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The definitions of Focal Length I saw seem to implicitly assume that the field of view is finite, which isn't true for 180-degree lenses, or presume parallel transport of certain rays passing through lenses, which is also not applicable. Thus I'm curious how Focal Length is defined for fisheyes.

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    The FoV for a 180° lens IS finite, even if the result of R = f tanΘ when Θ=180° is not. Collimated light (parallel rays striking the lens from a singular point source from a distance of infinity) can pass through a 180° fisheye lens and converge to a point behind the lens at a distance equal to the lens' focal length. – Michael C Oct 26 '17 at 20:20
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Unlike "normal" (rectilinear) lens, there are many types of fisheye lenses, using different projections. The same focal length (f) can give different image size and/or different distortions, depending on the type.

Example:

Assuming your 8mm lens uses the most basic equidistant fisheye (180° degrees field of view = -90°..+90°)

R = f · θ

8mm · π/2 ≈ 12.56mm

So, rays from -90°..+90° map to -12.56...+12.56mm on the image sensor or film, creating a 25mm circle. From calculation we can learn that such lens would not cover the 36x24mm full frame image area.

Fisheye projections from https://wiki.panotools.org/Fisheye_Projection

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The parallel rays (collimated light) referenced in your question that are used to determine focal length are rays that all originate from a single point source of light at a distance of infinity from the camera, not from all of the infinite number of different points within the lens' field of view. This is just as true for a very long focal length telephoto lens as it is for a very wide angle lens.

For a 'perfect' lens, collimated rays from any point source at a distance of infinity within the field of view will converge at a distance behind the lens equal to the focal length of the lens. Thus, focal length is the distance behind the lens at which collimated light striking the lens will converge.

For compound lenses (lenses with more than one lens element with real thickness - pretty much every modern photographic lens), the distance a theoretical thin lens having the same refractive properties would need to be in front of the focal plane for collimated rays striking that lens to converge to a point is used as the lens' focal length.

No lens is perfect, though. Most of the time rays from the edge of a lens' FoV don't converge as sharply as rays from the center of the optical axis do. This can be particularly the case with wide angle lenses that attempt a rectilinear projection. Fisheye lenses, which can have one of several different projections, give up a rectilinear projection that preserves straight lines in exchange for fewer aberrations at the edges of the lens' field of view.

Infinity does not refer to the lens' field of view (the angle between the camera and two objects that a lens can show in opposite corners), it refers to a distance away from the camera's focal plane at which light from a single point source will strike every point on the surface of the lens as collimated light (parallel rays) and converge back to a point at a distance behind the lens equal to the lens' focal length.

It seems you are confusing ray diagrams that show rays of light from a single point source striking every part of the front of the lens with diagrams that show the range of angles with which light can strike a lens and be included in the image it projects at the rear of the lens. The two types of diagrams are showing entirely different properties of a lens. The first shows various rays from a single point source of light that (hopefully) converge back to a single point behind the lens. The second shows the range of angles from which light can strike a lens and be included in the lens' field of view.

  • I wasn't talking about imperfections of real lenses by rather about the inconsistency of the usual definitions of focal length f when applied to 180 degree fisheye lenses. Specifically, the f for usual lenses can be derived from their projection formula, R = f tan theta, which makes no sense when theta = 90. As @szulat explained in his answer, the projection formulas for fisheye lenses are different from the usual lenses. – Michael Oct 26 '17 at 20:09
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    @Michael This answer is not primarily about the imperfections of real lenses. It is about the way terms used in the question are properly defined. If the result of a formula is an irrational number when a specific angle is used, that does not mean that the angle used to get that result equals infinity. That's the main point of this answer. A lens with a 180° FoV does not have an infinite FoV. Also, any lens with a FoV of any angle can be focused at infinity if it can converge collimated light to a point at a specific distance behind the lens. – Michael C Oct 26 '17 at 20:14
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To mount an extreme wide-angle, the optician will likely choose a retro-focus design. You might want to know that the focal length is a measurement of the distance from the rear nodal of the lens to the film or digital image sensor, when an object at infinity is sharply defined.

To make a wide-angle, the focal length is short thus the distance lens-to-film/sensor is contracted. Now the optician is forced to make a retro-focus lens.

A retro-focus design is actually an inverted telephoto. This strategy creates a longer than normal back focus distance (lens-to-sensor). In other words, the retro-focus design shifts the lens barrel forward allowing more room for mirror swing and mounting hardware.

The retro-focus design is accomplished by grinding the up-front lens group concave, this gives negative power i.e. divergent lenses. The rear grouping is ground strongly positive (convergent). The two groups work together to deliver a wide-angle with an elongated back focuses distance. This asymmetry causes barrel distortion. The good news is, an extreme wide-angle can now be accommodated.

  • Thanks for the description of the optical system design, but could you add a definition of what Focal Length is that would be applicable to extreme wide angle lenses? For example, what 8 mm and 15 mm refers to in the description of Cannon 8-15mm Zoom Fisheye lens that has 180 degree view angle? – Michael Oct 26 '17 at 18:56
  • @Michael The 8-15mm Fisheye is a zoom lens, It can only be at a single focal length between 8mm and 15mm at any given time, depending on the position of the various lens elements based on the position of the zoom ring. – Michael C Oct 26 '17 at 19:51
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    A lens with extreme angle of coverage. The geometry of a conventional lens allows delivery of undistorted image up to about 120°. For 180° coverage, extreme retro-focus design with barrel distortion permitted. Two versions a. quasi-fisheye fills the format and delivers 180° diagonal angle of view. True fisheye delivers circular image that does not fill the frame. Focal length = rear nodal to image. Fisheye is an equidistant projector where image height = focal length multiplied by the angle of coverage. Convention lens image height = focal length multiplied by tan of the angel. – Alan Marcus Oct 26 '17 at 20:04

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