The Mathematics of Exposure

Ben Schwartz · November 9, 2004

I'm reading about exposure and having trouble wrapping my head around the numbers. I never was very good at math. So bear with me if this seems elementary to you. I understand the general principles individually, but I'm still having trouble connecting the dots. Layman's terms are what I need. :blink:

Ahem. The inverse square law says that if you double the distance of a subject from its light source, that subject will receive 1/4 the light. But if you double the distance of a light source, the exposure loss is one stop? For example, let's say you're metering at f/5.6 with a source 16 feet away. You move the source to 32 feet away. Now you're metering at f/4. How is it that doubling the distance of a source requires the aperture to be opened one full stop, yet doubling the distance of a subject from its source means that it will receive 1/4 the fc's? Shouldn't something that receives 1/2 the fc's require a full stop of compensation?

Next, the cosine law. If you turn a subject away from a source, the decrease in exposure is equal to the cosine of the angle of the surface. Of course I haven't studied geometry since high school, so again, can someone explain this to me?

Of course with light meters, I suppose it isn't absolutely necessary to grasp the mathematics of exposure, but I'd like to anyway...

Edited November 9, 2004 by Ben Schwartz

Mike Panczenko · November 9, 2004

I don't know about the cosine law, but this visual might help with the inverse square:

http://hyperphysics.phy-astr.gsu.edu/hbase/forces/isq.html

Dominic Case · November 9, 2004

The inverse square law says that if you double the distance of a subject from its light source, that subject will receive 1/4 the light. But if you double the distance of a light source, the exposure loss is one stop?

No!

You've got the second rule wrong. You are quite right about the inverse square law: distance x2, light x 1/4. Correct.

Now for every 1/2 x light, you need one extra stop of exposure. So for 1/4 x light, you need two extra stops.

This only works for flooded lights. If they are spotlights or focussed beams, the light fall-off doesn't obey the inverse square law.

The cosine law: Basically, if you have a flat surface facing a light source, then a beam of light is spread over - say - one square meter of the surface. If you turn the surface at an angle to the light (or move the light round to one side), then the same beam of light is spread over a wider area of the surface. (This one IS easier to see with a spotlight, and the law does work better for spots or distant sources). If the same light is spread over a larger area, then it can't be as bright in any one spot. The greater the angle, the more spread-out the light. It turns out that the cosine is the factor you need. Easier to get a spotmeter and measure it.

Clue for understanding cosines (if you insist!): to reach a 10 ft high window, you need a 10 ft ladder at 0 degrees, but if you lean the ladder at an angle, you can't reach as high. The height you can actually reach, divided by the length of the ladder, is the cosine of the angle it's leaning at.

Ben Schwartz · November 9, 2004

Thank you Dominic for your response. Here is why I'm confused. On page 105 of Blain Brown's book "Cinematography: Theory and Practice", table 6.1 shows the relationship between lighting source distance and f-stop. It shows an f-stop of 4 with the source 64 feet away, and a f-stop of 5.6 with the source 32 feet away. Is this a mistake by Mr. Brown? If the distance is doubled, shouldn't the f-stop at 64 feet be 2.8, and not 4? :huh:

John Sprung · November 9, 2004

... an f-stop of 4 with the source 64 feet away, and a f-stop of 5.6 with the source 32 feet away. Is this a mistake... :huh:

<{POST_SNAPBACK}>

Yes, you have found an error in the book. It should be either f/4 and f/8, or f/2.8 and f/5.6.

-- J.S.

Ben Schwartz · November 9, 2004

Thank you! This answers all my questions. Mr. Brown's book is interesting, well-designed and informative, but so far I have found numerous errors in it -- both factual and typographical -- and I hope it is corrected in future editions.

Dominic Case · November 9, 2004

If you have time, it would be worth sending a note to Blain Brown (via the publishers, Focal Press) to draw his attention to this error. It's incredible how simple errors like this can slip through every check and double check in the system, (and not even be noticed by any reader for a couple of years!). These days publishers can often fix up minor typo corrections like this in the next print run (I'm sure there will be one for this book), without waiting for a new edition.

I'm saying this as an author who has also had errors pointed out in my books. It's mortifying to discover them, but better in the long run!

Matt Pacini · November 10, 2004

OK, this brings something to mind.

I had a friendly disagreement with a DP that shot some stuff for me, who is without question more knowledgable than myself, but still we didn't agreed over this.

We were shooting exteriors by sunlight only (with shiny reflector boards), and needed a bit more fill, so he told the grip to move the reflector in closer to the actor.

I said it wouldn't matter, because of the inverse square law, the distance from the light source wasn't the difference from the reflector to the actor, but is the distance from the sun to the reflector + the distance from the reflector to the actor, therefore moving the reflector closer would result in increasing the light level by about .000000000000000001%.

I realize the SIZE of the source would be a bit larger (the reflector would effectively be bigger in relation to the actor), but it wouldn't actually be any brighter, right?

(This was a diffused reflector panel, by the way, you know, with crinkly foil on it)

What do you guys think?

Matt Pacini

Alex Opdam · November 10, 2004

Surely though, Matt, the reflector effectively becomes the new light source once it has light bounced off it, and thus obeys the same rules as normal sources?

You say that the light on the subject gets 'larger' on the subject as you move it closer, isn't this effectively concentrating and increasing the light levels? Maybe i have missed something...

edit: also, you say that the final part of the light's journey to the actor is negligible... keep in mind this is also the part of the journey when it loses a lot of it's light (since it is being bounced from a diffusing surface).

Edited November 10, 2004 by Alex Opdam

John Sprung · November 10, 2004

If your reflector were a perfect mirror, uniformly changing the direction of all the rays from the sun in exactly the same way, your theory would be correct. But a diffuse surface changes the direction of the rays randomly.

The inverse square law only applies to point sources. For a soft source you have to treat every point on the surface as a source, and add up all the contributions from all the points on the surface as seen from the position of the subject. Either that, or just meter it and go shoot a movie. ;-)

In any case, your DP was right. Moving the soft reflector in closer makes the fill light both brighter and softer.

-- J.S.

Dominic Case · November 10, 2004

Moving the soft reflector in closer makes the fill light both brighter and softer.

Yes.

You can often answer these questions intuitively by going to the extreme. For example, moving the reflector a great deal further away would quite obviously reduce its effect (for instance, 50 meters away!)

So distance IS a factor. But it's not quite the same as the normal inverse square law for point sources. In this case we are talking about a large field of illumination. Your reflector behaves the same way as a bank of lights forming a softlight. What is important is how large an area it appears to the subject. Move it closer, or make it larger, either way you get more light on your subject.