# Crop factor and Short Focal Lengths



## unpopular (Jul 8, 2016)

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):




 

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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## Tim Tucker (Jul 8, 2016)

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.


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## john.margetts (Jul 8, 2016)

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?


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## Braineack (Jul 8, 2016)

I get this:


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## petrochemist (Jul 8, 2016)

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.


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## unpopular (Jul 8, 2016)

Tim Tucker said:


> 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.



john.margetts said:


> 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.



Braineack said:


> I get this:
> 
> View attachment 124516



I plotted that as well, it is correct. I am plotting the factor, not the focal length.


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## Braineack (Jul 8, 2016)

here's a really good formula:


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## john.margetts (Jul 8, 2016)

unpopular said:


> john.margetts said:
> 
> 
> > gging the formula' into?
> ...


That was not clear either from the words or the graphs so I asked.


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## unpopular (Jul 8, 2016)

john.margetts said:


> unpopular said:
> 
> 
> > john.margetts said:
> ...




No. Thats fine. I just wasn't sure what you needed clarification on. Sorry. I posted this late.


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## Bebulamar (Jul 10, 2016)

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


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## unpopular (Jul 10, 2016)

Bebulamar said:


> 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.


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## Bebulamar (Jul 10, 2016)

What the FOV measured with? Angle in degrees? If so then it won't be linear.


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## Bebulamar (Jul 10, 2016)

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.


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## unpopular (Jul 10, 2016)

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.


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## unpopular (Jul 10, 2016)

Bebulamar said:


> 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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## Bebulamar (Jul 10, 2016)

What I am saying is that the angle is not directly proportional to the crop factor. The linear dimension is. For example when you look thru the viewfinder and a length of say 10ft. fill your frame. If you change the FOV so that only 5ft. fill your frame the FOV in angle isn't half but is greater than half.


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## john.margetts (Jul 11, 2016)

The joys of geometry!  The crop factor will be working on the tangent of the angle of view, not the angle itself (might be sine or cosine - it's 45 years since I learnt my geometry).


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## Bebulamar (Jul 11, 2016)

unpopular said:


> 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.



It does. The 100mm on FF has the same FOV as the 50mm on 4/3. Just that the 50mm on FF has more than twice the FOV in angle as the 100mm on FF.


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## unpopular (Jul 11, 2016)

Bebulamar said:


> unpopular said:
> 
> 
> > Well. Exactly. That's my point.
> ...



50mm is a bad example. The FOV factor starts to correlate with the crop factor at this point, not precisely, but fairly close.

Lets look at 20mm instead. Using the Arctan FOV formula for 4/3 an ideal 20mm lens will have a an equivalent FOV of a 33.32mm lens on FF sensor, which depending on who you're talking to is pretty significantly different from 40mm.


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## unpopular (Jul 11, 2016)

john.margetts said:


> The joys of geometry!  The crop factor will be working on the tangent of the angle of view, not the angle itself (might be sine or cosine - it's 45 years since I learnt my geometry).



Well that sure makes things less complicated, doesn't it?


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## Bebulamar (Jul 12, 2016)

unpopular said:


> Bebulamar said:
> 
> 
> > unpopular said:
> ...



The 20mm lens on 4/3 has the FOV of 56.82 Degrees and the 40mm lens on FF has the FOV of 56.81 Degrees. You must make an error somewhere in the calculation.


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## unpopular (Jul 12, 2016)

@Bebulamar - you're missing the point. What I am saying is that a 20mm 4/3 lens will have a similar FOV of a 33mm FF lens, while the crop factor of 2.0 predicts it will be more similar to a 40mm lens.

I am saying that for shorter focal lengths, the crop factor is not accurate when estimating FOV.


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## Bebulamar (Jul 12, 2016)

unpopular said:


> @Bebulamar - you're missing the point. What I am saying is that a 20mm 4/3 lens will have a similar FOV of a 33mm FF lens, while the crop factor of 2.0 predicts it will be more similar to a 40mm lens.
> 
> I am saying that for shorter focal lengths, the crop factor is not accurate when estimating FOV.



That what I said 33mm on FF the FOV is 66.49 Degree not the same as 20mm on 4/3 which is 56.82 Degrees which is the same as 40mm on FF with the FOV of 56.81 Degrees.


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## unpopular (Jul 12, 2016)

Bebulamar said:


> unpopular said:
> 
> 
> > @Bebulamar - you're missing the point. What I am saying is that a 20mm 4/3 lens will have a similar FOV of a 33mm FF lens, while the crop factor of 2.0 predicts it will be more similar to a 40mm lens.
> ...



What formula are you using to calculate FOV?


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## Bebulamar (Jul 12, 2016)

Your formula.
FOV=2*Atan( Diagonal of Sensor / (2* Focal Length))


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## table1349 (Jul 12, 2016)




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## unpopular (Jul 12, 2016)

@Bebulamar  yep. you're right. I'm too tired to try and figure what I did wrong. But obviously I did something wrong.


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## Bebulamar (Jul 13, 2016)

So the crop factor is correct. The crop factor can not be applied to the FOV in angle but only to the FOV in linear dimension.


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