1] is called osculating curve of the first or second kind if its position vector (with respect to some chosen origin) always lies in the orthogonal complement [B.

Consequently, the position vector of the null osculating curve of the first and second kind satisfies respectively the equations

In this section we characterize null osculating curves of the first kind in the Minkowski space-time.

1] Then [alpha] is congruent to an osculating curve of the first kind if and only if its third curvature [[kappa].

First assume that [alpha] is an osculating curve of the first kind in [E.

2] is a constant vector, it follows that a is congruent to an osculating curve of the first kind.

1] if and only if it is an osculating curve of the first kind.

If its pseudo-Galilean trihedron {t(x), n(x), b(x)}; then the center of

osculating sphere of r at the point r(x) is given by

Historically, osculating bundles were introduced under various names long before the bundles [T.

In particular, exterior differentiation is an operation on the set of sector-forms that are constant on the osculating bundles.

The Hamiltonian system reduces to Lagrange's equations on the osculating bundle OscM.

k]M and the Lagrangian geometry on the osculating bundles [Osc.

At each of the curve, the planes spanned by {T, N}, {T, B} and {N, B} are known respectively as the osculating plane, the rectifying plane and the normal plane[5].

Let [phi] = [phi](s) be a unit speed osculating curve with constant first curvature in [E.

Here we have only used asymptotic

osculating hyperbolic trajectory data from [1].