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Alignment

Alignment. Which way is up?. w. Local Plane Coordinate System. q. v. r. u. z. r i. Plane i. r 0,i. Tower Origin. z’. y’. r 0. x’. y. x. Coordinate Systems.

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Alignment

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  1. Alignment Which way is up?

  2. w Local Plane Coordinate System q v r u z ri Plane i r0,i Tower Origin z’ y’ r0 x’ y x Coordinate Systems R = Rotation matrix transforming from local to global systemr0 = Position of Tower originr0,i = Translation vector from global to local originri = Position, relative to tower origin, of plane origin Local to Global: r = Rq + r0,i Global to Local: q = R-1(r – r0,i) Global Coordinate System

  3. w Local Plane Coordinate System q v r u z ri Plane i r0,i Tower Origin z’ y’ r0 x’ y x Coordinate Systems(another view) R = Rotation matrix transforming from local to global systemr0 = Position of Tower originr0,i = Translation vector from global to local originri = Position, relative to tower origin, of plane origin Local to Global: r = Rq + r0,i Global to Local: q = R-1(r – r0,i) Global Coordinate System

  4. 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)

  5. cos β 0 sin β 0 1 0 -sin β 0 cos β Ry(β)= • 0 0 • 0 cos α -sin α • 0 sin α cos α Cos γ -sin γ 0 Sin γ cos γ 0 0 0 1 Rx(α) = Rz(γ ) = 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

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

  7. 1 -γβ γ 1 -α -βα 1 x0 + x0 y0 + y0 z0 + z0 ui vi zi + r= 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: x = ui - γvi + βzi + x0 + x0y = vi + γui - αzi + y0 + y0z = zi - βui + αvi + z0 + z0

  8. 1 γ -β -γ 1 α β -α 1 x - x0 - x0 - βzi y - y0 + y0 + αzi z - z0 + z0 - zi q= 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): 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

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