The ratio between the number of the unit vectors
that meet the conditions and the total number of vector is similarity between two sequences.
are components of unit vectors
[u.sub.1], [u.sub.2], and [u.sub.3] in x, y, and z direction, respectively.
The unit vector
[[??].sub.3] is aligned with the Earth axis of rotation and is positive northward, whereas [[??].sub.1] is aligned with the vernal axis, which corresponds to the intersection of the Earth equatorial plane with the ecliptic plane.
Similarly, we set [e.sub.2] to an arbitrary unit vector
with an orthogonal projection on the subspace < [e.sub.-1], [e.sub.0], [e.sub.1] > lying on the line with directing vector [r.sub.1] and the angle between this vector and [r.sub.1] equal [[alpha].sub.1], and select the coefficient [c.sub.2] in such a way that for [r.sub.2] = [r.sub.1]-[c.sub.2][e.sub.2] scalar products ([r.sub.2], -[e.sub.2]), ([r.sub.2], -[e.sub.1]), ([r.sub.2], [e.sub.0]), ([r.sub.2], [e.sub.-1]) are equal or, equivalently, angles between [r.sub.2] and the vectors [e.sub.-1], [e.sub.0], -[e.sub.1], -[e.sub.2] are equal.
[[??].sub.0] is a unit vector
for [y.sub.0] axis, [[??].sub.0] a unit vector
for [z.sub.0], axis, [[??].sub.c] a unit vector
for [y.sub.c] axis and [[??].sub.c] a unit vector
for [Z.sub.c] axis.
where the dot x means the scalar product of vectors [??] and [??] In the fixed global coordinate system (x, y, z), these vectors are given in terms of its scalar components and the unit vectors
[mathematical expression not reproducible] along the coordinates x, y, z, respectively:
Here, [[??].sub.i] is the unit vector
along the i-th primal edge [l.sub.i].
where a and b are scalars of which b [not equal to] 0 and A is non-zero 1-form such that g(X, U) = A(X) for all vector field X and U is a unit vector
i, j, K - unit vectors
of the wheel coordinate frame
In case the direction of a three-dimensional unit vector
in the complex-quaternion wave function, is incapable of playing a major role, the unit vector
[i.sub.q] will be degenerated into the imaginary unit i or one new three-dimensional unit vector
[I'.sub.q], which is independent of the unit vector
Translation them to unit vector
using Euclidean norm:
where [n.sub.1] and [n.sub.2] are normal unit vectors
at the two sides of the wall and [C.sup.+](x) and [C.sup.-](x) are the two constants that depend on the position of x