Page 49 - Basic Structured Grid Generation
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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
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