E.2 Exemple of a sphere at low energy

The simulation was done with the following parameters:

Table E.2: Parameters used for the benchmarking of the rotation algorithm - exemple of a sphere at low energy

Parameter	Value
Distribution type	Uniform ellipsoid
Distribution $\sigma_x$	1 mm
Distribution $\sigma_y$	1 mm
Distribution $\sigma _z$	3 mm
Distribution energy $\mathcal{E}$	0.1 MeV
Distribution charge	-0.25 nC
Rotation axis	X
Rotation angle	$\pi/6$
3D grid used	$N\ast_0=5$ , $N\ast_f$ =20, $N\ast_2$ =32

**Figure E.4:** Sphere at low energy: canonical directions

**Figure E.5:** Rotation algorithm - Longitudinal Field along longitudinal direction. (a) Before the rotation of the fields - (b) After the rotation of the fields - (c) Fields from a distribution not rotated
(a) (b) (c)

**Figure E.6:** Rotation algorithm - Longitudinal Field along transverse direction. (a) Before the rotation of the fields - (b) After the rotation of the fields - (c) Fields from a distribution not rotated
(a) (b) (c)

**Figure E.7:** Rotation algorithm - Transverse Field in y direction. (a) Before the rotation of the fields - (b) After the rotation of the fields - (c) Fields from a distribution not rotated
(a) (b) (c)