Geometric interpretation of the vectors

**a**, **b**, **c**. (a) Imaging equations. Only the

*u* direction is considered here. Recall that by definition

*c* =

*n̂*/(−

*n̂* ·

*s*),

*a* = ((

*u*_{s} −

*u*_{c})

*n̂* +

*f* **û**)/(−

*n̂* ·

*s*) and that

*r*_{0} =

**0** is the center of the lab frame. Note that

*f* > 0 in the diagram since we have a direct imaging system. From its definition, vector

*a* is perpendicular to the line joining

*s* to

*r*_{0} (and is also perpendicular to

*v̂*), and lies in the direction shown, where

**â** =

*a*/‖

*a*‖. Note that

*l*_{1} =

**â** ·

*r* is negative, as is

*l*_{2} = (

*f/d*)

*l*_{1}, because

*d* =

*n̂* · (

*r* −

*s*) > 0. Finally,

and

*u* =

*l*_{3} is also negative (recalling that

*u* = ũ −

*u*_{c}). Assembling these pieces we have

(because −

*n̂* ·

*s* > 0). Thus we obtain the imaging equations directly from the definitions of

**a**, **b**, **c** and the imaging geometry (the diagram). (b) Directions and lengths. A rectangle with sides parallel to

**û** and

*v̂* and with diagonally opposing points (

*u*_{s},

*v*_{s}) and (

*u*_{c},

*v*_{c}) is formed on the detector. The pyramid with this rectangular base and vertex at

*s* is shown. Two of the sides rise perpendicularly from the detector, with unit normal vectors

**û** and

*v̂*. The other two sides (shaded) are slightly inclined, and their unit normal vectors are

**â** and

*b̂* for the more hidden vertical side and for the nearly horizontal side, respectively. The rectangular base has a unit normal vector

**ĉ**. For the lengths of

**a**, **b**, **c**, we note that 1/‖

*c*‖ = |

*n̂* ·

*s*| is the length from

*s* to

*r*_{0} projected along the vertical edge of the pyramid. We also note that

which is the full length of the pyramid edge with base point (

*u*_{s},

*v*_{c}). Similarly ‖

*a*‖/‖

*c*‖ is the length of the pyramid edge with base point (

*u*_{c},

*v*_{s}).

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