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Anamorphic filming technique was developed to make widescreen movies using 4:3 film. An anamorphic lens distorts the image picked up by the camera before it reaches the film. By using a similar lens when projecting the film back on screen, the correct, intended aspect ratio is restored.
Similar technique is used to store 16:9 video material at 4:3 aspect ratios. Anamorphic 16:9 video is common in DVD movies and DVB broadcasts.
When movies with an aspect ratio above 1.80:1 (1.78:1 to be exact) are converted to anamorphic widescreen format, they need to be letterboxed to maintain a correct aspect ratio without cropping.
For a visual, easy to understand explanation of anamorphic video, please see the World's Easiest Explanation of Anamorphic 16:9 WidescreenEnhancement in DVDs.