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More Math like Inverse Square Theory?


Max Field

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A couple years ago I learned about Inverse Square Theory from this forum and it's really helped me get down the math of setting up my lighting.

Does anyone have a list or link to other little mathematical tricks like that used in cinema everyday?

Thanks!

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100 foot-candles = f/2.8 (at 24 fps / 180 degree shutter) at 100 ASA.  Remembering that will make it easier to break down photometric data on lamps.

Every f-stop number is the double or the half the number two stops over, i.e. f/1.4 to f/2.8 is a two-stop jump. f/2 to f/4 is a two-stop jump.

Sunny 16 rule, that the exposure is f/16 when the shutter time value under 1/ is the same as the ASA, so 1/50 at 50 ASA = f/16 if shooting under direct sun on a clear day.

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Mired shifts better represent the apparent difference between colour temperatures. Divide 1,000,000 by the CT to get the mired value of any light source; compare two by subtracting them (order the calculation so that negative values mean bluer.)

The exposure value required to properly expose a scene is the binary logarithm of the light level in lux multiplied by the ISO, divided by C, where C is the meter calibration constant usually equal to about 330. For 1000 lux at 100 ISO, calculate:

1000 × 100 ÷ 330 = ln ÷ 2 ln = 8.24

The exposure value of a camera setup is the binary logarithm of the the aperture squared over the shutter time in seconds. For 1/48s at f/2.8, calculate:

2.82 ÷ (1 ÷ 48) = ln ÷ 2 ln = 8.56

(The calculations above are given in reverse polish notation, that is, that's the order you'd hit the keys on a scientific calculator.)

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> Every f-stop number is the double or the half the number two stops over, i.e. f/1.4 to f/2.8 is a two-stop jump. f/2 to f/4 is a two-stop jump.
 
This is another application of an inverse square law, and closely connected to the one Max is probably referring to in the original post, about light fall off with distance from the source.
 
 
"Every f-stop number is the double or the half the number two stops over,” is equivalent to saying "Every f-stop number is √2 or 1/√2 times the f-stop number one stop away from it"
 
 
For a given focal length, f-number is proportional to the inverse of the diameter of the entrance pupil of the lens (just using the formula for f-number). The amount of light the lens passes is proportional to the area of the entrance pupil (a circle), which is in turn proportional to the square of the circle's diameter (because area of a circle is pi r^2).
 
So overall the amount of light that passes is proportional to the inverse square of the f-number.
E.g. double the f-number and the amount of light drops to a quarter.
 
 
Realising this made it easier for me to understand the seemingly strange pattern of f-numbers. What is marked on lenses is usually an approximation of the true sequence which is:
 
1, √2, 2, 2√2, 4, 4√2, 8, 8√2, 16, 16√2…
 
 
 
The reason for the inverse square law of light fall is, roughly, because light emanates from its source in a sphere. At any given distance from the point of emanation the area of the sphere, over which the light is spread out, is proportional to the square of that distance, because the area of a sphere is 4πr^2, and the brightness of the light is inversely proportional to that area (you can think of  the same amount of light being spread out over a larger area).
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