1. Alignment Which way is up?

2. Coordinate Systems Global Coordinate System x z y Global to Local: q = R -1 (r – r 0,i ) Local to Global: r = Rq + r 0,i R = Rotation matrix transforming from local to global system r 0 = Position of Tower origin r 0,i = Translation vector from global to local origin r i = Position, relative to tower origin, of plane origin r 0,i r q u v w Local Plane Coordinate System Plane i x’ y’ z’ Tower Origin r0r0 riri

3. Coordinate Systems (another view) Global Coordinate System x z y Global to Local: q = R -1 (r – r 0,i ) Local to Global: r = Rq + r 0,i R = Rotation matrix transforming from local to global system r 0 = Position of Tower origin r 0,i = Translation vector from global to local origin r i = Position, relative to tower origin, of plane origin r 0,i r q u v w Local Plane Coordinate System Plane i x’ y’ z’ Tower Origin r0r0 riri

4. Transformations Rotation R and translation r 0,i are for a perfectly aligned detector –For Glast, we can take R = I (the identity matrix) Vector to origin of the ith plane: –r 0,i = r 0 + r i –For Glast we have r i = (0,0,  z i ) Corrections to perfect alignment will be small, above are modified by and incremental rotation  R and translation  r: –R →  RR –r 0 →  r 0 + r 0 These corrections give: –r 0,I = r 0 +  r 0 +  Rr i –r =  RRq +  r 0 + r 0 +  Rr i =  R(Rq + r i ) + r 0 +  r 0 –q = (  RR) -1 (r - r 0 -  r 0 -  Rr i )

5. Express the incremental rotation matrix as:  R = R x (  α)R y (  β) R z (  γ ) where R x (  α), R y (  β) and R z (  γ ) are small rotations by  α,  β,  γ about the x-axis, y-axis and z-axis, respectively In General Incremental Rotation Matrix 1 0 0 0 cos  α -sin  α 0 sin  α cos  α R x (  α) = cos  β 0 sin  β 0 1 0 -sin  β 0 cos  β Ry(β) =Ry(β) = Cos  γ -sin  γ 0 Sin  γ cos  γ 0 0 0 1 R z (  γ ) =

6. Multiplying it out, we get: Taking  α,  β and  γ to be small (and ignoring terms above 1 st order) gives: Incremental Rotation Matrix (continued)  R = 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  R = 1 -  γ  β  γ 1 -  α -  β  α 1

7. Start with: r =  R(Rq + r i ) + r 0 +  r 0 For Glast, R = I, the identity matrix  R as given on the previous page r i = (0, 0,  z i ) since, for Glast, the silicon planes are parallel to x-y plane q = (u i, v i, 0) since the measurement is in the sense plane (no z coordinate) This gives: Local to Global Transformation x = u i -  γv i +  β  z i + x 0 +  x 0 y = v i +  γu i -  α  z i + y 0 +  y 0 z =  z i -  βu i +  αv i + z 0 +  z 0 r =r = 1 -  γ  β  γ 1 -  α -  β  α 1 uiviziuivizi x 0 +  x 0 y 0 +  y 0 z 0 +  z 0 +

8. Start with: q = (  RR) -1 (r - r 0 -  r 0 -  Rr i ) For Glast, R = I, the identity matrix  R as given on the previous page, to 1 st order  R -1 =  R T  Rr i = (  β  z i, -  α  z i,  z i ) This gives (keeping terms to 1 st order only): Global to Local Transformation q =q = 1  γ -  β -  γ 1  α  β -  α 1 x - x 0 -  x 0 -  β  z i y - y 0 +  y 0 +  α  z i z - z 0 +  z 0 -  z i u i = x – x 0 –  x 0 +  γ (y – y 0 –  y 0 ) –  β (z – z 0 –  z 0 ) v i = y – y 0 –  y 0 –  γ (x – x 0 –  x 0 ) +  α (z – z 0 –  z 0 ) w i = z – z 0 –  z 0 –  β (x – x 0 –  x 0 ) +  α (y – y 0 –  y 0 ) –  z i