What does the focal length of a fisheye lens mean?

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?"

• 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. Commented Apr 27, 2022 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. Commented Apr 28, 2022 at 3:51
• @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. Commented Apr 28, 2022 at 3:54
• "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... Commented Apr 28, 2022 at 3:57
• ... lens on a FF camera, but the lens is still a 4.2mm lens when used on an APS-C camera. Commented Apr 28, 2022 at 3:58