PENELOPE 2003 - OECD Nuclear Energy Agency

5.1. Rotations and translations 155
the geometry on the computer screen. These programs use specific computer graphics
software and, therefore, they are not portable. The executable files included in the
penelope distribution package run on personal computers under Microsoft Windows;
they are simple and effective tools for debugging the geometry definition file.
5.1 Rotations and translations
The definition of various parts of the material system (quadric surfaces in reduced form
and modules) involves rotations and translations. To describe these transformations,
we shall adopt the active point of view: the reference frame remains fixed and only the
space points (vectors) are translated or rotated.
In what follows, and in the computer programs, all lengths are in cm. The position
and direction of movement of a particle are referred to the laboratory coordinate system,
a Cartesian reference frame which is defined by the position of its origin of coordinates
and the unit vectors ˆx = (1, 0, 0), ŷ = (0, 1, 0) and ẑ = (0, 0, 1) along the directions of
its axes.
A translation T (t), defined by the displacement vector t = (t x , t y , t z ), transforms
the vector r = (x, y, z) into
T (t) r = r + t = (x + t x , y + t y , z + t z ). (5.1)
Evidently, the inverse translation T −1 (t) corresponds to the displacement vector −t,
i.e. T −1 (t) = T (−t).
A rotation R is defined through the Euler angles ω, θ and φ, which specify a sequence
of rotations about the coordinate axes 1 : first a rotation of angle ω about the z-axis,
followed by a rotation of angle θ about the y-axis and, finally, a rotation of angle φ
about the z-axis. A positive rotation about a given axis would carry a right-handed
screw in the positive direction along that axis. Positive (negative) angles define positive
(negative) rotations.
The rotation R(ω, θ, φ) transforms the vector r = (x, y, z) into a vector
whose coordinates are given by
r ′ = R(ω, θ, φ) r = (x ′ , y ′ , z ′ ), (5.2)
⎛
x ′ ⎞
⎛
⎜
⎝ y ′
⎟
⎠ = R(ω, θ, φ) ⎜
⎝
z ′
⎞
x
y
z
⎟
⎠ , (5.3)
1 This definition of the Euler angles is the one usually adopted in Quantum Mechanics (see e.g.
Edmonds, 1960).