E.1 Exemple of an ellipsoid at rest

The simulation was done with the following parameters:

Table E.1: Parameters used for the benchmarking of the rotation algorithm - exemple of an ellipsoid at rest with two rotations

Parameter	Value
Distribution type	Uniform ellipsoid
Distribution $\sigma_x$	1 mm
Distribution $\sigma_y$	1 mm
Distribution $\sigma _z$	3 mm
Distribution energy $\mathcal{E}$	0 MeV
Distribution charge	-0.25 nC
Rotation 1 axis	Y
Rotation 1 angle	$-\pi/3$
Rotation 2 axis	Z
Rotation 2 angle	$-\pi/4$
3D grid used	$N\ast_0=5$ , $N\ast_f$ =20, $N\ast_2$ =32

**Figure E.1:** Ellipsoid at rest with two rotations: the two rotation steps

**Figure E.2:** Ellipsoid at rest with two rotations: canonical directions

**Figure E.3:** Rotation algorithm - Longitudinal Field. (a) Before the rotation of the fields - (b) After the rotation of the fields - (c) Fields from a distribution not rotated
(a) (b) (c)