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The term "anamorphic" has its origin in cinematography. In the context of DVDs, it has a slightly different meaning, so there may be some confusion because people use the same word in two different contexts.
In the early days of film making, a lot of different formats were used, but 35mm film was adopted fairly quickly. In 1930, a number of studios, including MGM, Universal, UA, and RKO settled on the 35mm format, which was then known as the "1930 standard". A couple of years later, this standard was further elaborated by fixing the frame size on the film at 22mm x 16mm, which produced a 1.37:1 ratio. This was then known as the Academy Standard. The 35mm standard is still around, and used very frequently for film.

Film stock has perforations along both edges, so the projection equipment can pull it past the lens at typical film speed, which is 24 frames per second. There are also embedded sound tracks on the film, which vary in number (1 = mono, 2 = stereo, more for Dolby ProLogic or 5.1). These soundtracks are also "burned in" optically, though in an analog fashion, not like the pits on a DVD.
The Academy Standard uses 4-pulldown, which simply means that there are four perforations per frame. This leaves enough space for a 22mm by 16mm frame per four perforations, as can be seen in the picture to the right. The exact aspect ratio works out to 1.375 to 1, but this is approximated to 1.37:1 in the Academy Standard.
Note that sometimes people refer to this as the 4:3 standard, but this is wrong. A 4:3 aspect ratio works out to 1:33 to 1, which is different from the Academy Standard of 1.37 to 1. There are some very old films shot at 1.33:1 (mostly from the silent film era), but the vast majority of films shot at Academy Standard follow the 1.37:1 ratio. The 4:3 standard is a much later product of television.
People realized early on that the Academy Standard doesn't match the human visual field very well. Humans have wide-angle vision, since we have two eyes, set on a horizontal plane. Our field of vision is therefore significantly wider than it is high. For this reason, photographs taken with a 1.37:1 aspect ratio don't adequately convey a sense of width. This is more noticable when photographing outdoors - scenes which have a wide, unobstructed field of vision. The 1:37:1 Academy Standard makes such scenes look horizontally constricted, as if you were looking at them through a window. Obviously, this detracts from the immersive quality of the film.
For this reason, even in the early days, film makers experimented with widening the field of view. Being constrained by the physical size of the 35mm negative, there was not a whole lot they could do. Since the width of the frame is restrained by the width of 35mm stock, that can't be changed much. It can't be increased beyond 22mm, because you have to leave room for the soundtrack, and for the perforations. So the only way to change the aspect ratio is to decrease the height of the frame. If you reduce the height from 16mm to 11mm, then you have a 22mm by 11mm frame, which gives a 2:1 ratio. This is much closer to the normal human field of vision. Of course, with the smaller frame, your frame resolution goes down, so it won't look quite so sharp and detailed when you blow it up to screen size. The are in use by the frame decreases from 22x16 = 352 mm2 to 22x11 = 242 mm2. That's a 30% loss, so the resolution (the amount of detail the frame can hold) is decreased by a third. Although improvements were continually made in film stock in terms of decreasing grain size and improving its intrinsic resolution, this loss of 1/3rd wasn't altogether satisfactory. So such early experiments were relatively rare and did not set any trends.
More dramatic changes are possible if we lay the frame on its side, rotating it by 90 degrees. The frame height then becomes 22 mm, and the width can be increased arbitrarily, since the width of the frame now lies along the long axis of the film stock. This was called "VistaVision", and it's a great system. This is actually what's used for true IMAX these days, except they go one further and use 70mm film stock instead of 35mm, and they turn the frame on its side. However, at the time, VistaVision did not become very popular. The main reason was because it required that theater owners across the country upgrade their projection equipment, and not many wanted to do that because of the costs.
Because of the resistance of the theater owners, there was another restriction, which is not immediately obvious. The Academy Standard, with its 22x16 frame, called for 4-pulldown, meaning that there are 4 perforations per frame on the film stock. This is something built into projection equipment, which pulls film through at a constant rate of 24 frames per second, or more literally, 24x4 = 96 perforations per second. So even if you changed the aspect ratio of your frames (by going to a 22mm x 11mm frame, for example), the projector would still pull the film through at the same rate. Obviously, a thinner frame means that there are black bars between frames. These bars needed to be masked during projection. Fortunately, this was cheap to do.
But as we can see, these were not satisfactory solutions. By changing aspect ratio, more film was wasted by the introduction of thick black bars between frames. As a consequence, the resolution decreased. Therefore, while film makers experimented with such wide aspect ratios even in the early days of film, they never became popular. Until the 1950's, and the introduction of television.
The introduction of TV in the late 40's changed the priorities of film makers. Very early TV screens were circular, since early CRT technology required a constant curvature across the whole screen, or the edges became fuzzy. However, since people don't like to watch stuff through portholes, manufacturers tried flattening their screens and squaring off the sides. Early "flat" screens were square, or slightly rectangular, and "flat" is just an approximation. They still had pronounced radial curvatures. As technology improved, the screens were widened, giving them a more rectangular look, wider than it was tall. TV manufacturers were trying to approximate the field of vision of the human eye, just as film makers had tried a few decades earlier. But this was a difficult problem to solve, and manufacturers were not able to produce flat wide aspect tubes cheaply until the 1980's, and the popularity of computer monitors. The vast majority of TVs produced from the 50's to 80's followed the NTSC standard, adopted in 1953, which had an aspect ratio of 4:3, or 1.33 to 1.
The popularity of TV scared the film industry, because of the concern that people might get used to watching movies in their homes, and stop visiting their local theater. This led the studios to focus effort on improving the theater experience. One way to do this was by going back to the old idea of wide angled viewing. This time, however, the situation was different. A lot of cash was hanging in balance. More importantly, theater owners were also worried about loss of revenues to TV, so they could be more easily persuaded to spend cash to upgrade their projection equipment.
The old problem of wasted space on the 35mm negative remained. As described earlier, if you increase the aspect ratio beyond the Academy Standard on 35mm film stock, you end up with a thinner, narrower frame, which you then have to pad with black bars at the top and bottom. To get around this problem, film makers turned to anamorphic lenses.
Anamorphic lenses had been invented much earlier (in WWI, by the military, to allow tank drivers to see a wider field of view). An anamorphic lens is one that breaks the normal rule of lenses, in that it does not produce the same magnification across the X and Y axes. An anamorphic lens produces a field of view that has a different magnification along the X axis than along the Y axis. You can consider this in two ways - either think of one axis as being squashed, or think of the other as being stretched. The easy way to think of it is to relate it to 35mm film. Since wide aspect ratios produce black bars at the top and bottom, the wasted space is at the top and bottom of each frame. In order to recover that wasted space, we need to stretch the vertical axis, so that the black bars become narrower, and some of the wasted space is reclaimed. In effect, this stretching increases the magnification of the lens along the vertical axis, in comparison to the horizontal axis.
Of course, film shot with anamorphic lenses must also be projected with anamorphic lenses. This, however, is not a huge expense, since all you need to do is to add a lens to the projection system which has the opposite curvature of the anamorphic lens used to shoot the movie. So if the movie was shot so that the vertical axis appeared elongated (people looked very thin and very tall), then you just have to stretch it out in width during projection, so that objects regain their proper proportions.
As a result of this process, numerous new formats appeared. In fact, "wide screen" became a selling point. As soon as someone introduced the 2:1 ratio, another studio would one-up them by producing an even wider ratio. There is a great profusion of very catchy and marketable terms from this period, referring to progressively wider aspect ratios, as studios competed with each other. Aspect ratios reached ridiculous proportions, up to 3.2 to 1. Not having had the opportunity to watch such a movie in a theater, I can only imagine what it must have looked like - a thin, narrow screen that only reached half way to the theater ceiling. It must have been hard to persuade the audience that chopping the screen in half was an improvement.
Fortunately, studios recognized the problem, and the format wars cooled down. The 2.4:1 standard became common for film (it's actually 2.39 to 1, but we approximate), and has continued to this day. A lot of films shot today still use the 2.4:1 aspect ratio, though the recent emergence of all-digital movie production is starting to change the situation again.
During the decades from the 50's through the 80's, a lot of material was also produced directly for TV. Material that was produced "direct for TV" followed the NTSC standard, which had been established in 1953. The NTSC standard called for 29.97 frames per second, with a fixed aspect ratio of 1.33 to 1. The number 29.97 may seem odd (why not just round up to 30?) but it has its basis in electronics rather than optics. It has to do with how bandwidth was assigned for a TV channel, how much data could be transmitted at whatever frequency they were using, and so the frame rate was calculated based on choosing nice, round numbers for the electronic signal characteristics.
At any rate, when the DVD standard was introduced, a huge amount of made-for-TV material had to be taken into account. Also, the vast majority of people at the time still had 4:3 TV sets. Widescreen TVs had just started to make an appearance. The bulk of made-for-TV programs were still being shot at 4:3 ratios. TV manufacturers had tentatively decided to adopt a 16:9 ratio for the new widescreen TVs, and some TV stations had started moving to new cameras that recorded in the 16:9 ratio. But the volume of such 16:9 programming was still very small.
This is why the DVD standard sometimes seems so unsuited for film. It really wasn't made with film in mind, it was made with a number of requirements in mind, which included the current NTSC standard, the forthcoming widescreen TVs, as well as film.
Since broadcast standards vary across the world (NTSC, PAL, SECAM), DVDs made for different regions also vary. In North America, DVDs follow the NTSC standard, which for the purpose of DVDs has been defined as a frame size of 720x480, at 29.97 frames per second. In Europe, DVDs follow the PAL standard, which records frames at 720x576 pixels, at 25 frames per second. There are various details which I have not mentioned here, which may make part of the frame unavailable for recording the picture. The rest of this explanation is therefore somewhat simplified to illustrate how the concept of anamorphism has been adapted to DVD movies. I've used numbers to illustrate the examples, but remember, these numbers are only approximate. Also, I'll only be talking about NTSC DVDs. The same principles apply for PAL as well, though actual numbers will be different.
Consider a typical movie, which has been shot at an aspect ratio of 2.4 to 1, at 24 frames per second. This movie needs to be sold on a DVD disc, which will be watched by people on at least two different kinds of TVs - regular TVs with a 4:3 aspect ratio, and widescreen TVs with a 16:9 aspect ratio. Since this discussion is about anamorphism, I'll only focus on aspect ratio concerns, and ignore the frame rate issue. There are well established methods (such as 3-2 pulldown) to do the frame rate conversion.
So, how do we record a movie with an aspect ratio of 2.4:1 on a 720x480 pixel DVD matrix?
The simplest way is to squeeze the movie down to regular TV width, and pad whatever needs to be padded with black borders. This is what it would look like on a 4:3 TV:

The advantage of this method is that the whole frame is visible, so nothing is lost. The disadvantage is that the picture is very small. This method is called "Letterboxed Widescreen".
Many people find this unpleasant, because of the small picture. To avoid this, we can use a technique called Pan and Scan. In this method, we overlay a 4:3 mask over the film frame, and simply record whatever appears inside the mask. Obviously, a 4:3 frame doesn't match the 2.4:1 ratio of the original film - it's not as wide - so the edges will be chopped off. The "pan" part of the name comes from the fact that the mask is panned across the frame as the movie progresses, to follow the action. Since we're chopping of the edges of the frame, we want to make sure that important action happening at the edges isn't chopped off. Therefore, the person operating the mask moves it side to side while the movie is scanned, making sure to keep in view whatever part of the screen has stuff going on that's relevant to the story.
This is what the same movie would look like in Pan and Scan, on a 4:3 TV:

As you can see, the picture is much larger, but the sides have been chopped off. This was considered acceptable in the early days of DVD, and was generally the preferred solution compared to letterboxing. However, many purists complain that a pan and scan movie no longer conveys the vision of the director, since some third party gets to decide what part of the frame is important and what gets chopped off. Also,wide screen TVs are much more common these days, so the huge black bars that were so objectionable in the past have been reduced in size somewhat, and are therefore less objectionable.
The adoption of widescreen TVs provided a solution, but also created a problem. The problem is this: the DVD standard has a resolution of 720 pixels wide by 480 pixels tall. This creates an aspect ratio of 1.5 to 1. However, a widescreen TV has an aspect ratio of 16:9, or 1.78 to 1. Obviously, the DVD format isn't wide enough for a widescreen TV.
As I explained above, the DVD standard wasn't specifically created for film or widescreen TVs, it was created with a lot of different things in mind, including a vast amount of existing programming created for the 4:3 NTSC standard. So the aspect ratio of 1.5 to 1, while not matching any of the intended uses exactly, was a good compromise.
So what would a letterboxed DVD movie look like on a widescreen TV? This is what it would look like:

Notice that now we have black bars on the sides as well. Why does this happen? The reason is because a widescreen TV is wider than the DVD standard. Widescreen TVs have an aspect ratio of 16:9, or 1.78 to 1. The DVD format for NTSC codes at 720 pixels by 480 pixels, which is an aspect ratio of 1.5 to 1. This is why we have space left over on the sides, which appears as black bars.
A regular 4:3 TV has an aspect ratio of 1.33 to 1. So, for example, if we standardize the height of the TV to 480 pixels (which is the vertical dimension of the DVD standard), then the width is 1.33 times that, or 640 pixels. On the other hand, if we standardize a widescreen TV to the same 480 pixel height, then its width would be 480 times 1.78, or 853 pixels.
Although it may look like the letterboxed pictures in the example above and letterboxed picture in the 4:3 TV example at the top are about the same size, the letterboxed picture on the widescreen TV is slightly bigger. The picture on the 4:3 TV is 640x267 pixels, while the picture on the widescreen set is 720x300 pixels. Why this difference? It's because the limiting factors in the two cases are different. In the case of the 4:3 TV, the limiting factor is the width of the TV, because it's less than the width of the DVD format. So you have to continue squeezing the movie until its width fits the width of the TV. In the case of the widescreen TV, the width of the TV is not the limiting factor. In fact, as we've seen, the TV is wider than the DVD standard, hence the black bars on the sides. In this case, the limiting factor is the NTSC format itself, which has a maximum width of 720 pixels.
Because of these different limiting factors, the widescreen movie looks bigger on the widescreen set than on the 4:3 set.
Ok, so we have a bigger letterboxed picture on a widescreen TV than on a 4:3 TV, but there is still a problem. The problem is the black borders on the sides. While it's natural that a 2.4:1 movie will have black borders at the top and bottom when shown on a 16:9 TV, there's no reason why we should have to put up with black borders on the sides. How do we get rid of them?
The obvious solution is just to zoom in. We can use circuitry in the TV to magnify the picture, so let's just keep magnifying until the borders at the sides disappear. This is what it would look like:

This is the ideal situation that we want to get to - we want to use the full available space horizontally, and then have black borders at the top and bottom only because the film had a wider aspect ratio than our widescreen TV, which can't be helped.
While this is the result we want, the method we used was not good. Zooming is simply magnifying, it's not increasing resolution. It's the same effect as putting your projection screen 30 feet from the projector instead of 25 feet. The picture will be bigger, but it will also be fuzzier. What we really want to do is to increase the resolution as well, so that enlarging the picture as shown above is possible while retaining sharpness and detail.
But here we run into a problem. In our original unzoomed picture with the black borders on the sides, the picture was already 720 pixels wide, which is the maximum horizontal resolution of DVD. How can you increase resolution beyond the DVD standard?
The answer is, you can't - at least, not horizontally. But here we can use the same trick the the film makers did, when they ran against this problem on their 35mm negatives. They used anamorphic lenses, to reclaim some of the vertical space at the top and bottom. In this way, they were able to increase vertical resolution, though not horizontal resolution. But hey, increasing it in one direction is better than increasing it in none, so we could do the same here.
What they did using lenses, DVDs do using software. The principle is the same, that you are using different resolutions across two different axes. Along the horizontal or X axis, you can't increase resolution any further, because you are already using 720 pixels, which is the maximum allowed by the DVD standard. But on the vertical or Y axis, you have a lot of unused pixels in those black bars. So how about using these pixels to increase vertical resolution a bit?
How would we do this? To continue with this example, we have a 2.4:1 movie, which we want to display so that its width equals the width of our widescreen TV. As shown in the examples above, a TV that was designed specifically with the DVD height in mind would have a vertical resolution of 480 pixels. If this TV also followed the widescreen TV standard, it would have an aspect ratio of 16:9, so it would be 853 pixels wide.
Our 2.4:1 movie on DVD uses only 720x300 pixels (720/300=2.4). We have no spare pixels on the sides, but we do have spare pixels on the top and bottom of each frame. We would like this movie to cover a screen that is 853 pixels wide, the width of a widescreen TV. By what factor do we need to magnify?
To get from 720 pixels wide to 853 pixels wide, you need to increase resolution by (853-720)/720 = 18.47%. However, you can't actually increase resolution along the horizontal axis, because you're already at the max so far as width goes. But you can increase resolution by 18.47% along the height, because we were using only 300 pixels of height, while the DVD resolution allows 480 pixels for height. Increasing resolution along one dimension, while not ideal, is at least better than increasing resolution along none. So if we increase resolution by 18.47% vertically, we get a height of 355 pixels, up from our original 300 pixels. That means we now have (355-300)x720 = 39,600 more pixels per frame that are carrying information, instead of going to waste. This is what happens in an anamorphic DVD.
Consider what we have done in making this anamorphic DVD. We have increased vertical resolution, but not horizontal resolution. Our movie now uses 720x355 pixels for each frame. However, if we were to play it as such on a TV which didn't understand anamorphism, this mismatched resolution across two axes would show up as a geometrical elongation along the axis which has the higher resolution (the vertical axis). This is what it would look like:

As you can see, the image is stretched vertically. The picture occupies 720x355 pixels of the frame. But if we could code our Y scale resolution factor of 18.47% into the file, and then display it on a TV capable of understanding anamorphism, the TV would stretch the image by 18.47% horizontally, but not vertically. By stretching the picture only horizontally, the width would increase from 720 pixels to 853 pixels, while the height would remain the same, at 480 pixels. Then the picture would look like the zoomed example earlier, except that it's been zoomed only in one direction, while the other direction didn't require zooming because it actually carried more information.
This is how anamorphic DVDs work. They have increased resolution on the Y or vertical axis. They also contain a multiplier (like 18.47% in our example), which tells the TV set to stretch the movie horizontally by 18.47% to compensate for the comparitively lower resolution of the X axis. You could think of it in the sense that this multiplier converts square pixels into pixels that are elongated vertically, and in order to square them again, the TV has to stretch them horizontally. Of course, non-square pixels exist only in our imagination. TV pixels can't change their shapes, they are physical things. What really happens is that more pixels are used, and the 720 pixel wide image on DVD is resampled to 853 pixels wide on the TV.
Couple of things to remember here. The numbers are used just for illustration purposes. I don't mean to imply that any given TV set is 853 pixels wide. The resolution of a TV set depends upon the manufacturer, but the important thing is that whatever the native resolution, it will scale to the 853 ratio described here. If you literally want to see the 853 pixels, you will have to do it on a computer. Take an anamorphic movie with an aspect ratio of 2.4 (they are very common). View it on a software player (such as VLC) that is capable of playing anamorphic DVDs. Pause the movie at any point and take a snapshot. The snapshot will be 853 pixels wide, even though the DVD itself only has 720 pixels of horizontal resolution.
The other thing to remember is that while I have talked about the TV recognizing anamorphic movies and doing the scaling, in actual fact it's rarely done by the TV. Recognizing the formats and doing the scaling or interpolation is the job of the DVD player.