Denote by e(t) := [gamma](t) the

unit tangent vector of [gamma](t).

In [5], the energy of a unit vector field X on a Riemannian manifold M is defined as the energy of the mapping X : M [right arrow] [T.sup.1] M, where the

unit tangent bundle [T.sup.1] M is equipped with the restriction of the Sasaki metric on TM.

The tangent plane [T.sub.p]M, which contains the velocity vector [gamma]'([mathematical expression not reproducible]) of the initial geodesic [gamma] at P, can be decomposed into two independent (not necessarily orthonormal)

unit tangent directions [mathematical expression not reproducible].

Next, we determine the

unit tangent vector at the endpoints and subsequently calculate the corresponding tangent angle by using specified formula as indicated in the algorithm.

where ([d.sup.k], [[tau].sub.k]) is the

unit tangent. When the path can be parameterized with respect to t, the step size h in the predictor step is usually referred to as [DELTA][t.sub.k]:

Since the curve [alpha](t) is also in space, there exists Frenet frame {T, N, B} at each points of the curve where T is

unit tangent vector, N is principal normal vector and B is binormal vector, respectively.

We denote v as the arc-length parameter of f .We denote t(v) = f'(v)and we call t(v) a

unit tangent vector of f at v.

which defines a

unit tangent vector field on M and consider [e.sub.2] a unit vector field on M orthogonal to [e.sub.1] in such a way that {[e.sub.1], [e.sub.2] [zeta]} defines a positively oriented unit orthonormal basis for every point of M.

where [c.sub.u] is the effective air-drag coefficient, d is the yarn diameter, [v.sub.n] = v - (v x t)t is the normal component of the yam velocity (t is the

unit tangent vector to the yarn at the given point).

Given a vector field v, the

unit tangent vector t, the unit normal n, and the unit bi-normal b can be constructed as

For a

unit tangent vector v [member of] UM, we denote by [[gamma].sub.v] the trajectory for a Kaahler magnetic field [B.sub.[kappa]] with initial condition [??](0) = v.