Maybe the following will clear some of this up (maybe it won't).

Most lenses are designed to image straight lines as straight lines on film, wherever they might appear. So generally, 'distortion' (ie, the imaging of straight lines as slightly bowed on the film) is supposed to be kept to a minimum. Most SLR wideangle (and other) lenses cannot do this perfectly, so compromises come into play. However, for most purposes they do a good job and we don't object (too much) to the slight curves we see, mostly near the edges, when we look at pictures of objects we know are blessed with straight edges in real life. These are normal, or 'rectilinear' wideangles. Even if we had a lens with essentially no 'distortion' in this technical sense, we would still perceive an imaging distortion in wideangles which causes close-up portraits to look hideous, causes round objects to look like ovals in the corners of pictures, and causes the members of a group photo who stand at the ends to look like prime candidates for Weight Watchers. These types of distortion are unavoidable due to the fact that we are trying to reproduce an image of the three dimensional world on a two dimensional surface, and then are trying to view the picture from the wrong distance (for correct perspective, and no distortion, we should view the picture from a distance so that the photo we are looking at covers the same angle of view that the taking lens did). Anyways, the lens is doing the job it was designed to do.

I won't go into the math here, but it can be easily shown that a rectilinear wideangle cannot cover 180 degrees. Wideangles with focal lengths equivalent to 10mm or so on 35mm film have been produced for large format, and Nikon makes a 13mm lens. This is about the maximum for rectilinear wideangles.

To make a single picture which includes, side-to-side, 180 degrees or more, a different lens or camera design is needed. One is the swing-lens camera, which in some forms can take a single picture encompassing 360 degrees (or more), and which produces pictures that many people consider the most distortion free extreme wideangle. But it cannot take instantaneous photos, ie, not all of the image gets exposed at the same time, and it cannot show 180 degrees vertically and horizontally at the same time. If you want that, you need a fisheye.

Most fisheyes produce their very wide angles of view by making the location of an image point directly proportional to the angle that the object is away from the optical axis of the lens. So... If an object is straight in front of you, and you point the fisheye lens directly at the object, the image of the object will be in the center of the film frame. With a 180 degree fisheye, if an object is at 45 degrees off to the side, to the top, bottom, etc., the image will appear 1/2 way to the edge of the image on the film. All fisheyes produce circular images, whether the film format, say 35mm, is large enough to cover the entire image (7.5 or 8mm lenses commonly), or can only use the middle, so that the edges of the cirle and the corners of the frame coincide (15 to 18mm lenses, usually). The above also applies to fisheyes that cover 220 degrees (6mm) and 170 degrees (the old 16/3.5 MF lens). All use the 'equidistant' projection to produce their image. One lens that was different was the Nikon 10mm OP lens, where OP stands for orthographic projection. The primary aim in the design of this lens was to produce a whole-sky image where the image density was directly proportional to the object light value. In other words, no light falloff, ever, under any circumstances. Other lenses, of whatever type, just do not do that. While regular, equidistant fisheyes produce their images through the formula Y = c * (Zenith angle), where the zenith is considered the lens axis, orthographic projection lenses use a formula Y = c * sin(Zenith angle), where Y is the distance of the image point from the picture center and c is a constant, dependant on the focal length. All fisheyes, by definition, curve lines that do not run through the zenith (lens axis), but the OP produces a more irregular pattern of curves.