PENELOPE 2003 - OECD Nuclear Energy Agency

156 Chapter 5. Constructive quadric geometry
where
⎛
⎞
R xx R xy R xz
R(ω, θ, φ) = ⎜
⎝ R yx R yy R ⎟ yz ⎠ (5.4)
R zx R zy R zz
is the rotation matrix. To obtain its explicit form, we recall that the matrices for
rotations about the z- and y-axes are
⎛
⎞
cos φ − sin φ 0
R z (φ) = ⎜
⎝ sin φ cos φ 0
0 0 1
respectively. Hence,
R(ω, θ, φ) = R z (φ)R y (θ)R z (ω)
⎟
⎠ and R y (θ) =
⎛
⎞
cos θ 0 sin θ
⎜
⎝ 0 1 0 ⎟
⎠ , (5.5)
− sin θ 0 cos θ
⎛
⎞ ⎛
⎞ ⎛
⎞
cos φ − sin φ 0 cos θ 0 sin θ cos ω − sin ω 0
= ⎜
⎝ sin φ cos φ 0 ⎟ ⎜
⎠ ⎝ 0 1 0 ⎟ ⎜
⎠ ⎝ sin ω cos ω 0 ⎟
⎠
0 0 1 − sin θ 0 cos θ 0 0 1
⎛
⎞
cos φ cos θ cos ω − sin φ sin ω − cos φ cos θ sin ω − sin φ cos ω cos φ sin θ
= ⎜
⎝ sin φ cos θ cos ω + cos φ sin ω − sin φ cos θ sin ω + cos φ cos ω sin φ sin θ ⎟
⎠ . (5.6)
− sin θ cos ω sin θ sin ω cos θ
The inverse of the rotation R(ω, θ, φ) is R(−φ, −θ, −ω) and its matrix is the transpose
of R(ω, θ, φ), i.e.
R −1 (ω, θ, φ) = R(−φ, −θ, −ω) = R z (−ω)R y (−θ)R z (−φ) = R T (ω, θ, φ). (5.7)
Let us now consider transformations C = T (t) R(ω, θ, φ) that are products of a
rotation R(ω, θ, φ) and a translation T (t). C transforms a point r into
or, in matrix form,
r ′ = C(r) = T (t) R(ω, θ, φ) r (5.8)
⎛
x ′ ⎞
⎛
⎜
⎝ y ′
⎟
⎠ = R(ω, θ, φ) ⎜
⎝
z ′
⎞ ⎛ ⎞
x t x
y ⎟
⎠ + ⎜
⎝ t ⎟ y ⎠ . (5.9)
z t z
Notice that the order of the factors does matter; the product of the same factors in
reverse order D = R(ω, θ, φ) T (t) transforms r into a point r ′ = D(r) with coordinates
⎛
x ′ ⎞
⎛
⎜
⎝ y ′
⎟
⎠ = R(ω, θ, φ) ⎜
⎝
z ′
⎞
x + t x
y + t ⎟ y ⎠ . (5.10)
z + t z