Crop factor and Short Focal Lengths

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Just for funzies, I took a look at the FOV formula and noticed something odd, that it's non-linear. This contrasts with the user-friendly notion of "crop factor" that we're all familiar with. So, I plugged in the formula using APS-C and 35mm and compared it against the standard crop factor of 1.6.

There is some discrepency, but it mostly settles out around 50mm. However, for wide focal lengths, the discrepency is more significant, and the problem is even larger for 4/3 (upper plot):

Screenshot (23).jpg


While this doesn't matter much at longer focal lengths, it'd seem that there is a pretty large discrepency in very short focal lengths, and at under 15mm the difference is pretty significant.

Is this right?
 

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But then the focal length marked on a lens is only an approximation, and it's only relevant at infinity focus. So is there really any point in exact maths (when you only deal with approximate values)?

Besides if you are using a smaller sensor came you could just look through the viewfinder? I've never understood why when using a crop sensor you need to know the exact focal length of the lens for the same FOV on a camera you're not using. ;)
 
So what is this you are 'plugging the formula' into?

The three graphs you present are too low resolution to be of any use. I was under the impression that the 'crop factor' applied to apparent focal length rather than FOV - or does apply to both?
 
In everyday use with rectilinear lenses the approximations are generally good enough. I've never tried measuring the FOV required to select the appropriate lens. A general feel for the angle tends to do the job quite well, perhaps aided by thoughts like 'I'd like the subject 5x bigger than I can get with this lens, so 1500mm would be ideal!' :)

The formula actually varies depending on the design of lens. Rectilinear lenses & sterographic fisheyes use (different) ArcTan functions, equisolid & orthogonal fisheye lenses use (different) ArcSin functions & equidistance type fisheye lenses are apparently linear. Part of my massive photography equipment list/spread sheet calculates FOV for them all at different focal lengths/frame sizes though I've never added graphs to it (yet).

The four types of classical fisheye design give very different FOV to focal length results, both compared to each other & to standard rectilinear designs. Real life lenses probably don't exactly conform to any of the classic calculations, they won't be perfectly rectilinear, and if a fisheye will generally not follow any of the classic types either.

I've never managed to find out which of the classic fisheye formula is the closest to what any of my fisheye lenses produce, I just know some are less useful/attractive than others.
 
But then the focal length marked on a lens is only an approximation, and it's only relevant at infinity focus. So is there really any point in exact maths (when you only deal with approximate values)?

Besides if you are using a smaller sensor came you could just look through the viewfinder? I've never understood why when using a crop sensor you need to know the exact focal length of the lens for the same FOV on a camera you're not using. ;)

I'm starting to agree. I always knew that the "crop factor" was approximate, but I am starting to really feel it's pretty useless.

gging the formula' into?

Uhm. I'm not sure I understand. x-axis is focal length, y-axis is is the relative AOV between APS-C (lower plot) or 4/3 (upper plot) and 35mm full frame.


I plotted that as well, it is correct. I am plotting the factor, not the focal length.
 
here's a really good formula:

pic_02.jpg
 
Can you clarify your graph. I don't understand.
 
Can you clarify your graph. I don't understand.

Basically they compare FOV of either APS-C or 4/3 with 35mm. So for an ideal rectilinear lens with a focal length of 15mm the FOV will be a factor or 1.3 on APS-C relative to 35mm. I did this to compare against the better known "crop" factor when estimating lens equivalency.

But as others had pointed out, neither is a particularly good estimate as every lens will behave uniquely.
 
What the FOV measured with? Angle in degrees? If so then it won't be linear.
 
I think I see where you went wrong. If the 50mm lens giving a FOV of 46 Degrees for 35mm then it's not 23 Degrees for the 4/3 format although the crop factor is 2.
 
Well. Exactly. That's my point.

Because when we're talking about focal length we're really discussing FOV there is a discrepancy. If a 50mm lens doesn't actually behave like a 100mm lens of 4/3, then approximation is inadequate.
 
What the FOV measured with? Angle in degrees? If so then it won't be linear.

It does not matter for this chart. If it were in degrees or radians the ratio will be the same.

I'm not sure if when I took this screen shot if I left it in degrees or radians, but regardless, the plot for focal length to FOV is non-linear using the rectilinear equation.
 

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