The Short Answer
Aspect ratio is used to describe an image's dimensions based on width to height. For example, standard television in the U.S. is 4x3, in other words, 4 units wide to 3 units tall. Hi Def television is 16x9, or 16 units wide to 9 units tall. 16x9 recordings can be SD or HD.

More Info
In this section, we'll go over how these aspect ratios are mixed and matched, the concept of anamorphic recording (which affects almost all widescreen recording formats), and the various aspect ratios used for film and television.
Standard aspect ratios in film and television:

The above illustrations are not drawn to scale. To keep things simple, and to be able to clearly see the difference in the aspect ratio only, the height is consistent in all of the above images. Notice that 1.78:1 (16x9) is very close to the U.S. theatrical standard of 1.85:1. 1.85:1 is slightly wider.
Displaying 16x9 Video
When SD or HD 16x9 video is viewed on television, the aspect ratio and options of the TV itself all affect how the image is displayed.
As can be seen from the above illustrations, when 16x9 video is displayed on a 4x3 monitor, it can be shown "correctly" when the video is letterboxed, in other words displayed with black bars at the top and bottom of the frame. If the video is displayed incorrectly, the image looks squeezed (tall and thin).
For display on 4x3 television, 16x9 video is letterboxed to display correctly. Although the letterboxing of 16x9 source material can be done automatically on some professional monitors and is an option on some newer televisions, a 16x9 program will be letterboxed in advance to ensure the program is seen in its correct proportions when viewed. The letterboxed video is actually a 4x3 overall frame with the 16x9 video placed inside.
U.S. Widescreen Films Shown On 16x9 TV
Notice that 1.78:1 (16x9) is very close to the U.S. theatrical standard of 1.85:1. The 1.85:1 aspect ratio is slightly wider. Current mass produced DVD's (as well as 1.85:1 films converted for HDTV) tend to scale these 1.85:1 formats slightly to fill the entire 16x9 frame, slightly cropping the original image. Although the frame isn't the same in theatres, it’s a much better compromise than the days of 4x3 TVs where the frame is substantially cropped and re-framed to fill the 4x3 frame, losing substantial amounts of the original image. On the other hand, U.S. widescreen films using the 2.35:1 aspect ratio are generally shown on DVD's and HDTV with a slight letterbox to preserve the aspect ratio of the original film.
Displaying 4x3 Video
When a 4x3 image is displayed on a 16x9 monitor, then the resulting image is "pillarboxed". This means that to display the image in its correct proportions, the image has black bars along the left and right sides of the frame.

Displaying Letterboxed Program In 4x3 And 16x9
If the 16x9 video program has been mastered in a 4x3 video format, then the black bars at the top and bottom of the frame are a part of the final image, and will always be there. For display on a 4x3 TV, you will get an image that is letterboxed as we looked at earlier.
On a 16x9 TV, the program view is definitely less than ideal. Since the actual frame of the video is 4x3, it will be pillarboxed on a 16x9 display (bars on the left and right). Because it was also letterboxed, there are also black bars embedded on the top and bottom. So the final display is both letterboxed and pillarboxed!
Here's a tip for creating a DVD of 16x9 content so that it shows up letterboxed on a 4x3 TV, and shows up full frame on a 16x9 TV. Instead of letterboxing the show in advance, create a 16x9 output of the show and create a 16x9 DVD. Standard 4x3 TVs will properly show this content as letterboxed, and of course 16x9 TVs will display a full frame 16x9 image.
Anamorphic
The term anamorphic is used with both film and video acquisition. It is a way of storing widescreen images on to a square like frame, in other words, "squeezing" the image horizontally. On display, the image is un-squeezed.
Widescreen Anamorphic Film
When working with 35mm film, the term anamorphic is also used, but its not quite the same as anamorphic video. For theatrical production in the U.S., there are two main aspect ratios used:

The 1.85:1 aspect ratio is filmed using spherical lenses, and there is no distortion or squeezing of the image. The wider of these two formats, 2.35:1, is often referred to as "Scope" referring to the old Cinemascope widescreen format. A 2.35:1 aspect ratio on film can be achieved using one of two techniques: Super 35 or Anamorphic.
For anamorphic filming:
- Anamorphic lenses are used.
- The image is "squeezed" on to the 35mm frame.
For display in theatres (from a 35mm print): Projector lenses are used to un-squeeze the image.

Unlike video, where anamorphic recordings are used to save on bandwidth by reducing the size of the frame for recording, in film, anamorphic lenses are used to make use of more of the full 35mm negative.

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