Transcript ppt

Alignment
Which way is up?
Coordinate Systems
R = Rotation matrix transforming
from local to global system
Local to Global:
r = Rq + r0,i
r0 = Position of Tower origin
r0,i = Translation vector from
global to local origin
w
Local Plane
Coordinate System
q
ri = Position, relative to tower
origin, of plane origin
v
r
u
ri
Plane i
z
r0,i
z’
r0
y
x
Global Coordinate System
Tower Origin
y’
x’
Global to Local:
q = R-1(r – r0,i)
Coordinate Systems
(another view)
R = Rotation matrix transforming
from local to global system
r0 = Position of Tower origin
r0,i = Translation vector from
global to local origin
ri = Position, relative to tower
origin, of plane origin
z
y
x
Global Coordinate System
Local to Global:
r = Rq + r0,i
Global to Local:
q = R-1(r – r0,i)
Transformations
•
Rotation R and translation r0,i are for a perfectly aligned detector
– For Glast, we can take R = I (the identity matrix)
•
Vector to origin of the ith plane:
– r0,i = r0 + ri
– For Glast we have ri = (0,0,zi)
•
Corrections to perfect alignment will be small, above are modified by and
incremental rotation R and translation r:
– R → RR
– r0 → r0 + r0
•
These corrections give:
– r0,I = r0 + r0 + Rri
– r = RRq + r0 + r0 + Rri
= R(Rq + ri) + r0 + r0
– q = (RR)-1(r - r0 - r0 - Rri)
Incremental Rotation Matrix
•
Express the incremental rotation matrix as:
R = Rx(α)Ry (β) Rz(γ )
where Rx(α), Ry(β) and Rz(γ ) are small rotations by α, β, γ about the
x-axis, y-axis and z-axis, respectively
•
In General
1
Rx(α) =
0
0
0 cos α -sin α
0 sin α
Rz(γ ) =
cos α
Cos γ -sin γ
0
Sin γ
0
0
cos γ
0
1
Ry(β) =
cos β
0
sin β
0
1
0
-sin β
0 cos β
Incremental Rotation Matrix
(continued)
•
R =
Multiplying it out, we get:
cos β cos γ
cos γ sin α sin β - cos α sin γ
cos α cos γ sin β + sin α sin γ
cos β sin γ
cos α cos γ + sin α sin β sin γ
-cos γ sin α + cos α sin β sin γ
- sin β
•
cos β sin α
cos  α cos β
Taking α, β and γ to be small (and ignoring terms above 1st order) gives:
R =
1
-γ
β
γ
1
-α
-β
α
1
Local to Global Transformation
•
Start with:
r = R(Rq + ri) + r0 + r0
•
•
•
•
•
For Glast, R = I, the identity matrix
R as given on the previous page
ri = (0, 0, zi) since, for Glast, the silicon planes are parallel to x-y plane
q = (ui, vi, 0) since the measurement is in the sense plane (no z coordinate)
This gives:
1
-γ
β
x0 + x0
ui
r=
γ
1
-α
-β
α
1
vi
zi
+
y0 + y0
z0 + z0
x = ui - γvi + βzi + x0 + x0
y = vi + γui - αzi + y0 + y0
z = zi - βui + αvi + z0 + z0
Global to Local Transformation
•
Start with:
q = (RR)-1(r - r0 - r0 - Rri)
•
•
•
•
For Glast, R = I, the identity matrix
R as given on the previous page, to 1st order R-1 = RT
Rri = (βzi, - αzi, zi)
This gives (keeping terms to 1st order only):
q=
1
γ
-β
x - x0 - x0 - βzi
-γ
1
α
y - y0 + y0 + αzi
β
-α
1
z - z0 + z0 - zi
ui = x – x0 –x0 + γ (y – y0 – y0) – β (z – z0 – z0)
vi = y – y0 – y0 – γ (x – x0 – x0) + α (z – z0 –z0)
wi = z – z0 –z0 – β (x – x0 – x0) + α (y – y0 – y0) – zi