Not Your Everyday Banana

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I read on the Lens Buying Guide @ neocamera:

Focal-length determines the angle-of-view seen through a lens for a given sensor-size. With a full-frame sensor a lens gives the same angle-of-view as it would on a 35mm film camera. With a smaller sensor, the angle-of-view becomes smaller. The crop-factor, also called FLM, is the ratio representing the difference in equivalent focal-lengths. So a 150mm on a full-frame DSLR such as the Nikon D700 gives the same angle-of-view as a 100mm on a D7000 since its FLM is 1.5X.

Short focal-lengths show a greater angle-of-view compared to longer ones. [...]

In layman's terms, what is angle-of-view? Is it the same as focal length? If not, how is it different? How is it used? Why do I need to know about it?

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In layman's terms (assuming a layman who knows some very basic geometry), imagine your nose as the point of a triangle. The left side of the triangle is the left edge of your peripheral vision, and the right side is the right edge. The horizontal angle of view is simply the angle between those edges, and the vertical angle of view is the same thing for up and down.

For a human eye, the angle of view happens to be about 95°, but since your eyes move around unconsciously and your brain fills in the details, it feels much wider than that.

The terms field of view and angle of view are basically interchangeable — angle of view is one way of measuring the field of view. (One could also say something like "10 meters at 20 meters away"... this describes different aspects of the same geometry, and with basic trig one can figure out one thing from the other.)

As the text you've quoted says, "Focal-length determines the angle-of-view seen through a lens for a given sensor-size." This is also basic trig, and you can actually plot it out on a piece of paper and measure for yourself. Obviously with a lens this is a three-dimensional problem, but we can just consider the horizontal dimension and reduce it to two. (Imagine this as a top-down cut-away view of the world.)

Draw a line 23.6mm long — the width of the sensor in your D7000 (and many similar cameras) — in the center at the bottom of a blank piece of paper.

You can just look at the images I've made below, but if you're a hands-on learner like I am, it's really useful to actually get out some real paper, colored pencils, and a ruler, and follow along in the physical world.

From the center of that line, draw a light perpendicular line out from that center dot out towards the middle of the page, so you have an inverted T shape. (This is for convenience. Think of it as "the line towards what you're pointing the camera at".)

Step 1 + Step 2: aps-c sensor with perpendicular guide line

Measure from your sensor along the center line you just drew. Put a dot at 35mm. Label this "35mm lens". This represents the pinhole aperture of an idealized 35mm lens.

Now measure from your sensor along the center line. Put a dot at 50mm. Label this "50mm lens". (And of course this represents the pinhole aperture of an idealized 50mm lens.)

Step 3 + Step 4: dots for lenses

With your straightedge, draw a line from the left edge of your sensor line through the 35mm aperture dot, and continue all the way through and on to the edge of the page. Then do the same thing from the right edge of the sensor line. This should produce a big X shape. Label both lines of the top cone of the X "35mm field of view".

Step 5: APS-C angle of view with a 35mm lens

Do the same thing with the 50mm lens dot. Label this, of course, "50mm field of view".

Step 6: APS-C angle of view with a 50mm lens

Now, you can directly see that a shorter focal length produces a wider field of view. Anything that's within those lines will be in your picture, and everything outside will be out of frame. Note that the lens may project a much wider cone of light that doesn't all fall on the sensor — the lines you drew ignore that, since light that isn't recorded doesn't really matter.

If you measure the angle, you should see that it's about 36.5° for the 35mm lens, and about 26° for the 50mm lens.

Then, two further experiments:

  • Choose some different focal lengths (15mm, 200mm) and see what those give you.

Step 8: a 15mm lens

Increase the size of the sensor line to 36mm, as in Nikon's "FX" full-frame cameras. Keep the line centered on the same dot, of course. Use your same lens dots but draw new X lines to the larger left and right edge of the sensor. It's immediately apparent that including this extra part of the light cone makes the recorded field of view of the same focal length much wider.

Notice that the 35mm on your D7000 gives roughly the field of view of the 50mm on FX — this is why people talk about "equivalent" lenses.

Step 10: Full-frame

You can see that the lines for APS-C 35mm and "Full-Frame" 50mm aren't right on top of each other, as one might expect for an "equivalent". That's because this breaks down a bit at macro distance. If you move back a few millimeters, it'll line up correctly (but change perspective ever so slightly). The lines are roughly parallel, though, so those few millimeters are still just a few millimeters across the room, where they're inconsequential. If you draw this on a really big piece of paper instead of this little on-screen one, that'll become clear. (And of course, they're not exactly parallel, because the lenses don't quite match the crop factor — 32.7777...mm and 50mm would be more exact. Ah, the real world, always getting in the way of explaining things simply. Other real-world factors apply as well; for example, focal length changes with focus distance, and also the focal length written on a lens is often rounded to a nice-looking number.)

This neatly (I hope) answers the question of the relationship between focal length and angle of view / field of view, and also explains the effect of different sensor sizes — and, as a bonus, shows how cropping is interchangeable with zooming (if you don't mind using less of your sensor).

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perfect. everything is crystal clear now! what a great explanation, a thousand thanks. this answers much of the ambiguity i face now with selecting my first lens purchase. im adding a new question about this now and your answer will make that question much more focused. the only other factor is getting the aperture nailed down as i want to look at a lens to help with indoor (low light) people shooting using hand held stabilization. –  kacalapy Dec 21 '10 at 19:01
    
great explanation. thanks!! –  Critical Skill Feb 3 '12 at 10:22
    
I went through couple of articles to get an idea about this..but was not clear enough....Thanks for this wonderful explanation and putting your time in it –  Umesh Awasthi Apr 25 at 2:41
    
Very neat, thank you for the details and your time! –  Andy M Jul 5 at 15:46
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The angle of view is a measurement of the view plane area of an optical system in arc degree.

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I wouldn't describe that as "in layman's terms". Also with my physicist hat on, the units of measurement are irrelevant. –  Philip Kendall Jul 17 at 19:09
    
I disagree with you to the fullest. It is very important to give the units when talking about the angle of view of a specific lens. The angle of view defines whether a lens is a fisheye, wide angle, standard, or telephoto lens. –  Julian Jul 17 at 19:38
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This is getting overly pedantic, but you're missing the point I'm trying to make. The definition of angle of view has nothing to do with degrees - it's equally valid in radians, gradians or any other unit of angle you want to use. When you do a specific measurement, of course you have to state what units you're using - but that's part of a specific measurement, not part of the definition of the concept. –  Philip Kendall Jul 17 at 20:54
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The last thing asked in the query was why should we care about this. Let me answer that with an example. I've been trying to decide if getting a 20mm lens for my 1.5 crop factor camera would make sense in light of the fact I've already got a 24mm. Here are the numbers (from an on line calculator). On the Dx body the 24 gives you 53.1 horizontal, 36.9 vertical, and 63 degrees diagonal. The 20 provides 61.9, 43.6, and 71.6, respectively. But when you look at what's in the frame of a picture those numbers add up. At 10 feet from the sensor, the 24 encompasses an area of 10 feet by 6.7 feet or 67 square feet. The 20mm frames 96 square feet (12x8) at that same distance. At 20 feet the difference in what's within the frame is 118 square feet (266 v 384). The 20mm lens thus encompasses 44.4 % more ground within its frame at 20 feet than the 24mm.

Still, if you can back up 2 feet from the 10 foot subject, increasing the subject to sensor distance to 12 feet, the exact same field of view is available with the 24mm as at 10 feet with the 20mm. At 20 feet, you'd have to back up 4 feet. So, in the situations you shoot, are those one or two steps backwards something you can take (keeping in mind that changing the distance to your subject changes perspective as well)?

The end result for me is I'm still undecided. But at least I quantified my dilemma. That's why angle and field of view matter. (Of course, a wide angle zoom would arguably solve my dilemma; but to keep equivalent speed, the price of poker goes up by more than $1,400.00 and 1.5 lbs, and avoiding THAT, is why I went back to primes in the first place.)

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To be precise, I think the problem is stereometric, and not trigonometric, but you are essentially right. However, I personally never bothered to point out this difference b/c we use to think "small angles" so problems are easily linearized. –  ysap May 26 '11 at 21:30
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One correction, though, it is not correct to say that "the exact same field of view is available". The FoV does not change with the subject distance. You can say that "the exact same framing of your subject is available". –  ysap May 26 '11 at 21:32
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The angle of view is related to the focal length, and the size of the sensor in use.

So, for a 50mm lens on a 35mm sized sensor (or film), you'd have a field of view of 46° on the diagonal. As the focal length doubles, the field of view halves, so a 100mm lens on a 35mm camera has a field of view of 24° on the diagonal.

If you use a smaller sensor, you're effectively cropping the image down, so whilst the lens may produce an image good enough for a 35mm frame, a smaller sensor will ignore the parts that fall off the edge. This crop factor divides the angle of view, so that 50mm lens would have a field of view of about 31° on the diagonal if you were using an APS-C sized sensor. Alternately, and how most people think about it is the equivalent focal length, had you been using 35mm, which would be case of multiplying the real focal length by your crop factor, so that 50mm lens on a D7000 is equivalent to a 75mm lens on a 35mm camera.

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can you translate this into layman's terms please? –  kacalapy Dec 21 '10 at 16:19
    
The math in this answer is wrong, as it relates to angle of view in degrees. You can't multiply it by the crop factor (or ratio of two focal lengths) in this way. It happens to work approximately as a rule of thumb (within a few degrees) at longer focal lengths (50mm and up, say), but totally breaks down at wide angle. –  mattdm Aug 19 '11 at 15:44
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Focal length is the property of lens.

Angle of view is essentially what its name says - the subset of space you can see, and it depends BOTH on focal length and frame size used. http://en.wikipedia.org/wiki/Angle_of_view

Angle of view is what actually matters in practice, lens focal length is just more convenient to use as an equivalent value when frame size is fixed and well-known (like the standard 35mm frame). As a logical consequence, these days sensor size is usually mentioned together with lens focal length in order to get understanding of the angle of view used.

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+1 Nice and simple. –  whuber Dec 21 '10 at 16:18
    
can you translate this into layman's terms please? or better yet provide an real world example –  kacalapy Dec 21 '10 at 16:19
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