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Alignment Which way is up?
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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
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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
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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 )
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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 ) =
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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
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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 uiviziuivizi x 0 + x 0 y 0 + y 0 z 0 + z 0 +
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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
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