Ahh, the good olâ€™ point source / area source debate once again. This discussion somehow always pops up in many differing fields of physics. I will try my best to straighten out the concepts.
|The illumination intensity of a plain does NOT change with distance.
This is not really relevant here but, granted - true â€“ if the area in question (plane) can be considered infinitely large. This is approximately the case if the distance is much smaller than the area dimensions. If this is the case you are in the so called near-field and total contribution (integrated over the area) intensity will be constant with varying distance. But this is not true in this case as the distance to the screen is even larger than the screen dimensions (and, much more importantly, a lot larger than the screen segments we are considering). Thus, we are in the near to far-field transition region and the statement above is not applicable even when considering total contribution illumination. Note that my ray-tracing algorithm will also produce approximately constant total screen contribution illumination for different distances close to the source or, in other words, as long as the source dimensions are large in comparison to the distance. This does not mean that the contribution from each screen segment is equal, however.
|If you used a spot meter to measure your screen intensity, you would get the same reading from 10 feet as 100 feet etc.
Only if you had one large muther of a screen â€“1000x1000 ft :D. This may well hold true for the planetarium where you work, but will very definitely not be true for any regular size screen, and the following explains why:
When we are far from a matte white screen, say 10 times screen width, it can be approximated as being a point source radiating into a half-sphere. So letâ€™s consider radiation from a point source. Intensity is defined as units/m2 or units/ft2 and thus falls by distance squared. Total radiated power will be the same for all distances, what we are doing is expanding the area of our â€œcontrol sphereâ€. In fact, if the intensity did not fall off this way we would increase the power output of the source just by increasing the distance. I hope there is no controversy here.
It is perfectly permissible that, without violating any physical principles, divide a large area into small segments (approximating point-sources) and consider the contribution from each source. This type of modeling is used extensively in many fields. Still no controversy, I am sure.
What my ray-traced results relate to is this situation:
1. Perfectly evenly illuminated screen. For example, if the projector has a light output of 1000 lumen, each screen segment is considered a source with a light output of 1000/20000 = 0.05 lumen. All segments are naturally equi-distant.
2. The illuminance [lux], [lumen/m2] or [foot-lamberts] from each screen segment is calculated assuming point source radiation, while also accounting for the gain characteristics.
3. The viewer is considered to be looking directly at all screen segments simultaneously (or in practice, scanning the screen area), otherwise there would be even more fall-off for off-center segments (cosine function).
|What your analysis doesn't take into effect is that the left side of the screen is SMALLER when the viewer is on the right side. Therefore the point charges are more densely concentrated.
The point charges or point sources have nothing to do with viewing position as they are only PJ-related, see 1. above.
Now letâ€™s build an example to illustrate my point:
Construct an imaginary array with the same dimensions as the screen we have been discussing in our previous posts. Cover it with 20000 50 W bulbs. Place yourself with a spot-meter 1.5X array width away, level with the right array edge. Turn off all bulbs except for the column on the very far left and measure the illumination. Now do the same but with the right edge bulbs active. The measured illuminances WILL differ.
|This is the same reason why your matte screen had a BR of 1.15 which should by definition have a perfect 1.00 BR (at least with a one gun digital projector, A 3 gun CRT can change things concievably in pratice).
You are comparing apples to oranges here. Brightness Ratio (BR) relates to the combined effects of distance and gain. It is perfectly possible for a perfect matte white material to have a constant gain of one over all angles while a particular setup with that material produces a BR > 1. See my bulb example above, the imaginary array is equivalent to a perfect flat matte white screen.
:D, Thanks for an interesting discussion.