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I'm trying to understand the physics of circular fish eye lenses. As far as I can see circular fish eye lenses create an circular image because the area of the sensor is larger than the projected image. But I need to validate my statement.

Can anybody explain why the circular fish eye lenses cast a circular image?

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All lenses create a circular image, it's just that most of them have an image circle large enough that it covers the entire sensor. Vignetting at wide apertures is a manifestation of the image circle encroaching on the corners of the sensor as the circle edge is not as sharp as it would be with a narrower aperture.

With a fish-eye lens, the image circle is restricted by the field of view of the lens, which tops out at about 180°. Once you get wide enough it's not possible to make it larger than the sensor - to do so would mean parts of the image are from behind the camera, which is physically impossible.

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    In theory I think you could install a ring of prisms partially overlapping the outer edge of the lens to capture light from behind and divert it into it. IN practice it sounds like something that only Mr. Goldberg would think was a good idea. – Dan Neely Jul 13 '14 at 21:14
  • I think you could have a secondary lens behind the fisheye that could magnify the image circle to cover the entire sensor, but why put a teleconverter on an 8mm lens to transform it into a 16mm lens when you can just build a 16mm lens much cheaper? – Michael C Jul 14 '14 at 0:42
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    @DanNeely It can be done without prisms: kenrockwell.com/nikon/8mm.htm 220 degree field of view. 6-figure pricetag. – Loren Pechtel Jul 14 '14 at 3:41
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Consider this review of the Canon 8-15 f/4L USM fisheye, which can shift from circular at 8mm to diagonal at 15mm. Yes, it's circular because the lens's entire image circle is inside the area of the sensor, rather than covering the entire sensor. I'm not sure there's going to be an entire explanatory webpage other than Wikipedia on this, because it's such a basic fact of fisheyes--most of the time you'll just see a comparison between circular and diagonal.

I have a full-frame Sigma 8mm circular fisheye. On my full-frame sensor, I get a circular image. If I mount it on my crop-body camera, it looks much like a diagonal fisheye with dark corners. The circular image is used to gain a much larger field-of-view (say 180° HFoV) than a diagonal could typically encompass.

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    This review has some easy to understand diagrams that demonstrates what is going on as you zoom the EF 8-15mm. It also has extensive mouse over photos that show the different focal length and sensor size combinations. IMHO it is a little more explanatory than the link you cited. Feel free to insert it into your answer if you wish. the-digital-picture.com/Reviews/… – Michael C Jul 14 '14 at 1:35
  • D'oh! Of course the-digital-picture review would be visually better. Thanks. Have updated the link. – inkista Jul 14 '14 at 2:00
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The answers above are correct but another way of looking at it is as mentioned: all lenses create a circular image. Most are designed so that the diameter of the image circle is greater than the diagonal of the sensor otherwise you are not filling the whole sensor.

Now fish eye lenses are designed to cover 180 degrees in all directions so to make the whole 180 degrees fit on the sensor the circle must cover no more than the shortest dimension of the sensor, which is top to bottom. So, you have a circular image that reaches from top to bottom of the sensor but not to the sides or corners.

One more part of the explanation: if the circle covered the entire sensor it would only cover the whole 180 degrees on the diagonals and not in any other direction.

OK did this help or did I get both of us hopelessly confused?

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