**Reflections and Trigonometry**

When I built my first scanner, I worked out the position of the lamp by trial and error. However, it should be possible to calculate this, and at the same time gain a better insight into how camera position and glass reflections influence performance.

This diagram shows the key angles...

C is the camera position

F is the lamp position

AB is one side of the glass platen

angle FAB is 50deg (in most scanners)

CDA is 90deg since we must avoid tombstone distortions

The most serious reflections into the camera lens follow the path FBC. That is when the lens can just see the lamp reflected in the edge of the platen. It is probably best to have the lamp a little bit higher, since internal reflections can occur in the camera lens if stray light enters at all. It should be noted that BD is larger than AD, since the platen must be able to handle book covers.

Finding the length AF now reduces to fairly simple trigonometry if we know the lengths CD, BD, DA and the angle FAB.

The steps in the calculation are..

angles DCB, CBE, and FBE are equal and can be calculated, since

tanDCB = BD/DC (both lengths are known, so the angle is known)

If we look at triangle ABF, we know all its angles and the length of side AB, so we can use the law of sines to calculate side AF.

(See http://en.wikipedia.org/wiki/Law_of_sines for more information)

AF/sinABF = AB/sinBFA

I put these calculations into a spreadsheet, and produced results that agreed with my measurements. (Please correct me if there are errors in my workings, since it is a very long time since I attended a trig lesson).

After drawing this diagram, it became clear that the light source should be as small as possible in the two dimensions here, but could be a strip of light in the third dimension. Although AF (the lamp height) can be reduced by moving the camera out (increasing DC), this can produce problems elsewhere.

Malcolm