Telescopes form images of infinity-foused objects and collimators project objects to infinity. Sounds simple, is simple. Except when it comes to distortions. Telescopes and collimators create the strangest “distortions” (like keystoning), that really are projections.
A telescopes with focal length In the simplest case, the telescope is an f-theta lens and linearly translates field angles (1) (2) How Telescopes Form Images
takes an infinity-focused object of field angle 
and creates an image with height 
, like this:
to image heights dictated by the relation
and thus an angular magnification of 
, or 
.
The whole selling point of such f-theta optics is that it is distortion free. It’s angular magnification is constant over the entire field. However, when one images a perfect rectangular field of view, say, 100 ° x 80 °, using this distortion-free telescope, then the image looks anything but free of distortion. It looks like this:
The image looks barrel distorted. And in practice, it is often treated like a barrel distortion and approximately “distortion-corrected” with various dot charts, line charts or similar patterns. Such efforts tend to be more-or-less succesful, but if anything, they are misguided.
The catch, of course, that this seeming barrel distortion is not a distortion. It is a map projection of angular space (that of the field) onto Euclidean space (that of the image). And angular space and Euclidean space follow different metrics. In Euclidean space, a rectangular object is described by its width 
, its height 
and its diagonal 
follows the Euclidean norm
(3) 
and height 
has a diagonal 
described by
(4) 
A similar projection-effect can be observed in the counterpart of the telescopes, the collimators.
How Collimators Focus To Infinity
A collimator is really just a telescope in reverse. Instead of taking an infinity-focused object and focusing it to an image, it takes some object and collimates it to infinity. Often, telescopes and collimators are optically perfectly identical; collimators just have some object in the place where the telescopes have sensors. A collimator looks like this:
For an f-theta lens, a collimator of focal length projects an object of height to a field angle described by
(5) 
Just like with telescopes, the Euclidean space to angular space projection rules apply. So, to crate a perfectly rectangular field, the object has to take this projection into account. If it fails to do so, then the field will not be rectangular, it will look like this:
Combining Telescope & Collimator
The takeaway is that one can get perfectly fooled by the Euclidean-angular space projection. If one uses a telescopes to verify the quality of the field created by some collimator, then seeing a non-distorted image is precisely the image one does not wish to see. It must appear barrel distorted for the field to be rectangular. This image illustrates this seeming paradox:
Keystoning
The Euclidean-angular projection becomes perhaps most noticeable when the object is not perfectly aligned to the collimator optical axis and the telescope is slightly titled to make up for this slight misalignment. Then the projection will cause an effect reminiscent of keystoning found in ceiling-mounted projectors:
Just like with ceiling-mounted projectors, the presence of keystoning is a tell-tale sign of misalignment. Eliminating it is a way to properly align a telescope / projector pair.
Keystoning is a direct consequence of the Euclidean-angular projection. Without taking the Euclidean-angular projection into account, keystoning cannot be understood and will appear as some odd unexplained distortion.
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