The local reference systems ( MRS and DRS)

As explained in [GEOM001], the setup is described via the definition of an initial volume inside which all the others will be positioned. In GEANT terminology, each time a volume has contents, created either via division or by positioning other volumes inside, it is called a MOTHER. The volumes contained are called DAUGHTERs, and they, in turn, can contain volumes to a depth of 15 levels. This is sometimes referred to as a Russian doll geometry.

Every volume defined in GEANT has a reference system attached to it (see GEOM section). When this volume has contents, this is referred to as the Mother Reference System ( MRS, with origin in $O$_m ). Daughters are positioned inside the mother with respect to the MRS. The MRS of the first volume defined, containing all the others, is nothing else than the MARS.

Each one of the daughters has its own reference system, which is referred to as the Daughter Reference System, or DRS with origin in $O$_d .

The transformation of a point from the MRS ($V$_m ) to the DRS ($V$_d ), at any level, is performed using a rotation matrix $[R]$ and a translation vector T via the relation :

$V$_d=[ R ](V_m-T)

The components of T are the projections of the vector $(O$_m, O_d)

onto the MRS axes. The rotation matrices are computed from the spherical angles of each of the axes of the daughter reference systems ( I, II, III) with respect to the mother reference system ( 1, 2, 3). The spherical angles $\Theta$ and $\Phi$ of a direction D are defined as follows :

$\Theta$

is the angle formed by the axis 3 and D ( ).

$\Phi$

is the angle formed by the axis 1 and the projection of D onto the plane defined by the axes 1 and 2 ( ).