Geometrical transformation package.
All geometrical transformations handled by the modeller are provided as a
built-in package. This was designed to minimize memory requirements and
optimize performance of point/vector master-to-local and local-to-master
computation. We need to have in mind that a transformation in TGeo has 2
major use-cases. The first one is for defining the placement of a volume
with respect to its container reference frame. This frame will be called
'master' and the frame of the positioned volume - 'local'. If T is a
transformation used for positioning volume daughters, then:
MASTER = T * LOCAL
Therefore a local-to-master conversion will be performed by using T, while
a master-to-local by using its inverse. The second use case is the computation
of the global transformation of a given object in the geometry. Since the
geometry is built as 'volumes-inside-volumes', this global transformation
represent the pile-up of all local transformations in the corresponding
branch. The conversion from the global reference frame and the given object
is also called master-to-local, but it is handled by the manager class.
A general homogenous transformation is defined as a 4x4 matrix embeeding
a rotation, a translation and a scale. The advantage of this description
is that each basic transformation can be represented as a homogenous matrix,
composition being performed as simple matrix multiplication.
Rotation: Inverse rotation:
r11 r12 r13 0 r11 r21 r31 0
r21 r22 r23 0 r12 r22 r32 0
r31 r32 r33 0 r13 r23 r33 0
0 0 0 1 0 0 0 1
Translation: Inverse translation:
1 0 0 tx 1 0 0 -tx
0 1 0 ty 0 1 0 -ty
0 0 1 tz 0 0 1 -tz
0 0 0 1 0 0 0 1
Scale: Inverse scale:
sx 0 0 0 1/sx 0 0 0
0 sy 0 0 0 1/sy 0 0
0 0 sz 0 0 0 1/sz 0
0 0 0 1 0 0 0 1
where: rij are the 3x3 rotation matrix components,
tx, ty, tz are the translation components
sx, sy, sz are arbitrary scale constants on the eacks axis,
The disadvantage in using this approach is that computation for 4x4 matrices
is expensive. Even combining two translation would become a multiplication
of their corresponding matrices, which is quite an undesired effect. On the
other hand, it is not a good idea to store a translation as a block of 16
numbers. We have therefore chosen to implement each basic transformation type
as a class deriving from the same basic abstract class and handling its specific
data and point/vector transformation algorithms.
/*
*/
The base class TGeoMatrix defines abstract metods for:
- translation, rotation and scale getters. Every derived class stores only
its specific data, e.g. a translation stores an array of 3 doubles and a
rotation an array of 9. However, asking which is the rotation array of a
TGeoTranslation through the base TGeoMatrix interface is a legal operation.
The answer in this case is a pointer to a global constant array representing
an identity rotation.
Double_t *TGeoMatrix::GetTranslation()
Double_t *TGeoMatrix::GetRotation()
Double_t *TGeoMatrix::GetScale()
- MasterToLocal() and LocalToMaster() point and vector transformations :
void TGeoMatrix::MasterToLocal(const Double_t *master, Double_t *local)
void TGeoMatrix::LocalToMaster(const Double_t *local, Double_t *master)
void TGeoMatrix::MasterToLocalVect(const Double_t *master, Double_t *local)
void TGeoMatrix::LocalToMasterVect(const Double_t *local, Double_t *master)
These allow correct conversion also for reflections.
- Transformation type getters :
Bool_t TGeoMatrix::IsIdentity()
Bool_t TGeoMatrix::IsTranslation()
Bool_t TGeoMatrix::IsRotation()
Bool_t TGeoMatrix::IsScale()
Bool_t TGeoMatrix::IsCombi() (translation + rotation)
Bool_t TGeoMatrix::IsGeneral() (translation + rotation + scale)
Combinations of basic transformations are represented by specific classes
deriving from TGeoMatrix. In order to define a matrix as a combination of several
others, a special class TGeoHMatrix is provided. Here is an example of matrix
creation :
Matrix creation example:
root[0] TGeoRotation r1,r2;
r1.SetAngles(90,0,30); // rotation defined by Euler angles
r2.SetAngles(90,90,90,180,0,0); // rotation defined by GEANT3 angles
TGeoTranslation t1(-10,10,0);
TGeoTranslation t2(10,-10,5);
TGeoCombiTrans c1(t1,r1);
TGeoCombiTrans c2(t2,r2);
TGeoHMatrix h = c1 * c2; // composition is done via TGeoHMatrix class
root[7] TGeoHMatrix *ph = new TGeoHMatrix(hm); // this is the one we want to
// use for positioning a volume
root[8] ph->Print();
pVolume->AddNode(pVolDaughter,id,ph) // now ph is owned by the manager
Rule for matrix creation:
- unless explicitly used for positioning nodes (TGeoVolume::AddNode()) all
matrices deletion have to be managed by users. Matrices passed to geometry
have to be created by using new() operator and their deletion is done by
TGeoManager class.
Available geometrical transformations
1. TGeoTranslation - represent a (dx,dy,dz) translation. Data members:
Double_t fTranslation[3]. Translations can be added/subtracted.
TGeoTranslation t1;
t1->SetTranslation(-5,10,4);
TGeoTranslation *t2 = new TGeoTranslation(4,3,10);
t2->Subtract(&t1);
2. Rotations - represent a pure rotation. Data members: Double_t fRotationMatrix[3*3].
Rotations can be defined either by Euler angles, either, by GEANT3 angles :
TGeoRotation *r1 = new TGeoRotation();
r1->SetAngles(phi, theta, psi); // all angles in degrees
This represent the composition of : first a rotation about Z axis with
angle phi, then a rotation with theta about the rotated X axis, and
finally a rotation with psi about the new Z axis.
r1->SetAngles(th1,phi1, th2,phi2, th3,phi3)
This is a rotation defined in GEANT3 style. Theta and phi are the spherical
angles of each axis of the rotated coordinate system with respect to the
initial one. This construction allows definition of malformed rotations,
e.g. not orthogonal. A check is performed and an error message is issued
in this case.
Specific utilities : determinant, inverse.
3. Scale transformations - represent a scale shrinking/enlargement. Data
members :Double_t fScale[3]. Not fully implemented yet.
4. Combined transformations - represent a rotation folowed by a translation.
Data members: Double_t fTranslation[3], TGeoRotation *fRotation.
TGeoRotation *rot = new TGeoRotation("rot",10,20,30);
TGeoTranslation trans;
TGeoCombiTrans *c1 = new TGeoCombiTrans(trans, rot);
TGeoCombiTrans *c2 = new TGeoCombiTrans("somename",10,20,30,rot)
5. TGeoGenTrans - combined transformations including a scale. Not implemented.
6. TGeoIdentity - a generic singleton matrix representing a identity transformation
NOTE: identified by the global variable gGeoIdentity.
