Focal length and the size of the image circle are two UTTERLY SEPARATE issues. Not. Even. Related. Two, totally different issues, entirely.
Hmm. could you go over that middle part again? I'm afraid you lost me.. lol
There is the angle of view that a lens exhibits. A wide-angle has a wide angle of view. A telephoto lens has a narrow angle of view.
There is also the image circle's coverage; large-format lenses can literally "cover" a LARGE piece of film with light rays. Lenses designed for APS-C cameras "cover" only a small circular area.
This entire area, of lenses, focal lengths, angles of view, and image circle sizes, is fraught with misunderstanding and misconceptions.
Check this thread for more information, and some numbers.
Image Circle Coverage and Angle of View - how to calculate?
I only mentioned this because the OP posed a question and it had some erros in understanding in this part: " Those full frame lenses are designed to cast a wider circle of light than the APS-C sensor. This is why you get a narrower field of view. But what about a lens designed
for APS-C? The image circle it produces is more focused/smaller than the full frame lens."
I don't even want to get into this whole issue, but the idea that the image circle affects the angle of view is NOT correct, at all. Not correct at all.
I was just reading this thread because of my interest in the the focal reducer (Speed booster), however, there has to be a simpler way of describing the difference effected to the "perceived" focal length of lenses when used on FF or APS-C.
So I'll take a stab at this.
Let's disregard the purpose built APS-C lens for a minute.
Let's say you are using a FF lens on a FF, AND an APS-C sensored camera.
First of all. The image sensor in your camera is essentially fitted upside down (a very simplistic view I know but bear with me). That means that the light must at some point cross-over or invert itself due to lens properties. This cross-over point is what determines the FOCAL LENTH of a lens and roughly means that the cross-over point is "x"mm ahead of the sensor plane, be it film or chip.
Now in the case of the FF x FF setup, if you were to scribe a simple line-diagram of the light path from the edges of the sensor to a cross-over point (somewhere in the lens) and on-ward to whatever subject, you'd effectively determine a certain angle (or AOV).
Now, if you change only the size of the sensor in the diagram leaving all other points as is and then re-scribe the the lines from this smaller sensor's edges, you'd find that the focal length is effectively the same but the AOV has changed.
The reason therefore, for so-called APS-C lenses becomes more obvious when one considers the cost of producing a FF lens that can project light onto a large sensor, and then end up being used on a smaller APS-C sensor. There is a LOT of light that is going to "waste" on an APS-C sensor, hence it is unnecessary to manufacture such an expensive lens when you only need a smaller project area of light.
Using the APS-C crop factor is simply to give an indication of what lens would give a similar AOV performance should you switch to FF overnight. Or a marketing ploy to get you to aim for the bigger bucks of the FF body and lenses.
So, Derrel, I agree with you but I don't really agree. You are right, the image circle doesn't affect the angle of view but the size of the sensor does, and THAT is what makes APS-C lenses "longer".
And the APS-C lenses DOES NOT produce a more "focused" light circle. It produces exactly what a FF lens does, it's just looking through a smaller (and more importantly - CHEAPER) hole.