Page 135 - Intro to Tensor Calculus

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The reciprocal of curvature is called the radius of curvature. The curvature measures the rate of change of
the tangent vector to the curve as the arc length varies. By differentiating intrinsically, with respect to arc
j
i
length s, the relation g ij T N = 0 we find that
δN j δT i
j
g ij T i + g ij N =0. (1.5.6)
δs δs
Consequently, the curvature κ can be determined from the relation
δN j δT i j i j
i
g ij T = −g ij N = −g ij κN N = −κ (1.5.7)
δs δs
which defines the sign of the curvature. In a similar fashion we differentiate the relation (1.5.5) and find that
δN j
i
g ij N =0. (1.5.8)
δs
i
This later equation indicates that the vector δN j is perpendicular to the unit normal N . The equation
δs
i
i
(1.5.3) indicates that T is also perpendicular to N and hence any linear combination of these vectors will
i
also be perpendicular to N . The unit binormal vector is defined by selecting the linear combination
δN j j
+ κT (1.5.9)
δs
and then scaling it into a unit vector by defining
j
1 δN j
j
B = + κT (1.5.10)
τ δs
i
i
i
where τ is a scalar called the torsion. The sign of τ is selected such that the vectors T ,N and B form a
i
j
i
k
right handed system with ijk T N B = 1 and the magnitude of τ is selected such that B is a unit vector
satisfying
i
j
g ij B B =1. (1.5.11)
i
i
i
i
i
The triad of vectors T ,N ,B at a point on the curve form three planes. The plane containing T and B is
i
i
called the rectifying plane. The plane containing N and B is called the normal plane. The plane containing
i
i
T and N is called the osculating plane. The reciprocal of the torsion is called the radius of torsion. The
i
i
i
torsion measures the rate of change of the osculating plane. The vectors T ,N and B form a right-handed
orthogonal system at a point on the space curve and satisfy the relation
i
B = ijk T j N k . (1.5.12)
i
i
i
By using the equation (1.5.10) it can be shown that B is perpendicular to both the vectors T and N since
j
i
j
i
g ij B T =0 and g ij B N =0.
i
i
It is left as an exercise to show that the binormal vector B satisfies the relation δB i = −τN . The three
δs
relations
δT i i
= κN
δs
δN i i i
= τB − κT (1.5.13)
δs
δB i i
= −τN
δs