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Say an ordinary lens, Sigma 30mm f1.4 for example, is designed for APS-C and thus produces images that have a 35mm-equivalence of 48mm.

But what about the fisheye lens?

From Wikipedia, it is said that for fisheye lenses:

The focal length is determined by the angular coverage, the specific mapping function used, and the required dimensions of the final image.

Also in the article, it is stated that a Circular, Stereographic fisheye lens that has a 4.2mm focal length for APS-C would be 6mm for a full-frame sensor.

So, what does the focal length mean for a fisheye lens? Does it have anything to do with the distance from the optical center to the sensor, or is it only a reference to the field of view this lens can provide assuming full sensor coverage?

Edit: As pointed out in the comment, the optical center is not the determinant for focal length. I guess a better-phrased question would be "Does the focal length on a fisheye lens have anything to do with its rectilinear counterpart?"

Amarth Gûl
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  • Even for ordinary lenses the focal length is determined by the angle of view, not by the distance to the sensor. DSLRs tend to have a flange to sensor distance of a little over 40mm, so shorter lenses must use a reverse telephoto system to get the image to focus on the sensor. – Ross Millikan Apr 27 '22 at 16:33
  • "The specific mapping function" for a fisheye lens is what defines it as a fisheye lens and a rectilinear mapping function is what defines your "ordinary" lens as an "ordinary" lens. – Michael C Apr 28 '22 at 03:51
  • @RossMillikan Even with a retrofocus lens, the rear nodal point is one focal length in front of the focal plane at infinity focus. The rear nodal point of a compound lens can be in front of the front of the lens (telephoto), somewhere inside (more or less "normal" if near the front element), or fully behind the flange (retrofocus), depending on the lens' design. – Michael C Apr 28 '22 at 03:54
  • "Also in the article, it is stated that a Circular, Stereographic fisheye lens that has a 4.2mm focal length for APS-C would be 6mm for a full-frame sensor." That's categorically incorrect. What size the sensor behind a lens is has absolutely no influence on that lens' focal length. It only affects the angle of view. But focal length is not defined by angle of view. Focal length is defined by how far behind a lens' rear nodal point collimated light entering the lens is focused behind the lens. A 4.2mm on a FF camera is still a 4.2mm lens on an APS-C camera. The AoV would be similar to a 6mm... – Michael C Apr 28 '22 at 03:57
  • ... lens on a FF camera, but the lens is still a 4.2mm lens when used on an APS-C camera. – Michael C Apr 28 '22 at 03:58
  • From the Wiki article: "Assumes 180° maximum angle of view for the mapping function, {\displaystyle \theta ={\frac {\pi }{2}}}{\displaystyle \theta ={\frac {\pi }{2}}}" In other words, it's not really expressing focal length of the lens, it's expressing angle of view and assigning a "focal length" that would give a rectilinear lens the same AoV. – Michael C Apr 28 '22 at 04:05
  • This idea of "circular" and "diagonal" lenses as distinct designs is absurd. The same lens can project an image circle large enough to give a "diagonal" image on that small sensor camera and the same circular image will only cover the center of the sensor in a camera with a much larger sensor. – Michael C Apr 28 '22 at 04:09
  • Does this answer your question? How is Focal Length defined for 180-degree fisheye lenses? – inkista May 01 '22 at 04:18

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