Field of View / Focal Lengths Table

[carllooper]

I computer generated this lookup table of horizontal field of view angles, and the required focal length of the lens, for various film formats, to obtain that field of view.

http://members.iinet.net.au/~carllooper ... fView.html

The dimensions of the film formats I used are from here: http://www.gcmstudio.com/filmspecs/filmspecs.html

The formula I used is:

F = w / 2 tan( a / 2)

F: Focal Length of Lens (mm)
w: width of film frame (mm)
a: horizontal field of view (degrees)

Diagrams:


Result of changing film format for a given focal length



Result of changing focal length for a given film format

[cameratech]

There are a number of field of view calculators and comparators online, for example:
http://www.abelcine.com/fov/
http://www.panavision.co.uk/services/im ... ulator.asp
or cinematography phone apps which do all that and more:
http://www.theblackandblue.com/2010/03/ ... ve-on-set/

It's not an exact science, since a lot of lenses are only approximately the focal length stated on the barrel, and if a lens breathes during its focus travel the field of view will be different depending on the subject distance.

[carllooper]

ELIZABETH
Yes, Dr Dee. I am following you.
So what does it all mean?

DEE
It means the rise of a great
empire, majesty. And it means
convulsions, also. The fall of an
empire.

Elizabeth listens, a faraway look in her eyes. She knows
nothing of the planets, but she feels it deep within
herself: her moment of destiny is near.





ELIZABETH
Which empire is to rise, and
which is to fall?



DEE
That I can't say. Astrology is,
as yet, more an art than a
science.


Printing focal lengths on a lens provides the most practical and accurate way lens markings can play their role in determining a field of view angle. To get more accuracy would require more involved methods than that which the focal length markings and film width provide, and would require the peculiarity of each lens be taken into account. In the absence of such, focal length numbers provide a generic common frame of reference across all lenses. And the focal length number has no other real purpose than providing for determination of field of view (ie. for any film/sensor formats). In other words there isn't much else you do with such a number.

The various web calculators available (and iphone apps) will use exactly the same equation as that provided here. But with the equation, one can use a $5 calculator instead. And indeed, without calculating anything at all, the equation (and diagrams which otherwise illustrate the equation) provide an exact definition of what the terms 'focal length' and 'field of view' mean.

In other words, to the extent that focal length numbers (and film format widths) can be accurately given in the first place, the resulting field of view angle, by means of the equation, can't actually be any more accurate.

None of this need be necessary of course. Looking through the viewfinder, or rather, shooting film and looking at the results, will give you completely accurate field of view results, and in a sensory form, rather than a numerical/theoretical form.

Carl

[cameratech]

..the focal length number has no other real purpose than providing for determination of field of view (ie. for any film/sensor formats). In other words there isn't much else you do with such a number.

On the contrary, there are many reasons why you need to know the focal length of a lens, beyond determining the field of view. Aside from actually describing a fundamental optical characteristic of the lens which is independent of other factors, you need to know the focal length to determine things like depth of field, depth of focus, hyperfocal point, nodal points, magnification ratios and other characteristics that might be helpful to know sometimes. You often need to know a few other variables in conjunction with the focal length, perhaps the format size, or the f-stop, or an appropriate circle of confusion figure.

..the equation (and diagrams which otherwise illustrate the equation) provide an exact definition of what the terms 'focal length' and 'field of view' mean.

An exact definition of focal length requires a bit more clarification than that. What does the centre of projection of a lens mean? How do you determine where that is? At what distance is the subject that is in focus at the film plane, or does it not matter?

From a design point, the field of view is actually the first criterion around which a design hinges. Lenses are always designed to cover a certain format, and optimised for that size image circle. A 25mm lens for full frame 35mm stills would be a wide angle, requiring different correction for aberrations, a different specification for fine detail acutance and a different type of optical construction than a 25mm made for standard 8mm, which would be be considered a medium long lens. You could of course use the larger format lens with an adapter on your 8mm camera, and get the same field of view as the one made for 8mm gave you. But far from using the "sweet spot" centre that people sometimes talk about, you would instead be getting unnecessary barrel distortion and other aberrations that were a compromise to achieve the far wider angle of view that the lens is capable of when used with the format it was designed for.

[carllooper]

..the focal length number has no other real purpose than providing for determination of field of view (ie. for any film/sensor formats). In other words there isn't much else you do with such a number.

On the contrary, there are many reasons why you need to know the focal length of a lens, beyond determining the field of view. Aside from actually describing a fundamental optical characteristic of the lens which is independent of other factors, you need to know the focal length to determine things like depth of field, depth of focus, hyperfocal point, nodal points, magnification ratios and other characteristics that might be helpful to know sometimes. You often need to know a few other variables in conjunction with the focal length, perhaps the format size, or the f-stop, or an appropriate circle of confusion figure.

..the equation (and diagrams which otherwise illustrate the equation) provide an exact definition of what the terms 'focal length' and 'field of view' mean.

An exact definition of focal length requires a bit more clarification than that. What does the centre of projection of a lens mean? How do you determine where that is? At what distance is the subject that is in focus at the film plane, or does it not matter?

From a design point, the field of view is actually the first criterion around which a design hinges. Lenses are always designed to cover a certain format, and optimised for that size image circle. A 25mm lens for full frame 35mm stills would be a wide angle, requiring different correction for aberrations, a different specification for fine detail acutance and a different type of optical construction than a 25mm made for standard 8mm, which would be be considered a medium long lens. You could of course use the larger format lens with an adapter on your 8mm camera, and get the same field of view as the one made for 8mm gave you. But far from using the "sweet spot" centre that people sometimes talk about, you would instead be getting unnecessary barrel distortion and other aberrations that were a compromise to achieve the far wider angle of view that the lens is capable of when used with the format it was designed for.

Yes, the definition of focal length given assumes, of course, that 'centre of projection' is understood. The concept of centre of projection can be completely understood in terms of a pin hole camera. The centre of projection is simply the location of the pin hole. Focal length is then the distance between the pin hole and the projection plane (ie. film plane).

To the extent that a lens can be conceived in terms of a pin hole is the extent to which centre of projection and the corresponding focal length can be conceived. A pin hole camera model provides a definition of focal length but it doesn't have anything to say about how you would specify a real lens in terms of such a model. However its important to note that the definition doesn't depend on real lenses. It is the description of real lenses, in terms of the model, that depends on the model. The model itself is exact. What is not so exact is how you describe a real lens with respect to such a model. To put it another way the "problem" with a real lens is that it is not a pin hole. To describe a real lens you need a more elaborate model. But in terms of what focal length means you don't.

The definition of centre of projection doesn't depend on whether you can measure it. The definition is what it is. It is so by definition, as they say. That said, how does one locate the centre of projection for a real lens? What you want to do is answer the question of where a pin hole camera would need to have it's pin hole located (with respect to the film plane) in order to obtain the same image at the film plane that a real lens would produce at the film plane. Once this correlation is obtained the distance between the pin hole and the film plane becomes the focal length you assign to the lens. To the extent this is possible is the extent to which a corresponding focal length number will have any use value. The act of doing this will generally reveal that it's not possible to get the image from a pin hole to exactly match an image from a lens. An important difference one will find is known as "lens distortion". Its not really distortion as such. A lens does what it does. It doesn't really distort anything. We could equally argue the pin hole camera model is the distortion. But we don't. By convention we say the lens is doing the distortion. The lens is "distorting" the image with respect to the image a pin hole camera produces. We can elaborate a more involved lens model in order to describe the difference between a real lens and the pin hole model. For example we can write a lens distortion model. And it too will be exact. Whether or not a real lens can be described in terms of such a model is, like focal length, a different sort of question.

That's the short answer.

A good book on optics (and machine vision) is the longer answer.

Carl

ps. yes there are many uses for focal length. I'm not sure what possessed me to suggest otherwise. Field of view is just one. Perhaps what I meant is that field of view is the only use value in terms of the model which establishes the meaning of focal length. To determine things like depth of field (etc) requires a more elaborate model. But yes, insofar as more elaborate models continue to involve focal length, focal length would then have more use value than just determining field of view.

pps. The equation provided doesn't determine what priorities are used in lens design. It simply defines the relationship between focal length, film format and field of view. The equation can be rearranged to make focal length a function of film format and field of view. As I did when computing the lookup table. Indeed it can be rearranged to make film format a function of focal length and field of view. None of this rearrangement has anything to do with the relative importance one might assign to any of these values during lens design. The equation describes the model which defines the relationship between the terms.

What I would stress is that field of view is completely unrelated to lens circles and their angle of view. Lens circles can vary all over the shop. Field of view doesn't. It is focal length and film format together that define field of view. Accordingly a lens without the film frame specified does not (and can not) have a field of view. This can be demonstrated by simply looking at the image a lens (or better: a pinhole) produces at the film plane. Without a film frame there to specify which part of the image will be cropped by the film frame, we can't describe what field of view a resulting film would exhibit.

[carllooper]

From a design point, the field of view is actually the first criterion around which a design hinges. Lenses are always designed to cover a certain format, and optimised for that size image circle. A 25mm lens for full frame 35mm stills would be a wide angle, requiring different correction for aberrations, a different specification for fine detail acutance and a different type of optical construction than a 25mm made for standard 8mm, which would be be considered a medium long lens. You could of course use the larger format lens with an adapter on your 8mm camera, and get the same field of view as the one made for 8mm gave you. But far from using the "sweet spot" centre that people sometimes talk about, you would instead be getting unnecessary barrel distortion and other aberrations that were a compromise to achieve the far wider angle of view that the lens is capable of when used with the format it was designed for.

As mentioned a lens doesn't have a field of view. What you must mean, of course, is the lens circle's angle of view - but as mentioned, this is completely unrelated to field of view.

Lenses are typically designed in relation to a particular film format. So the lens circle would play a correspondingly important role here. But the film format to which the lens is designed is just the maximum size format the lens will support. It doesn't stop use of the lens on anything smaller. As you say a 25mm lens designed for use on a 35mm camera doesn't mean you can't use it on an 8mm camera (by means of an adapter). The adapter doesn't alter the focal length. It just ensures the lens is the same distance from the film as it would be on a larger format camera - and therefore that the focal length remains the same. And because the focal length is the same it will give the same field of view as a 25mm designed specifically for an 8mm camera. Which can surprise a number of people. Which is why it is worth mentioning and re-iterating. What it does mean is that you probably can't use a lens designed for 35mm on a larger format camera, as the lens circle would probably not be large enough for what a larger format camera requires. But this is the important point. Lens circles (and their arbitrary angle of view) have nothing whatsoever to do with field of view. When we say all 25mm lenses have the same field of view on the same camera we do not mean they have the same lens circles. The diameter of the lens circles will vary but the field of view won't, ie. they are not the same thing.

Would a 25mm lens designed for a 35mm camera, when used on an 8mm camera, exhibit more distortion compared to using a 25mm lens designed for an 8mm camera? I don't think one can really answer such a question. There are lenses which exhibit more distortion and lenses that exhibit less, regardless of format. What one could say, is that a lens designed for 35mm would tend to exhibit less distortion if used on 8mm, than it did if used on 35mm, insofar as typical distortions increase as a function of distance from the centre of the image. The corners of the 8mm frame are closer to the centre of the image than the corners of the 35mm frame would be - so the 8mm frame should see less distortion, be it barrel distortion, pin cushion distortion, chromatic distortion (etc). I guess if you scale the resulting projected image up to the same size as a 35mm frame the distortion would also scale up. It is, of course, conceivable that a lens might be designable to have more distortion in the centre than it does at the perimeter! However such a lens would be the exception rather than the rule.

[carllooper]

I should say the focal length used to specify a lens differs slightly from the definition given.

The definitions will agree with each other but only when a lens is focused at infinity (and the corresponding image is in focus). One might call the focal length for a lens, given with focus at infinity (and corresponding image in focus), as a normalised focal length. For any other focus, the focal length of the pin hole model will disagree with this normalised focal length.

However this doesn't matter. The definition of "focal length" given by the pin hole camera model (ie. the centre of projection to film plane distance), while not the same as normalised focal length (other than at infinity), is the definition you actually require for determining an exact field of view.

In other words, if you had a lens which could show the exact centre of projection to film plane distance, for whatever focus you had the lens set, then plugging this distance into the model as the "focal length" (and assuming no lens distortion) will give you an exact field of view (for whatever is in focus).

But in the absence of such information (as is typical) you can just use the normalised focal length specified for the lens, instead. It will be "close enough". The resulting field of view value won't be as accurate but the important point here is that without more information it can't be any more accurate. The model is exact (yay), but the information you can typically plug into it, isn't (bummer).

Carl

ps. I imagine a more elaborate model could be written, using a definition of normalised focal length and focus as input variables, from which it is conceivable the centre of projection to film plane distance could then be computable, and from which (as per current model) the corresponding field of view becomes computable. However going this far, and short of testing a lens with appropriate tools (or writing out tables of measurements for every focus), the direction you would start to take, from a generic computation point of view, would be towards lens calibration algorithms, as used in machine vision libraries (such as OpenCV). Parameters such as focal length and lens distortion co-efficients can be reverse engineered with respect to test charts photographed through the lens in question.