There's a Depth of Field (DOF) calculator below, with some new features.
This technique is nothing new, it has always been well known to pros. Using f/1.8 on photo work they hope to sell seems not their best choice. Suggesting consideration of a good alternative is the point of this article. This calculator is an interactive comparison of two lens choices, allowing making decisions about how to blur the background.
This subject is of course about adjustable cameras. Determining sensor size (or focal length) is difficult for most compact or phone cameras. The sensor is so small that their lens is necessarily very short, which ensures a great depth of field with little choice. Setting a wide aperture may not be a choice, but choosing more dim lighting can help that. The best chance to blur the background some with a compact camera is to zoom in significantly, and then stand back as necessary (not closer than 5 or 6 feet for a portrait). Choose a background that is very much more distant, hopefully a few hundred feet.
Both focal length and subject distance are Depth of Field (DOF) factors. We can use them both for our goal. My notion of a portrait at f/1.8 is that there will of course be DOF problems, usually about the hardest possible way to make a good picture, and the last thing I want if I can prevent it. Notions may vary, but studio portraits likely work at f/8 or f/11 (the usual goal is so that the picture will sell well). We do like the sharpness of depth of field. Pros would likely see the advantage of maybe a 200 mm at f/4 for this purpose (of hiding the background in head and shoulders portrait).
This is the old adage about the "same DOF for same size image". In this situation with both lens with distances adjusted to be the same Field of View at the subject, if both are also at the same aperture, then they have the Same Depth of Field span too (at the subject they do, it is the same picture, but the background is a different picture). And the longer lens has advantage of being able to stop down a little more, winning with more DOF at the subject, and winning with still less DOF at the background too (if it is far enough back to be a different picture). Where we stand also affects the portrait perspective,. And the long lens, most of the wide background is zoomed out and gone missing, but what's left is even more blurred focus (assuming that is to be a plus here). This standing back at greater distance is little problem to do outdoors, and it possibly need not be that extreme. If both could use the same f/1.8 then, the depth of field at the subject is the Same, both 0.29 feet of DOF in this example (which is only 3.5 inches). DOF does not describe the sharpest necessarily, instead it defines the maximum blur that we can accept. But we don't have to use the same aperture, the 200 mm lens at 24 feet can use say f/4, which has 0.66 feet of DOF (8 inches). That's still not much, but it's sure a lot better, more than twice as much DOF. But again, it is even more blur at the background.
And the overwhelming advantage is even much better yet: We said this Field of View (FOV) at the subject would be the same in either situation (about 2.8x1.9 feet, about right for head and shoulders). But the background field at 40 feet of the 50 mm lens is over 21 feet wide. 21 feet of stuff you want blurred away. However, the field of view of the 200 mm lens is only 7.5 feet wide at 40 feet (behind the subject). So most of the objectionable junk you want to blur is simply missing, simply gone, removed in the best possible way. And better, you can surely simply move the camera slightly to one side to choose to align the best 7.5 feet of background decently enough, probably even if it were not blurred. But in fact, it is blurred at 200 mm. Probably blurred more, but 1) the subject DOF is so much better, and 2) there is much less of the background even showing. And 3) usually the background that is visible is blurred even more.
Maybe I'm a purist, but a "portrait lens" means a longer lens to force standing back for proper portrait perspective. Newbies may get strange notions, but "portrait lens" absolutely does NOT mean a f/1.8 lens. f/1.8 is more for low light levels, but today, improved high ISO does that well. f/1.8 might be about blurring backgrounds, but we're describing a better choice. A portrait studio (with goal hoping to sell the photo) will be using around f/8, maybe f/11, and will provide the proper light for it. A "portrait lens" for "head and shoulders" means 70 to 85 mm for 1.5x or 1.6x crop APS, or 105 to 135 mm for full size 35 mm frame. That longer length forces us to stand back for better perspective, to NOT enlarge noses, etc. The 50 mm lens standing back properly might do full length well, but is simply too short and close (not the best try) for tighter head and shoulders portraits. A cardinal rule of "Portrait" includes standing back for proper portrait perspective. Which is very important. We guys are often too dumb to see or realize it, but the wives (may not know why, but) will tell us they don't like their too-close portraits. Backing up a few steps and then zooming in is always a good plan.
OK, the pro might instead choose 100 mm f/2.8 at 12 feet, but it still offers all of the several advantages over 50 mm f/1.8. It should be obvious that this is a really big deal to know. For head and shoulders portraits, there are four strong properties often offered by standing back with the longer lens:
What's not to like? The only downside is we need to bring the longer lens, and have room to stand back.
There are many numerical combinations where the longer lens is simply better. And even with a close background, a few more where a property or two is still worth consideration. If you also find f/1.8 distasteful, there is this better way.
So the calculator below computes the FOV and sensor CoC at both subject and background distances, to make this point.
Identify your camera sensor size by entering either actual Sensor Size or Film Size, or Crop Factor or even a final CoC value. Any of those can calculate sensor size. Sensor size can be hard to know, but CoC (and standard divisor) also determines a sensor size, because CoC is about the standard enlargement of sensor size. You can see how to determine your Crop Factor. It's hard to beat precise actual sensor size specifications though.
Film or Sensor Size dropdown box: I have NOT researched every possible camera models sensor size. The film sizes are known good, but the "1/inches digital sensor size" system is a crude approximation, because actuals instead depend on the specific camera models chip. Especially the compact and phone sizes like 1/1.8" CCD are vague (actual sensor sizes are instead described as specifications of XxY mm). The approximated sensor size used is shown in results. If actual sensor size is not known, I suggest the Crop Factor option may be more accurate.
It will be appreciated if you would please report (Here) any problems with the calculator, or with any aspect of this or any page.
About the DOF calculator:
If you see results of NaN, it's an error meaning an input is Not A Number (periods are OK, but don't use commas).
DOF is Depth of Field, CoC is Circle of Confusion, and FOV is Field of View.
The next page has photo examples of these two initial default cases.
The feet/meters selection is which distance units you are using (the DOF and FOV results are these same units). When it is changed, the checked Convert checkbox will convert previous numbers to keep the same distances. Otherwise that feet/meters change will leave distance values numerically unchanged (but feet and meters are different distance values affecting DOF).
The background distance behind subject will normally be the same for both lenses, since that's where the subject is standing. A relatively long distance behind is good if the goal is for the background.
CoC is Circle of Confusion. It is used two ways.
Background CoC is the computed actual CoC at the BackGround distance. A larger multiplier means greater relative blur. Normal Depth of Field computes the distance limits where the blur becomes as large as the maximum acceptable CoC limit. Background CoC is shown in the calculator as "X times CoC", meaning actual CoC there is X times size of that maximum acceptable CoC limit entered. You could multiply it out, but this is a relative scale of bokeh and blurring there at the background distance, relative to the just-acceptable CoC at the limit of DOF.
The DOF concept implies that if the background were located exactly at the computed far limit of DOF, the blur diameter there would be exactly equal to CoC (1X CoC). Saying, in the default case B above, 200 mm f/4 at 24 feet, the DOF zone extends 0.34 feet behind. If we put the background 0.34 feet behind, the Background CoC necessarily computes exactly 1x CoC. Should the background be closer than the far DOF limit, then the multiplier will of course be less than 1 (and within the DOF range). A larger multiple is a multiplied greater blur. The Background Distance is input here as the distance Behind The Subject, not from the camera. It assumes the subject still stands where they were (with respect to background), but the longer lens steps back.
The classic "Same DOF for same picture image" rule of thumb: If standing back with longer lens at the longer distance that gives the same Field of View, and IF the two cameras have same sensor size and use the same f/stop, then the Depth of Field "span" can be the same for both lenses. This works better for telephoto lens, it being true if focus distance is less than 1/4 of hyperfocal of shorter lens, with longer lens at the longer corresponding distance for same picture. Example for the calculator, same sensor size, same f/stop, 75 mm f/5.6 at 10 feet, vs 150 mm f/5.6 at 20 feet, for same DOF range span (see Google). The background CoC will be quite different however (background will not be the same size).
The point: In many cases when wanting to hide the background, standing back with a longer lens can provide the same field of view of subject, but with much less view width of the background, and which also allows stopping down a bit more to provide greater depth of field at at the subject, but while still offering greater blur at the smaller background area. Standing back with the longer lens offers better portrait perspective too. These factors can make a significant difference.
CoC Divisor: Maximum CoC limit in DOF calculators is usually computed as (sensor diagonal mm / 1442), which is default here unless CoC or Divisor is directly specified. It is a little arbitrary, and if you decide you want it different, you can change it, and get different results.
Full frame 35 mm cameras often use 0.03 mm for CoC, and APS cameras often use 0.02 mm (due to crop factor). A compact camera or smart phone might have CoC = 0.004 to 0.007 mm (much more enlargement is necessary). For 35 mm film, Zeiss has recommended CoC = Sensor Diagonal / 1500 (CoC = 0.029 mm), but to get 0.03 mm requires Sensor Diagonal / 1442. So 1442 is a standard usual value, considered appropriate for viewing a standard 8x10 inch print size. However, Depth of Field is somewhat arbitrary.
Largest Print Dimension is about the relative enlargement of your viewed image. The meaning is that is the "largest dimension" of 8x10 is 10. When we enlarge the viewed image, we enlarge the CoC too, so it's easier to see the blur then, which becomes no longer a suitable indictor. The DOF concept is all about the CoC we can perceive, when enlarged from the sensor size we use. If we are going to enlarge our view more, then we need to start with a smaller CoC. Standard DOF calculations assume viewing a standard 8x10 print size from 10 inches, which is the 10 inch default here (254 mm). This feature is to describe a different image size that you may view, to account for the effect of your enlargement on the Depth of Field calculation.
The CoC used is shown in bold, if and when modified by the Largest Print not being the standard 8x10 inches (254 mm largest). Because, CoC is the largest allowable blur, to still not be perceptible by our eye. If we're going to view an image enlarged bigger, then maximum allowable CoC at the sensor has to be smaller (to not exceed what our eye can perceive). Or vice versa.
However, you can specify any CoC limit directly. Yes, it will then compute and show a sensor diagonal size based on (CoC x Divisor), which is as accurate as those terms. But that sensor size is just for reference, and is used for FOV, but is not further used for DOF. It does Not affect DOF now, since CoC has already been specified directly. The DOF formula computes with only CoC, focal length, f/stop, and focus distance. Sensor size is not in the DOF formula, except that it should have of course defined CoC. So, bottom line, you need to know what you're doing if you specify CoC directly. Just because you saw someplace use 0.03 mm CoC, this does NOT mean that is a proper number for your camera and its sensor size.
If comparing results with numbers from other calculators, make sure the CoC and Sensor Size used are the same value. For those reading this far, Zeiss and Wikipedia suggest the CoC divisor should be 1500 today, a slightly tighter limit on sharpness, but 1442 still seems clearly the standard on the internet, so I went with the flow to avoid confusion. The diagonal of 35 mm film is 43.267 mm, and it divided by 1442 is what makes CoC be 0.03. You can change the divisor as desired. CoC is a little arbitrary anyway, and there's not much difference, being 0.03 or 0.029 mm CoC for full frame 35 mm, perhaps a 4% change in DOF calculations. Other factors like focal length, distance and viewing size probably are larger issues.
Rounding: Note that numerically, real world APS sensors are slightly smaller than 24x16 mm, and their crop factors are actually slightly larger than 1.5 or 1.6. Just for example, the Nikon D5300 DSLR camera manual provides specifications:
Macro: Depth of Field calculators are not accurate for macro situations. Macro calculations are inaccurate because we don't know extended focal length, and maybe not f/stop reduction, and probably not the location of the front nodal point of the lens to know distance. At the close focus point, these are large factors. Accuracy depends on knowing the numbers. Macro instead computes DOF from measured magnification. Macro 1:1 means the object image is the same size on the sensor as the object in real life, true regardless of sensor size. For DOF calculators, distances of at least a few feet will be most accurate in any lens calculation.
Circle of Confusion (CoC) is theoretically zero diameter (a point) at the focus point. But this blur circle grows larger when not in focus, and the DOF range is calculated to not exceed the standard limit (sensor diagonal/1442) of acceptable CoC. CoC (and therefore Depth of Field) definitely also depends on the current enlarged viewing size, which is magnification of the DOF blur. We should know that standard CoC is considered to view as acceptable sharpness in the standard 8x10 inch enlarged print viewed at 10 inches.
Depth of Field (DOF) is certainly not ONLY about aperture. DOF is an extremely important basic of photography, however IMO, exact DOF calculators may not be great practical use, other than to get a rough idea. But we certainly do need to know the concept. We need to know this, it should be second nature to you.
Statistical tests have said the average resolution of our eye is to perceive 6 mm of detail at 6 meters distance, called 6/6 vision in Europe, or 20/20 vision in the US (feet = meters x 3.28). This scales to other similar ratios, like 0.025 mm at 25 cm is familiar. For DOF to be judged in a standard way, that was standardized as perceiving 0.025 mm of film detail on an 8x10 inch print when viewed at 25 cm (ten inches). This size print represents substantial enlargement of the small sensor image, so CoC limits at the sensor must be divided by the enlargement factor (from sensor size). Eyes do vary, but someone established this ballpark number, used for DOF as the limit of acceptable CoC diameter (that blurriness limit that we still call sharp). So this 8x10 print viewed at 10 inches is our standard for calculating the DOF blur that will be created. Today, this judgment is contained in the CoC = Sensor diagonal mm / divisor definition. History has used /1730 and then /1000 and /1442 and /1500, and likely others. The divisor 1442 is most common today. This number is hard to verify in results. The DOF formula details the geometry involved in the lens, and one factor is the CoC value which is determined by sensor size, for the purpose to scale it to what our eye can perceive at this standard 8x10 enlargement. The CoC number is defined as sensor diagonal / divisor (often 1442), but which is the CoC diameter as perceived in the standard 8x10 enlargement. It will vary in other enlargement scales.
The Depth of Field is the computed distance zone around the focus point, the span where the CoC remains less than our arbitrary limit for the size of CoC, considered to be in focus. The image is only focused at one distance, and gradually degrades away from that point. Focus just outside the DOF calculation will be hardly different than the focus just inside the DOF calculation. For example, maybe the DOF limit computes 20 feet. But then you probably cannot detect much difference a couple of feet either side of 20 feet, but the exact focus point will be better. DOF is NOT at all magic numbers, it's just where the math precisely computes the CoC size crossed an arbitrary threshold size boundary. The boundary is very vague to our eyes. Sharpest focus is at the one distance where we actually focus. Depth of Field is a vague concept.
If we can see these blurred blobs in the results, that's normally considered bad, when that distance is not in focus well enough. If only slightly out of focus, it may not enough for us to even notice it, much less object about it. Which is good, and while standards vary, DOF is a way to judge it. The exact calculated numbers are rarely important (just a simple guide). But the zone of DOF we perceive is certainly really important, and the big thing to know is what the controls are, and to know how to adjust it. With a little experience, we know what to expect, and this works pretty well.
The name Circle of Confusion is from another era, and Wikipedia quotes work in 1829 and 1832 calculating Circle of Confusion. They had microscope and telescope lenses then, but this was before cameras or film. Still same concept, but maybe if invented today, we might pick a simpler name for CoC (it is the diameter of the blurred circle of an out of focus point source). Camera sensor size is a factor of enlargement. Older work used CoC = (sensor diagonal / 1730), or 0.025 mm for 35 mm film. Today, we often use the computation (sensor diagonal mm / 1442) for acceptable maximum CoC in the final standard print size. These are often rounded numbers, or CoC = 0.03 mm for full frame 35 mm sensors, and CoC = 0.02 mm for smaller APS sensors (because the smaller sensor requires half again greater enlargement).
CoC is arbitrary, and professional level might prefer it smaller, with larger safety factor. Our CoC number choice does not affect the image in any way, it only affects how we might judge it, or plan it. It is an arbitrary notion about when out of focus is judged to become too noticeable. And DOF very definitely also depends on how large you enlarge the image to view it.
What makes DOF even more arbitrary is that the larger we enlarge and view the image, the more noticeable becomes the blur blob of CoC. View it small, and we may not even notice it. The standard of viewing DOF is considered to be an 8x10 inch print viewed from 10 inches. That's about a 9x enlargement of 35 mm film (CoC 0.03 mm), and so the CoC we see then is the 0.03 mm x 9 = 0.27 mm in this print. We enlarge our smaller digital sensors even more to see 8x10, so allowable CoC has to be smaller. Every sensor size has a different CoC (from sensor diagonal mm / divisor) - because we assume to enlarge each to the standard 8x10 inch print to judge it. DOF is a different number after enlargement, BUT the standard maximum CoC value was chosen to be acceptable when viewing this standard print size. Today, we view first on the computer screen, or even a cell phone. But we view different sizes, and this also affects the acceptable CoC goal. There is NOT just one number for DOF of a situation.
But, you can use the Largest Print Size parameter above to describe a print size (specify the largest dimension, like 20 for a 16x20 print), and it will calculate DOF based on that instead of the standard 8x10 inch print.
The Hyperfocal distance is a special idea of DOF. It is sometimes used for landscape photography with wide angle short lenses, when we want an extreme DOF range, extending to infinity, and also back to a rather near foreground object (to emphasize depth). For example, an APS-C sensor (1.5x crop factor) with 24 mm lens at f/16 computes Hyperfocal distance as about 6 feet (depending on precise sensor dimensions). What that means is this:
Just saying, even if your hyperfocal is 6 feet, but your subject is at 20 feet, and there really isn't anything much closer, then of course focus at 20 feet instead. Infinity is still good. But if you have do something at 3 feet that's important, then of course the idea is to focus at hyperfocal at 6 feet, for DOF from 3 feet to infinity. It just won't be the sharpest possible focus for 20 feet, but 20 is well within DOF limits of 3 feet to infinity.
Hyperfocal varies with the square of focal length ratio. Doubling focal length gives 4x hyperfocal distance, or 10x focal length gives 100x hyperfocal.
Stopping down two stops more gives half of hyperfocal distance (stopping down one more stop is 0.707 x hyperfocal).
So, doubling focal length AND stopping down four more stops is the same hyperfocal distance.
These are basic ideas which have been known for maybe 150 years. The alternative of simply focusing on the near side of the subject zone typically wastes much of the depth of field range in the empty space out in front of the focus point, where there may be nothing of interest. Focusing more into the depth centers and maximizes the DOF range, generally more useful. We hear it said about moderate distance scenes (not including infinity) that focusing at a point 1/3 of the way into the depth range works for this, which is often near true, maybe a little crude, better than knowing nothing, but some situations do vary from that 1/3 depth (below). Close and macro focus situations are closer to the middle at 1/2 way in, and don't include infinity.
The crude distance marking today on our lenses make it hard to set a specific focus distance. If your lens only has a mark at 10 and at 5 feet, setting 6.117 feet won't be easy. But we can approximate it to about 6, at least closer to 5 than to 10, normally close enough. It's all a little vague anyway. Sometimes it might be easy to focus on something at an estimated 6 feet, and then shift your camera aim to the real subject.
Every Depth of Field calculator should show hyperfocal focus distance.
Many prime lenses have a DOF calculator built into them. Speaking of prime lenses (i.e. those lenses that are not zoom lenses) which normally have marks at the distance scale showing the depth of field range at the critical aperture f/stops. However, this tremendous feature is becoming a lost art today. Zoom lenses cannot mark this for their many focal lengths. Also todays faster AF-S focusing rates can put the marks pretty close together. These 85 mm and 105 mm lenses are AF-S, but it still gives a DOF clue. (the "dots" are the focus mark correction for infrared.)
For example of hyperfocal distance, at right is an older 50 mm FX lens, with focus adjusted to place the f/22 DOF mark at the middle of the infinity mark, which then actually focuses at about 12 feet, and the other f/22 DOF mark predicts depth of field from about six feet to infinity (assuming we do stop down to f/22). This places the focus at about 12 feet. The DOF calculator says this example (FX, 50 mm, f/22, 12.3 feet) DOF range is 6.1 feet to infinity.
Or another case, one not including infinity. If we instead focus this 50 mm lens at 7 feet, then the FX f/11 marks suggest DOF from about 5.5 to 10 feet (at f/11, which is about 1/3 back in this case). The idea of the markings (which appear on prime lenses, zooms are too complex to mark) is to indicate the extents of the DOF range. And done because it can be very helpful. Sometimes f/22 is the best idea, sometimes it is not. f/22 causes a little more diffraction, but it can also create a lot more depth of field. And of course, the lens markings apply to the expected sensor size for that lens.
Those DOF end point extremes will of course Not be as sharply focused as the actual focus point, but they will still satisfy the standard CoC specified. Do realize that DOF just means barely tolerable limits, where the CoC has grown to the maximum limit specified. Focus is always of course sharpest at the exact focused distance. Focus is not necessarily perfect if inside DOF, instead it is assumed unacceptable if outside DOF, but there is no sharp dividing line. If you want really sharp images, include ample safety factor for DOF; pay attention to enlargement size, stopping down at least one more f/stop, and if really important, focus on the important spot that needs to be sharp.
Your DOF calculations may not exactly be realized particularly close in practice, due to your own degree of enlargement, and your viewing distance, and your own eyes, or an inaccurately specified sensor size, and how accurately you guess the actual distances. It is just a large ballpark. You'll have to decide for yourself if your images are as sharp as you want. What you specifically need to know are the factors to increase DOF (stopping down, a shorter lens, and longer distance).
A couple of tricks are to plan on having sufficient DOF with ample safety factor, and then learn to center that DOF around your subject depth. If DOF is limited, don't focus on the nose if you want the ears sharp too. Repeat this to yourself: Focusing on the closest point wastes the half of the DOF range in front of that point (where there is nothing). Instead, you can plan to better center the DOF zone around your subject.
To do that centering, we hear about the simple (rough) guide of focusing 1/3 of the way into the scene depth (1/3 of scene in front of focus point, and 2/3 behind). If we think that 1/3 of the DOF range is in front of subject, then it makes sense to focus 1/3 into the scene, instead of at front point, and instead of half way back. There is no good argument for the front point, and half way is true if up focusing pretty close. That focus point may not be where the subject is, and of course that subject will always be sharpest if you actually focus on it (so there are trade offs).
Regardless, hyperfocal becomes interesting:
Without knowing hyperfocal, we could debate if 1/3 or 1/2 into scene depth was generally more often about right? Focusing 1/3 into the scene depth can sometimes be true enough, often not greatly wrong, and can be a better rough guide than knowing nothing. But usually, focusing somewhere between 1/3 and 1/2 into the scene depth is likely very reasonable in the general case.
Situations will vary, and the DOF in front of focus might be from 0% to 50% (at extremes). Otherwise, 1/3 is not the worst guess (we are not actually measuring distances anyway). Generally, short lenses have closer hyperfocal, and stopping down any lens brings hyperfocal back closer to us (and brings a short lens back very near). That's a lot to know. Frankly, in practice, we never compute the hyperfocal number, so we just soon learn the general idea of what we need to do when DOF is important. Stopping down some, and focusing somewhat into the scene depth can usually help considerably. Just standing closer with a shorter lens can help DOF, and as discussed here, standing back with a longer lens can reduce DOF range (specifically, will be same DOF at the subject with same f/stop, but greatly different at the background).
For portraits at around 8 or 10 feet, I think a good tip is to focus on the near eye, after ensuring ample DOF, like f/8. IMO, f/1.8 is never the best try, and this article is about an alternative. For full frame portraits, I like about 120 mm at around 10 feet. For DX or APS crop cameras, that would be about 80 mm around 10 feet. Ten feet is very good portrait perspective, and at f/8, that's about a 2x3 foot FOV with around a one foot zone of DOF (again, of course speaking about the standard 8x10 inch print viewed at 10 inches).
Depth of Field is NOT an exact number. Depth of Field is computed based on the Circle of Confusion (CoC), which is the arbitrary criteria defining the maximum acceptable blur circle (to be small, not quite perceptible) due to being out of focus. CoC is the diameter of the smallest possible theoretical point after it is defocused to be seen as a larger blur circle (because it focuses in front of, or behind, the sensor plane - then causing a larger out of focus circle on that plane). CoC is the maximum permissible diameter of this blur circle, arbitrarily still judged to be imperceptible in our vision (also assuming a standard viewing enlargement). If the blurred circle is too small for us to perceive it, then we imagine it's not blurred.
Carl Friedrich Gauss 1777-1855 was a most brilliant mathematician (in a class with Newton) who did many amazing great things, one of which was to formulate optical theory (by 1840) that is still used today. Gauss thought the eye's criteria of visibility of focus blur ought to be a CoC of (frame diagonal divided by 1730, in mm), which computes 0.025 mm today for 35 mm film size. But today, CoC of diagonal divided by 1442 is a common universal value (0.03 mm for 35 mm film). The sensor diagonal is involved because it is a factor of the final print enlargement required, where we see and judge the perception of the enlarged CoC. Enlargement is a big factor of perception. But it's still an arbitrary guess about blur, about what our eyes see after enlargement. Blur diameter cannot be precisely defined... kinda depends. And so a CoC limit is somewhat arbitrary, there's been a few choices. CoC is just a rough guess attempting to measure focus blur, which makes DOF numbers be a vague thing.
The DOF Standard of Viewing is in an enlargement of an 8x10 inch print (near A4 size) when viewed at a distance of 10 inches (25 cm). You should know that DOF calculators use a CoC which assumes this standard enlargement, regardless if you assume it or not. If you view it up close on a large HDTV screen, DOF will appear much less than you calculated. If you view it on a smaller wallet or 4x5 inch photo, DOF should appear better than you calculated.
Viewing the enlargement size is an important factor in what we see, and in CoC and the Depth of Field calculations. This viewing enlargement factor makes small sensor diagonal be an important factor of DOF. It's the reason smaller sensors have a smaller CoC, and larger sensors have a larger CoC (sensor size requires enlargement of CoC). But standard DOF calculations assume a standard 8x10 inch print is viewed. So this affects viewing a smaller print or a larger print:
Computing on the diagonal attempts to equalize for different sensor or print shapes, but many vague assumptions are included. You should include a safety factor, especially for large prints, one extra f/stop for safety.
Depth of Field is an angular size concept, and the math is very precise, EXCEPT for the main factor of CoC, which is rather vague and arbitrary. So there are no hard answers about Depth of Field. And since Depth of Field GRADUALLY changes with distance, there is of course no sharp line at the computed limit. There will be virtually no difference seen slightly either side of the computed limit. Numerical Depth of Field is at very best, an extremely rough guide.
Depth of Field is a fundamentally important principle of photography. However using it is MUCH LESS ABOUT any computed numbers, and VERY MUCH MORE ABOUT understanding how to use the factors that increase or decrease it (above, f/stop, distance, focal length, and sensor size). Normal situations are not much concern, but sometimes we're aware we want a lot of depth of field, or don't want much of it, and we should know how to control that, to do what we can.
Continued - Part Two, Examples about background