Page 49 - Basic Structured Grid Generation
P. 49
38 Basic Structured Grid Generation
Summarizing, we have the Serret-Frenet formulas in covariant form:
δt i
= κn i
δs
δn i
=−κt i + τb i (2.43)
δs
δb i
= −τn i .
δs
Alternatively, having established that {t, n, b} form an orthonormal set, we can write
the vector product b = t × n in component form as
i ijk
b = ε t j n k ,
and, making use of eqn (1.130), we have
δb i ijk δn k ijk δt j ijk ijk
= ε t j + ε n k = ε t j (−κt k + τb k ) + κε n j n k
δs δs δs
ijk
= τε t j b k , (2.44)
the other terms vanishing due to the symmetry of t j t k and n j n k and the skew-symmetry
of ε ijk in the summed indices j, k. The right-hand side now gives the scalar τ multiplied
by the contravariant component of the vector product t×b, which because of the choice
i
of the sense of b is equal to −n, with contravariant component n .
Hence
δb i i
=−τn ,
δs
with associated covariant components
δb i
=−τn i
δs
as expected.
2.4 Metric tensor of a space-curve
In this section we parametrise a space-curve C directly by a curvilinear co-ordinate ξ,
so that
r = (x(ξ), y(ξ), z(ξ)) (2.45)
on the curve. In the context of grid generation, space-curves appear as boundaries of
surfaces and as edges of three-dimensional blocks, and it is convenient to map a given
finite length of space-curve onto an interval of the ξ-axis, say 0 ξ 1. A uniformly
spaced set of points in the ξ-interval will then map to a set of points along the curve.
Arc-length along the curve is then defined by
dr dr
2 2 2
(ds) = dr · dr = · (dξ) = g 11 (dξ) ,
dξ dξ