Aperature

gdw · January 27, 2009

Pete, being an very old oldie myself I will address you question as diplomatically as I can. I have never heard the term maths, however, I can see the appropriateness since most "math" requires more than on step/function/formula etc. to arrive at anything useful. The plural may be more grammatically correct although to me it sounds a little strange.

peter_howard2 · January 27, 2009

Thanks for your reply Gary, it occurs to me there may also be a difference depending on which side of the "pond" you are situated in. Me . . . I'm a good ol' Essex boy in U.K. Thanks for your input, appreciate your time. Pete

William Michael · January 27, 2009

Aside comments:

"it occurs to me there may also be a difference depending on which side of the "pond" you are situated in."

Correct. That is the differentiating factor.

"Maths" here too: noun, a contraction of the word “Math(ematic)s”: it is the English heritage.

We (should) say "do the maths", they say "do the math" (both can be singular number)

We (should) say "I will sit FOR the maths exam", they say "I will sit the math exam". Both singular number.

Note also, we were taught to use the correct preposition, according the phase in which it is contained, specifically to accord with the predicate, in that phrase: . . . . “sit FOR the EXAM” . . . “sit ON the COMMITTEE” . . . “sit AS his ASSISTANT” . . . "sit IN place of the CAPTAIN" or "sit IN for the CAPTAIN"

We (should) say "he won a medal in his event" they say "he medalled in his event" . . . Noun to Verb conversion is quite dynamic in the language's movement, at the moment. So it is possible we might have:

"I will math those aperture equations."

Language is just one of my pet pedantic pastimes - as well as meandering alliterations.

[The words “we” and “they” merely separate Pure English and the American changes to it.]

WW

jenkins · January 27, 2009

I know what you meant William :)

That comment was tongue in cheek, when Photography becomes maths and Physics, i'm out of here.. :(

William Michael · January 27, 2009

"I know what you meant William :)"

Hi Simon,

Yes. That's good. When I wrote it, I had just returned from "inside the pub with the window light and the really nice looking woman . . . " :)

So in the cold light of morning, I was a little concerned I sounded terse, rather than jovial.

BTW I posted an idea about your Bridge / Sky issue - I really do not yet understand all the technicalities of CS3, but I am still studying. I do revert to the old fashioned darkroom and play with the filters in my enlarger and then just apply that theory to Photoshop . . . have a look . . . ?

Cheers,

WW

Alan Marcus · January 28, 2009

The f/numbers explained:

The camera lens gets its name from the lentil seed it resembles. The lens projects an image of the outside world, focusing at the focal plane which is the position occupied by the surface of film or electronic chip. This image is allowed to play briefly on these surfaces by the shutters which acts as a gate. A lens is much like a funnel in that it gathers light. The larger its diameter the more light it gathers. Thus the diameter of the lens is a key controlling factor regarding how bright the projected image will be. In addition to diameter, image brightness is a product of scene brightness and lens focal length.

Needed is a precision way to control image brightness. The optical solution is adjustable control over the working diameter of the lens. This takes the form of a washer shaped restriction or stop called an aperture. This mechanism is technically known as the Iris, so named after the Greek god of the rainbow (colored portion of the human eye). Sometimes its called a stop, so named after a set of thin metal slides, with different size holes, invented by John Waterhouse in 1858, they were inserted into the lens barrel, each stopped a different amount of light allowing repeatable adjustment of image brightness.

Photo scientists concluded the logical sequence of adjustment should be in 2 x increments i.e. twofold brightness change stop-to-stop. To accomplish, the diameter of the aperture hole can be enlarged or reduced by a calculated amount thus causing a shift in the area (square measure) of the hole in twofold increments. Scientists recognized that when dealing with a circle you must vary the diameter multiplying it by the square root of 2 which is 1.4142. This math calculates a revised diameter that generates a circle with a twofold area increase. Conversely if you multiply the diameter of a circle by 0.7071, you calculated a revised circle with a twofold area decrease.

Using 1.4142 (rounded to 1.4) a number set emerged:

This is set is called the full stop set or the f/stops:

1 – 1.4 – 2 – 2.8 – 4 – 5.6 - 8 – 11 – 16 – 22 – 32 – 64

Note each number is its neighbor on its left multiplied by 1.4

Note each number is its neighbor on the right multiplied by 0.7

As time passed a twofold adjustment increment proved to be too course. A revised set was calculated that generate changes half again as fine. The number set that satisfies this requirement is based on the fourth root of 2 which is 1.1892

Thus the finer number set is in ½ stop increments:

1 – 1.2 – 1.4 – 1.7 – 2 – 2.4 – 2.8 – 3.5 – 4 – 4.5 – 5.6 – 6.3 – 8 – 9.8 – 11 – 13.5 – 16 – 19 – 22 – 26.9 - 32

As more time passed and light meters came into common usage, the f/number set was again made finer incrementing in 1/3 f/stop progression. This 1/3 f/stop set is based on the sixth root of 2 which is 1.1225

This set completely satisfies photo scientists as shutters accuracy, mechanical apertures gear backlash and ISO values are marginally able to hold this tolerance.

The 1/3 f/number set is:

1 – 1.1 – 1.2 – 1.4 – 1.6 – 1.8 – 2 – 2.2 – 2.5 – 2.8 – 3.2 – 3.5 – 4 – 5 – 5.6 – 6.3 – 7 – 8 – 10 – 11 – 12.6 – 14 – 16 – 18 – 20 – 22 – 25 – 40 – 45 – 50 – 57 – 64

As to image brightness at the film/chip plane (focal plane):

Within the camera, the two most significant factors are the working lens diameter and the lens focal length. Image brightness decreases as focal length increases. Conversely as focal length decreased image brightness increases. Thus both diameter and focal length must be taken into account.

With thousands or perhaps millions of camera designs that bring into play a hodgepodge of lens dimensional combinations, each will presents a different brightness at the focal plane. You guessed it – disorder results. Without a solution, most pictures you take will would be under or over exposed.

Ratio to the rescue: The ratio of two numbers is pure because it is devoid of dimension. In the case of the camera lens we divide the focal length by the working diameter to get a ratio which we call the focal ratio or in other words the f/number. The beauty of the f/number system is, when any lens is set to a specific f/number it delivers the same image brightness at the image plane as any other lens positioned to the same f/number regardless of differences in dimension.. Simply stated, the focal ratio or f/number gets rid of the chaos revolving around setting the lens aperture. Other systems have been proposed and tried but we still elect to keep the one that has works best. (You’re free to invent a better system).

Sorry to report that we photographers must cherish the lowly focal ratio (f/number) system until you think up something better.

peter_howard2 · January 28, 2009

William, Thank you for the clarification re "math" & "maths" - yep it's just like I thought, the old "you say tomaaaatoe" we say "tomarrrrto" sydrome - whatever. . . as long as we are all talking about the same fruit . . .

Simon, I'm with you here, not because I'm necessarily a "rebel", but simply because I ain't clever enough to understand all that "math"!!!! So I'll just meander about in ignorant bliss punching in the numbers and being amazed at the results.

Alan, I'm in total awe ! ! ! you certainly know the "biz".

James G. Dainis · January 28, 2009

Alan,

Thank you for providing the simple method - simply multiply by 1.1225 to get the next 1/3rd stop. It makes sense since the cube root of a square root is the sixth root

(x1/2 )1/3 = x1/6

It is a bit difficult trying to answer a question that requires higher mathematics and make it understandable for the majority of people who consider "higher mathematics" to be doing long division by hand.

William Michael · January 28, 2009

"Alan, I'm in total awe ! ! ! you certainly know the "biz"."

Alan also wrote a really interesting piece about Colour Temperature, Lord Kelvin and Cannon Balls - it was in response to a question about a 20D and setting Manual Colour Temperature . . .

***

"the majority of people who consider "higher mathematics" to be doing long division by hand."

Sadly, Manual Long Division is no longer part of the Mathematics Curriculum here - we were taught it in Primary School - (about age 8) . . .

and so I ask, what do we do if the batteries run out in the calculator?

IMO kinda why I like having my 303b handy - it is basically clockwork, the battery just supplies the light meter power, and I can reasonable guess the exposure and I know how to develop the negs in a jam jar . . . geez I sound like an old F4%T . . .

But I am attempting to make a logical point; I think if we understand the fundamental "why" and the "how" of all the automatic functions of the modern digital cameras, our photography just might be better and also more rewarding.

It is not about doing the maths whilst taking the photos - it is about understanding the processes, better.

WW

Alan Marcus · January 28, 2009

I know the vast majority of photographers could care less about the ins & outs of f/numbers. However, I write to address the minority, those that might be the future scientists and engineers designing the equipment we love to use.

Again and again the number two (2) or one of its derivatives is intertwined with our science. So as a teaching aid I fancied this little story.

The job of the lens is to project an image of the outside world onto the film or digital chip. Thus the camera system acts like a slide projector backwards, the film/chip being the screen. Now how bright the image on the screen will be is a function of several factors. For this discussion we are only interested in the lens’s working diameter which is defined as the lens’s aperture. We need the ability to change the working diameter as this is an important tool to make the screen image brighter or dimmer. Years ago it was determined that the best way to do this was to use an increment that either doubles or halves the screen brightness.

Now changing the subject (maybe). You are the captain of Cavalry “A” Troop. One hundred men with horses marching through the American South Western Desert on patrol. Water is a problem but you expect rain. You order the troops to bivouac for the night. You order the men to dig a circular pit 8 feet in diameter and line it with their canvas tent fabric. It rains as expected and the pit begins to collect rainwater. Due to your West Point training, you know an 8 foot diameter pit is adequate to collect rain water for your needs. Unexpectedly a lookout spots “B” Troop approaching -- another 100 men with horses. You order your men to expand the diameter of the circular pit to accumulate 200 men and horses.

How big must the revised pit be to double the amount of collected rain water?

Answer: You multiply the pit diameter (8 feet) by 1.4142. This value is the square root of 2. The answer is 11.3 (rounded it’s 11 feet). You order the pit expanded to 11 feet diameter. Surprise, this new value causes the pit to accumulate twice as much water as before. Why? The surface area (catch basin) now has double the surface area; thus it can capture twice the amount of rain.

The lens opening or aperture is also a circular geometric figure. The area of any circle (thus its ability to collect rain or light) is doubled if you multiply its diameter by 1.4 (1.4142 rounded). Using this factor a number set emerges:

1 – 1.4 – 2 – 2.8 – 4 - 5.6 – 8 – 11 – 16 – 22 – 32 – 45 – 64

Note each number to the right is its neighbor on the left multiplied by 1.4 and then rounded. Each number to the left is its neighbor on the right divided by 1.4 and then rounded.

These are the mysterious values engraved on the lens barrel. With geometric precision they allow the adjustment of the working diameter of a lens, making it smaller or larger. We need this number set because it allows even and logical and predictable changes to be made in image brightness in an increment that either doubles or halves the amount of light allowed to play on the film or digital chip.

William Michael · January 28, 2009

. . . your stories certainly have a military bent . . . Cannon Balls to describe Colour Temperature and brining in the Cavalry to rescue the Aperture . . .

what a nice read. :)

James G. Dainis · January 29, 2009

If you get one inch of rainfall you would gather about 1-1/4 quarts per man. But what about the horses, eh? Are the poor beasties to stand with their tongues lolling about watching the soldiers swilling down the aqua pura*? Take pity on the poor dumb brutes (the horses not the troops) and make the hole big enough to gather at least five gallons per horse. Or should one expect at least 4 inches of rainfall?

*water

One shouldn't dismiss math as unnecessary in general photography. With a 400mm lens how far should you plan to be in order to take a frame filling shot of a 6 foot tall athlete, assuming that worthy is leaning on his bat or niblick at an angle so as to fill the diagonal of the vertical frame?

Empirically, the angle of view is 2400/focal length. 2400/400 = 6 degree angle of view.

6 feet/tan6 = 57 feet. Now you have a general idea of how far to position yourself from the pitch or green.

William Michael · January 29, 2009

Nah! THAT it is a trick question . . . James :) I will answer (all guesses): . . . um about 70ft (for my 20D) . . . or . . . um 45ft for my 645 . . . but that is a trick answer because I don't have a 400mm for the 645, only a 500mm but if I did have a 400mm for my 645 I would guess about 35ft.

At those distances I reckon I could to nail the 6ft Batsman leaning on his bat (Cricket) - with enough scope to catch him in a Full Blooded Drive for each of the different camera formats I noted.

Now I gotta go and check the maths - they were all guesses, though I could have cheated a bit as I did take 60ft on a 135 format camera as a reference point. And I know from experience that the EF400F2.8L on a 20D is about half an olympic pool (25m / 80ft) to frame a person, standing.

WW

allardk · January 29, 2009

Pick one.

allardk · January 29, 2009

BTW James you take a bit of a detour back through the tangent, which is at the basis of the 'empirical' rule for angle of view. Doesn't work for wide angle either. There's a much simpler way. Assuming 35mm film format (ff digital), you want to make your subject about 50x smaller to fit exactly in the vertical, so about 40x to fit exactly in the diagonal.

Magnification is focal length divided by subject distance. That's the simple rule .

That means you need a distance of 40 times the focal length or 16 meters. No need to memorize tangents or approximating them in a small angle limit.

Personally, I'd just move backwards (watching where I go) until the subject fills the frame. Math in the field is limited to: I only see half of him, so I need to go twice as far.

gdw · January 29, 2009

Ding, Ding, Ding!!!!! and the winner is Allard K. At last, a common sense, workable answer from someone who most likely actually takes photographs. Allard, your are a light in a sea of darkness. Look in the viewfinder, walk forward, walk backward as needed now that is what I call a tangent!!!!

James G. Dainis · January 29, 2009

I shan't elect to go stomping about a muddy field using trial and error when five simple key strokes would give me the answer I need. Pure math, you see. One would not tell Euclid (and I make no comparison between myself and that ancient) "Trying to square the circle, old boy? Simply wrap a string around a cylinder, lay it out and measure it with a ruler." Gad, what cheek! Pure geometers would use straight edge and compass alone. None of this filthy, "Lay it on the ground and measure it with a stick" business.

Okay, enough joking. We have digressed a bit from the original question concerning the math involved for 1/2, 1/3rd etc. apertures. My concern is the difference between having to know something and wanting to know something. Certainly one doesn't have to know the math involved when going up 1/3rd stop but if one wants to know it, one shouldn't be discouraged.

As a further digression, I seem to have picked up a bit of an British accent. Lest I come off as a cheeky colonial, I must explain. I have been immersed in reading the recently discovered memoirs of Sir Harry Flashman, V.C. K.C.B. as edited by George MacDonald Fraser. Those of you familiar with Tom Brown's School Days will recall Flashman as a cowardly bully of low moral character. He hadn't changed as an adult but that never stopped him from becoming one of the most honored and decorated heroes of the Empire. While battles raged around him he would hide under a gun carriage ("The only sensible thing to do") and when the blood letting was over, he would emerge, pick up a bloody saber and pose heroically amidst the carnage while disclaiming "By gad, we gave them what for that time, eh lads?"

deborah_ormsby · February 1, 2009

very informative forum, enjoyed the sparring. I am leaving with more knowledge then I came with and that is always good. Thank you for taking the time to answer our questions!

rnemtpski · February 2, 2009

I read this whole thing and laughed the entire time. This was informative for me, however, I really got a good laugh also. Thanks to all. Linda

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