Example
Using the above information we can calculate, for example, the field of view of a full frame fisheye lens designed for 35mm use when used on an APS-C camera. Lets take the example of a 15mm fisheye lens. Let's assume it uses equisolid angle projection, so the FOV is given by 4 * arcsin (frame size/(focal length * 4)).
For a 24 x36mm frame this gives a horizontal FOV of 147.5degrees, a vertical FOV of 94.3 degrees and a diagonal FOV of 185 degrees. Canon give numbers of 142, 92 and 180 for their 15/2.8 fisheye lens, so the mapping isn't exactly equisolid angle, but it's a typical full frame fisheye with approximately 180 degrees diagonal coverage
For a 22.7 x 15.1mm sensor (APS-C) the numbers become: Horizontal FOV = 88.9 degrees, Vertical FOV = 58.3 degrees, diagonal FOV = 108.1degrees. If you "defish" a fisheye image, i.e. convert the image to rectilinear mapping, you keep the horizontal and vertical FOV, stretch the edges of the image and reduce the diagonal FOV. So if you "defished" the image you'd get an image with approximately and 88 degree horizontal FOV and a 58 degree vertical FOV. This corresponds to the horizontal FOV of a 19mm lens and the vertical FOV of a 22mm lens. How is this possible? Well the 1:1.5 ratio of vertical to horizontal if the APS-C sensor is changed when the image is "defished" and becomes closer to 1:1.7
