Page 46 - Basic Structured Grid Generation
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Classical differential geometry of space-curves 35
which can also be expressed as
x y z
τ = ρ 2 x y z . (2.25)
x y z
Exercise 2. If a curve is given in terms of a parameter t, show, making use of
...
eqns (2.12), (2.13), and a similar equation for r ,that
... ...
˙ r · (¨ r × r ) ˙ r · (¨ r × r )
2 2
τ = ρ = ρ , (2.26)
˙ s 6 (˙ r · ˙ r) 3
where ρ may be obtained from eqns (2.15) and (2.18).
Exercise 3. For the curve given by the parametric form
2
3
3
x = a(3t − t ), y = 3at , z = a(3t + t ),
2 −2
1 −1
where a is a constant, show that κ = τ = a (1 + t ) .
3
2.3 Generalized co-ordinate approach
It may be instructive to derive the Serret-Frenet formulas using generalized co-ordi-
nates. In the process we introduce the concept of intrinsic differentiation. Given a set
3
2
1
of curvilinear co-ordinates x , x , x , a space-curve may be specified in terms of a
parameter t and the functions
i
i
x = x (t), i = 1, 2, 3.
Arc-length is given in terms of the basic metric
2
i
j
ds = g ij dx dx , (2.27)
or in terms of the parameter t as
t i j
dx dx
s = g ij dt, (2.28)
dt dt
t 0
measured from some point on the curve where t = t 0 . Equation (2.27) can be re-
written as
i
dx dx j
g ij = 1, (2.29)
ds ds
which according to eqn (1.55) may be interpreted as implying that the vector whose
i
contravariant components are dx /ds, i = 1, 2, 3, has magnitude unity. Of course, this
is just the unit tangent vector t to the curve; here we write
dx i
i
t = . (2.30)
ds
For any vector field u we have by the Chain Rule
du ∂u dx j i dx j dx j i
= = u ,j g i = u i,j g
ds ∂x j ds ds ds