CERN Program Library Long Writeup W5013 - CERNLIB ...

Geant 3.16 GEANT User’s Guide GEOM200
Origin : R.Brun, F.Carena Submitted: 01.06.83
Revision : Revised: 14.12.93
Documentation :
Rotation matrices
The relative position of a volume inside its mother is expressed in GEANT by a translation vector and a
rotation matrix which are arguments of the routines GSPOS and GSPOSP. The rotation matrix expresses the
transformation from the Mother Reference System to the Daughter Reference System.
A rotation matrix is described to GEANT by giving the polar and azimuthal angles of the axes of the DRS
(x ′ ,y ′ ,z ′ ) in the MRS via the routine GSROTM.
CALL GSROTM
(IROT,THETA1,PHI1,THETA2,PHI2,THETA3,PHI3)
IROT (INTEGER) number of the rotation matrix;
THETA1 (REAL) polar angle for axis x ′ ;
PHI1 (REAL) azimuthal angle for axis x ′ ;
THETA2 (REAL) polar angle for axis y ′ ;
THI2 (REAL) azimuthal angle for axis y ′ ;
THETA3 (REAL) polar angle for axis z ′ ;
PHI3 (REAL) azimuthal angle for axis z ′ .
Stores rotation matrix IROT in the data structure JROTM. If the matrix is not orthonormal, it will be corrected
by setting y ′ ⊥ x ′ and then z ′ = x ′ × y ′ . A warning message is printed in this case.
Note: the angles THETA and PHI must be given in degrees.
Examples of use
The unit matrix is defined in the following way:
x ′ ‖ x
y ′ ‖ y
z ′ ‖ z
⎫ ⎧
⎪⎬ ⎪⎨
⇒
⎪⎭ ⎪⎩
θ 1 = 90 ◦ ; φ 1 = 0 ◦
θ 2 = 90 ◦ ; φ 2 = 90 ◦
θ 3 = 0 ◦ ; φ 3 = 0 ◦
This is just an example. There is in fact no need to define a unit rotation matrix. Giving the value 0 to the
rotation matrix number in the call to GSPOS and GSPOSP is equivalent to a positioning without rotation and
it improves tracking performance.
The result of a 90 ◦ counterclockwise rotation around z, followed by a 90 ◦ counterclockwise rotation around
the new x is a cyclic shift of the axes: x → z ′ ,y→ x ′ ,z→ y ′ . This is expressed by the following rotation
matrix:
x ′ ‖ y
y ′ ‖ z
z ′ ‖ x
⎫ ⎧
⎪⎬ ⎪⎨
⇒
⎪⎭ ⎪⎩
θ 1 = 90 ◦ ; φ 1 = 90 ◦
θ 2 = 0 ◦ ; φ 2 = 0 ◦
θ 3 = 90 ◦ ; φ 3 = 0 ◦
Sometimes the rotation matrix is known or it can be constructed. In this case the arguments to the routine
GSROTM can be calculated with the help of the routine GFANG in the following way:
133 GEOM200 – 1