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Anywhere in the web it's stated that the image circle diameter depends on the lens design. I'm curious to understand it a bit better.

Let's start with basics. Let's consider a single thin lens creating an image starting from the input rays emitted by a point-source.

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Will this lens create a circular image if there are more emitting point-sources? Is its finite size the reason, and is its physical diameter the same of the image circle diameter?

In case of a point-source object, if it is moved upwards a lot in the previous picture, then less of its rays will enter the lens. But there will always be at least one ray entering the lens. Why should the image be cropped (and why be cropped circularly)?

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Ultimately, the question, "why is the image circle, circular?" is because lenses are round. The vast majority of elements in lenses are circular because the lens surfaces are ground to a truncated spherical profile. This is because the lens maker equations dictate a specific radius of curvature of the lens elements.

Even for lens elements that aren't truncated spheres, so-called aspheric elements, they are still surfaces of revolution about the optical axis. Their off-axis (optical axis) refractive properties are radially symmetric around the center of the image.

Now, regarding the actual diameter of the image circle, let's first stipulate that the image circle diameter scales with the actual size of the lens. That is, for a given lens of focal length F, if you scale up the lens by a factor of say, 2, the lens's diameter and thickness will be twice as large. So will the focal length of the new lens, with a focal length of 2・F. And also, the image circle diameter will be twice as large.

But what determines that diameter? Well, it's a bit arbitrary. The lens manufacturer wants to design a lens that projects the best possible image across the entire image area. That means they want the least amount of optical distortion from center to edge of the image circle. The center of lenses are almost always the sharpest. The edges of the image area show the most optical aberrations. At a certain point, the distortions and aberrations are unacceptable, and anything beyond that is unacceptable. So why not just call that limit the image circle?

Now, manufacturers don't quite approach lens design that way. Firstly, they design lenses to cover whatever film or sensor size the lens is being made for. So for 35mm format cameras, the lenses must cover the area of the 35mm sensor, which has a diameter of about 43.26 mm. Thus, the image circles of their lenses are at least 42.26 mm. Some of those lenses might project a much larger image circle, but it's immaterial in the context of being mounted on a 35mm camera. This is a similar case with lenses designed for 35mm full frame bodies mounted on APS-C (1.5 to 1.6 crop) camera bodies that accept them. Even though the lens could cover an entire 24mm × 36mm frame of the 35mm format, the body's sensor only cares about the central 16mm × 24mm of the image circle.

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