kT is called the geodesic normal  to   on  .

The fundamental angle  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  N  sin  B
 g   =  sin  N  cos  B

Let's introduce therotation vector of thetrihedron with respect to  s  as:

_g T  _k g + _g k

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

Darboux Formulas :

dT / ds =    g g   +  k k   dg / ds = g T   + g k dk / ds = k T g g

The three new quantities so introducedcan be expressed in terms of the curve's own curvature   and torsion  ,  namely:

 Those expressions of _k  and _g  are easily established byderiving the first Darboux formula from the first Frenet formula:

dT/ds  =   N    using    N  =  cos   k  +  sin    g.

_g  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 , N  andB) and substitute in the result the values given bytheFrenet formulas for  dN/ds and  dB/ds :

dg / ds   =  sin   dN/ds  cos   dB/ds  +  d/ds (cos  N + sin  B )
=  sin  ( T  +  B )  cos   (N )  +  d/ds (cos  N + sin  B )
=  (  sin ) T  +   ( +d/ds ) ( cos  N + sin  B )
=  _g T  +  _g k 

As a mere check,  let's also obtain the last formulaby differentiating  k :

dk / ds   =  cos   dN/ds + sin   dB/ds  +  d/ds (cos  B  sin  N )
=  cos  ( T  +  B ) + sin   (N )  +  d/ds (cos  B  sin  N )
=  (  cos ) T  +   ( +d/ds ) ( cos  B  sin  N )
=  _k T   _g g

(2012-03-17)  
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 canbe defined  via the following differential relations:

dM   =  M'_u du  + M'_v dv
d²M   =  M''_uu (du)² +  2M''_uv du dv + M''_vv (dv)²

The 1^st and 2^nd fundamental quadratic forms  are traditionallydenoted:

₁(du,dv)  =     (dM)²   =   E (du)²  +  2 F du dv  +  G (dv)²
₂(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 useP,Q,R instead of E,F,G.  Others use e,f,g instead of L,M,N.

Normal curvature _k :

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   =  _k  (dM)²

_k is the ratio of the two fundamental forms :

k    =   k . d2M     =     2(du,dv) (dM)2 1(du,dv)

Unless it is constant, the normal curvature _k  takes on twodistinct extreme values ₁  and ₂ 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 becauseof 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 ₁  and ₂ (extremes of_k )

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

Mean Curvature   H

1 +2     =     GL + EN 2 FM 2 2 ( EG F2)

Gaussian Curvature   K

1 2      =    LN M2     EG F2

 

(2008-12-09)   (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 themeridians 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 ifthere 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 ona 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)  theirthree tangent planes are pairwise othogonal.

The theorem of Dupin  says that the intersectionof two surfaces from such a system is a line of curvature of both surfaces.

(2008-12-09)  
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 toJules Tannery)  of equation: 8a2(x2 + y2) = (a2 z2) z2 A convenient parametrization is:
		x   =  (a / 32)  sin u  cos v       y   =  (a / 32)  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:

(2008-12-09)  
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. 

(2016-01-10)  
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   =    1 +2     =     GL + EN 2 FM 2 2 ( EG F2)

Minimal Surfaces  (Lagrange, 1762):

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

G L  +  E N   =   2 F M

(2008-11-30)  
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   =  12  =    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 trianglewith two sides oflength  s  on perpendicular geodesics intersecting at  P.

h(s)   =   2 [ 1    K s² / 12  +  O(s³) ]

Likewise, the perimeter of a small circle of radius  r  centered on  P  is:

2 s [ 1    K s² / 6  +  O(s³) ]

One way to check or memorize that formula is to consider the special case of a sphere ofradius  R  (with  K = 1/R² )  where the exact  circumference is:

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

Sectional curvature :

(2009-07-22)  
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 surfacebordered by the loop.

(2003-11-15)  
Statements related to the Fary-Milnor Theorem  (1949, 1950).

The integral of the curvature of a closed 3-dimensional curve is no lessthan  2. This minimum is achieved for any simple convex planar curve.

times an integer called the "turning number"of the curve  (which is, loosely speaking, the number of times the extremity ofits 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 absolutevalue of its integral (2) is indeed the integralof its absolute value  1/R, as advertised.

For a knotted  curve, the integral of the curvature is no lessthan  4. This statement is the Fary-Milnor theorem which was proved independently in 1949 and 1950, respectively, byIstvánFáry (1922-1984) and John Milnor (1931-).

It's natural to ask whether the integral of any combination ofcurvature and torsion can remain invariant by homotopy among 3D loops, in thesame 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 Kds over the whole curve   is a vector of length  2, whenever    happens to bea simple closed planar curve...

(2009-07-22)  

In  n  dimensions,the Riemann curvature tensor  is a tensor of rank 4whose  n⁴  covariant coordinatesobey 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 1has 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:

0, 1, 3, 14, 40,   90, 175, 308, 504, 780, 1155, ...  (A050297)

For n > 2 ,  this differs from the previoussequence by   ½ n (n-1).

That numerical evidence suggests that thecurvature information which cannot be specified by scalarscorresponds to a single antisymmetrical  tensorof rank 2  which is not  definedat all for 2-dimensional surfaces...