Transformation matrices Emmanuel Branlard January 2011 The laboratory frame R, the rest frame R0 . We will note respectivly (e), (e0 ) and (e0b ) the canonical bases adapted to the laboratory frame, the rest frame and the main directions of the bunch. To adapt the meshing to the distribution, one need to detect the distribution main axis. For this, we used the covariance matrix of the particles. Let C being this 3×3-matrix. For this method, there should be no difference between using coordinates in the rest frame or in the laboratory frame. In our case, we chose the coordinates in the rest frame R0 . C = Cov x0i , yi0 , zi0
(1)
C is by definition a symmetric real matrix, hence diagonalizable to a matrix D. Once we find the eigen vectors of this matrix, we have the principal axis of the distribution. Thus diagonalizing the covariance matrix is an elegant method to further adapt the meshing on the distribution. Rotations and base transformation can be confusing, but not if one takes the time to write the proper formalism: e0
P |eb0 e0 e0b
P|
= Mat Id, e0b , e0
(2) e0b −1 e0
0
= Mat Id, e0 , eb = P |
(3)
e0
Where P |eb0 is the transformation matrix from the base (e0 ) to the base (e0b ). We will immediatly note that these matrix are rotations, i.e. orthogonal matrix, such that P −1 = P T and det P = 1. The diagonalization of C can be writen: 0 0 e0 D|e0b = P |ee0 · C|ee0 · P |eb0 (4) b
0 | 0 , E 0 | 0 , E 0 | 0 the three normalised eigen vectors of C Let us define Eb,1 e b,2 e b,3 e forming the base (e0b ), which coordinates are expressed in the base (e0 ). The 0 is: 3×3-matrix formed by the three column vectors Eb,i e0
0 0 0 P |eb0 = Eb,1 Eb,2 Eb,3
1
(5)
And to simplify notations we will further write it R: According to the previous formalism we then have: e0
R = P |eb0 T
D = R ·C ·R
(6) (7)
And eventually, let’s write down the fact that, for a given vector X expressed in the rest base (e0 ), its coordinaates in (e0b ) are: 0
X|e0b = P |ee0 · X|e0 = RT · X|e0 b
2
(8)
