Posts Tagged ‘theory’

Photography formulas – Macro

Thursday, February 10th, 2011 by

Recently I was doing some research regarding calculations for close-focus photography. Since I didn’t come across a place that collected all the relevant formulas (aside from a few forum posts), I thought I’d compile them a bit.

F: Focal length
X: Extension
D: Focused distance (measured from the lens nodal point)
P: Focused distance (measured from the film plane)
M: Magnification
B: Bellows factor
O: Object size
I: Image size (size of O on film)

Focal length, extension and distance

First, the relationship between focal length, extension and distance. Extension is how far the nodal point of the lens is from the film plane. Note that this is at least the focal length of the lens1. An extension of F gives you infinity focus, more extension gets you closer.

\frac{1}{F} = \frac{1}{X} + \frac{1}{D}


Magnification is the relationship between the size of the subject, and the size it will be on film. With a magnification of 1, a 10mm subject will be 10mm on the film. Note that for macro lenses, magnification is often given in the form 1:(1/M), so a magnification of 0.5 would be 1:2, and life-size would be 1:1. magnifications bigger than 1 will have the bigger number in front, so a magnification of 2 would be 2:1.

M = \frac{X}{F} - 1 = \frac{F}{D-F} = \frac{I}{O}

Note that you can invert this to find how small an item you can photograph with a given magnification. Using a 1:2 lens on 35mm film will allow you to photograph a scene of 72mm wide and 48mm high, for example.

Bellows factor

Bellows factor determines how much extra exposure is needed when photographing close to the camera2. Multiply the `normal’ exposure of the scene by the bellows factor to get the correct exposure.

B = (\frac{X}{F})^2 = (\frac{I}{O} + 1)^2 = (M + 1)^2

Note that this only applies if you are using a camera without TTL metering. A TTL meter will automatically correct for this.

Some examples

Say I have a 135mm lens, that normally focusses down to 0.9 meters from the lens. The maximum magnification is then:

M = \frac{F}{D-F} = \frac{135}{900-135} = \frac{135}{765} = 0.18

I can calculate the lens extension at closest focus:

\frac{1}{X} = \frac{1}{F} - \frac{1}{D} = \frac{1}{135} - \frac{1}{900} = \frac{1}{159} \Rightarrow X = 159

If I add a 25mm extension ring between the lens and the camera, I can calculate the new magnification at infinity:

M = \frac{X}{F} - 1 = \frac{135 + 25}{135} - 1 = 0.19

And the new magnification at closest focus:

M = \frac{X}{F} - 1 = \frac{159 + 25}{135} - 1 = 0.36

So with the extension ring, the lens becomes capable of roughly 1:3 macro.

If I were using a camera without TTL metering, I’d have to calculate the new exposure:

B = (M + 1)^2 = 1.36^2 = 1.85

So I’d have to add about two-thirds of a stop to the exposure my meter sees in order to get a correctly exposed picture.

Without the macro ring, at closest focus the bellows factor would be:

B = (\frac{X}{F})^2 = (\frac{159}{135})^2 = 1.39

Which would be roughly one-third of a stop.

You can see that bellows factor doesn’t really become an issue until you get at 1:2 magnification or greater, from which point is increases very quickly.

It’s worth to note that the distance scales on most lenses indicate distance from the film plane, not from the focal point of the lens. This makes the calculations a lot more complex, as

P = D + X

which converts the first formula above to:

\frac{1}{F} = \frac{1}{X} + \frac{1}{P - X}

Imagine the lens mentioned above doen’t focus 0.9 meters from the lens, but from the film plane. It’s now not as easy to determine X. Rewriting the formula above, we get a quadratic formula:

-X^2 + PX - FP = 0

which our math teacher taught us means that:

X = \frac{-P \pm \sqrt{b^2 - 4 \cdot -1  \cdot -FP}}{-2}

And thus:

X = 165

from there on, we can use the formulas above to determine magnification, bellows factor and everything else as normal.

Note that the quadratic formula also gives us

X = 735

which is another extension that would give a subject to film-plane distance of 0.9 meters, though that would take us very solidly into macro territory, with a magnification of:

M = \frac{X}{F} - 1 = \frac{735}{135} - 1 = 4.44

In other words: Well over four times life-size, and the exact reciprocal of the magnification at 165mm.
This is logical, since

735 = 900 - 165

This is simply the entire optical system reversed, with the subject and film switching places.

  1. Telephoto and retrofocus lens designs can be shorter or longer than their own focal length, but they `cast’ a nodal point outside (or behind) themselves, as it were []
  2. Due to the inverse square law, the further you rack out your lens, the less light hits the film/sensor []

Filter factors

Wednesday, September 9th, 2009 by

Sorry for the recent draught in updates. I haven’t been home a lot, and as such haven’t had time to scan any pictures. I’ve shot plenty though.

To compensate for the lack of imagery, a bit of text today.

I’m by now not half-bad at guessing exposure with the M3, and when I’m wrong, I’m usually at least close enough that the exposure latitude of the negative can give me an acceptable picture. One thing is really throwing me though: Filters.

I’ve been experimenting a bit with differently coloured filters, and this has led to a lot of overexposure. This is mostly caused by trusting the filter factor of the filter.

Example: I’m trying to accentuate something in a picture, say, a red flower, so I take out my red filter1. The flower is in direct sunlight, so I set 1/500th of a second at f16 to start with. The filter has a 5x filter factor, so I lower my shutter speed to say, 1/125th (or open up to f8).
Result: well exposed background, over-exposed flower.

The filter doesn’t do much for its own colour, yet blocks its complementary colours. So the filter factor is in fact mostly applicable for colours other than the filter colour.

The right way to do it would be to maybe open up one stop from the base exposure, to highlight the primary subject. For lower filter factors, don’t open up at all.

Of course, if your primary subject isn’t the filter colour, all bets are off (and I still have to experiment with that).

But the short version is: If you’re using a filter to boost contrast by making colours darker, don’t negate that by making everything brighter.

Dante Stella has an excelent article on filters that covers this as well.

Thus ends todays lesson.

  1. B+W 090 []
  2. If I know why this is, why didn’t I do it right the first time? Because I figured out the why from looking at my mistakes, silly. That’s why I’m experimenting. []


Tuesday, June 30th, 2009 by

One of the reasons I choose to do this project/training/exercise, is that it combines nicely with another exercise set out by Mike Johnston (years) earlier: GSOTPANWASTOTZSS1. Being able to guess exposure is a skill that’s always allured to me.

This is one of those skills that I think will be incredibly useful, even if you’re using a top-of-the-line, auto-everything DSLR body. Knowing when to apply exposure compensation, selecting you base exposure when shooting TTL flash, there are plenty of aspects that will yield better results if the photographer knows what he’s doing.

It’s also an incredibly cool thing to be able to do2.

Unfortunately, this will make the exercise just a bit harder, as in addition to using a rangefinder for the first time and shooting dedicated B&W for the first time, I’ll have only the sunny sixteen rule and my knowledge of fluorescent-lit swimming pools to guide me in getting a proper exposure.

I guess Yoda was right: “You must unlearn what you have learned”.

Another rule should thus be added to the earlier post: No exposure meters of any kind.

  1. I’m still working on memorizing that one. []
  2. To me that is. Some people may prefer owning a larger telephoto. []