Page 48 - Basic Structured Grid Generation

Classical differential geometry of space-curves 37

Further intrinsic differentiation gives

ij δn j
g n i = 0, (2.38)
δs
which implies that δn j /δs is a vector orthogonal to n i ; moreover,

ij δn j ij δt i ij δn j ij ij δn j
g t i + g n j = g t i + g (κn i )n j = g t i + κ
δs δs δs δs

ij δn j ij ij δn j
= g t i + κg t i t j = g t i + κt j = 0.
δs δs

We deduce that δn j + κt j are covariant components of a vector orthogonal to t,
δs
and define
δn i
+ κt i = τb i , (2.39)
δs
where b is a unit vector satisfying
ij
g b i b j = 1 (2.40)
and
ij
g t i b j = 0, (2.41)
but with sense yet to be defined. Now b is orthogonal to n as well as t, since from
eqn (2.39)
1 δn i
ij ij ij
g n i b j = g n i + κg n i t j = 0, (2.42)
τ δs
using eqns (2.37) and (2.38), assuming that the scalar τ is non-zero. So we can choose
the sense of b such that {t, n, b} form a right-handed set of unit vectors.
Any vector u can be expanded on a rectangular cartesian basis as
u = (u · i)i + (u · j)j + (u · k)k

and, in the same way,
u = (u · t)t + (u · n)n + (u · b)b.

Applying this to the vector δb/δs, taking covariant components with respect to the
given curvilinear co-ordinates, we get

δb i δb k δb k δb k
jk jk jk
= g t j t i + g n j n i + g b j b i
δs δs δs δs

jk δt j jk δn j
=− g b k t i − g b k n i + 0,
δs δs
after differentiating eqns (2.40), (2.41), and (2.42). Hence from eqns (2.35) and (2.39)
δb i jk jk
=−κ(g n j b k )t i − g (−κt j + τb j )b k n i =−τn i ,
δs
again making use of eqns (2.40), (2.41), and (2.42).