156
I. P. Markov and M. V. Markina
Fig. 11.1 Geometry of
vectors p, q, d, n and e
Fig. 11.2 Relations between
a, b, ϕ, ψ and vectors e, p, q,
d, n
n(ϕ, b) = ad(ϕ) + be = [n 1 , n 2 , n 3 ]
T
,
(11.6)
a =
1 − b 2 , d(ϕ) · r = 0, e =
r
|r |
, e = [e 1 , e 2 , e 3 ]
T
.
(11.7)
To define vector d(ϕ), we introduce a set of orthonormal vectors p and q
p =
[e 2 , −e 1 , 0]
T
1 − e
2
3
, q = e × p =
[e 1 e 3 , e 2 e 3 , −
1 − e
2
3
]
T
1 − e
2
3
,
(11.8)
p · e = 0, q · e = 0, p · q = 0.
(11.9)
That way, we can express vector d(ϕ) as
d(ϕ) = p cos ϕ + q sin ϕ, 0 ≤ ϕ ≤ 2π,
(11.10)
d(ϕ) =
e 2 cos ϕ + e 1 e 3 sin ϕ, −e 1 cos ϕ + e 2 e 3 sin ϕ, −
1 − e
2
3
sinϕ
T
1 − e
2
3
. (11.11)