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• P – translation vector pointing from IP to the detector origin,
• e_x – unit vector along the detector frame x axis,

• e_y – unit vector along the detector frame y axis

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Third unit vector e_z – is assumed to be right-hand triplet component

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,

               ez=[

...

ex

...

× ey

...

].

Components of these three unit vectors forms the matrix eij, where indexes "i" and "j" corresponds to the vector components in the setup and detector local coordinate frames, respectively. Within this definition 3-d pixel coordinate "c" from in the detector frame can be easily transformed to the setup coordinate "C":

                              Ci=eij·cj + Pi,where indexes "_i" and "j" corresponds to the vector components in the setup and detector local coordinate frames, respectively.

Tile ideal geometry

• Assume that the detector image is produced by a set of tiles (a.k.a., segment, section, 2x1, sensor, Si-pixel matrix, etc).
• Each tile has well defined by design geometry of pixels, which does not need in separate calibration.
• Pixel (x,y) center coordinate at z=0 for each tile can be defined as a look-up table in its "natural" matrix-style coordinate system.
• For example, CSPAD 2x1 tile pixel geometry is defined as

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