I am aware there are multiple projection types, and a complicated relationship between imaging circle, focal length and how a crop will look like.

How to choose a fisheye that is useful on a full frame camera as a circular, and useful as a semi-circular (or with some cropping, rectangular) on APS-C without just looking like a broken ultrawide?

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    Do you have sample images to illustrate "useful as a semi-circular" and "just looking like a broken ultrawide"? – xiota Apr 9 '19 at 21:05
  • My (mis?)understanding is that fisheyes are supposed to compress the corners in a certain "correct" way, depending on the type of fisheye... so won't I be cutting the good crust off the loaf cropping the wrong kind of fisheye? – rackandboneman Apr 9 '19 at 21:45
  • True fisheyes don't really compress anything, corners or center. Rectilinear wide angle lenses expand corners. – Michael C Apr 10 '19 at 4:04
  • So the classic fisheye projections are kind of fractal-like? – rackandboneman Apr 10 '19 at 12:18
  • Classic fisheye projections are more or less the natural result of making a conventional lens with a very wide angle of view without altering the projection to make it rectilinear. – Michael C Apr 10 '19 at 17:58

The usual routine for selecting lenses doesn't really change with the type of lens.

  1. Read lens reviews.
  2. Evaluate sample images.
  3. Try out the lens yourself.
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Get the one that gives you the projection and angle of view that you want on APS-C. Using it on full frame will give you the same image in the center, with the additional vertical field of view, and whatever little additional horizontal the image circle provides.

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