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04-28-2010, 10:17 AM   #1
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plots of focal length vs. field of view

There has been some discussion of how focal length relates to field of view, so I thought I'd get out Python and make a chart or two. Here's the process I used, so people can catch any mistakes I may have made.

First, for a rectilinear lens, the equation relating focal length (f) to angle of view (a) is well known. The only other parameter it depends on is the image dimension (d), meaning the linear size of the film or sensor. We can choose this to be width, height or diagonal -- I have produced charts with all three.

The formula for the angle in radians is:
a = 2 arctan(d/2f)

(Note that f is technically the "effective focal length", but unless we are doing macro photography this is the same as the focal length stated on the lens.)

For the Pentax APS-C sensor we have these dimensions in millimetres:
w = 23.6
h = 15.7
d = 28.34

Lastly, we can convert radians to degrees by multiplying by 360 / pi.

Using the math library in Python to get the atan function, this reduces to these three equations:
a_w = 114.6 * math.atan(11.8 / f)
a_h = 114.6 * math.atan( 7.85 / f)
a_d = 114.6 * math.atan(14.17 / f)

I plotted them on two scales, for convenience.







I hope this is of use to someone.

04-28-2010, 10:48 AM   #2
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And here are the focal length vs. angle of view plots.




04-28-2010, 11:52 AM   #3
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Thanks, Robin--this is really helpful in getting an important concept through my thick skull.
04-29-2010, 12:12 AM   #4
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Thanks, this is interesting. If it's not too much of a problem it would be great to have the same charts made for all pentax formats, i.e. 35mm, 645 and 67. So that one can compare between formats.

PS. What size is the 110 film frame?

04-29-2010, 05:56 AM   #5
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QuoteOriginally posted by Jimfear Quote
Thanks, this is interesting. If it's not too much of a problem it would be great to have the same charts made for all pentax formats, i.e. 35mm, 645 and 67. So that one can compare between formats.

PS. What size is the 110 film frame?
110 film was 17mm x 13mm.

Not sure about producing more of these as they take a while to make pretty. But will assess demand.
04-29-2010, 03:02 PM   #6
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What about a double-logarithmic plot, like this? (a quick shot of mine, ugly scales)

Last edited by falconeye; 06-15-2011 at 05:29 AM.
04-29-2010, 05:40 PM   #7
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QuoteOriginally posted by falconeye Quote
What about a double-logarithmic plot, like this? (a quick shot of mine, ugly scales)

Arithmetic I can handle: logarithms, not so much.
04-30-2010, 06:49 AM   #8
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QuoteOriginally posted by Jimfear Quote
If it's not too much of a problem it would be great to have the same charts made for all pentax formats, i.e. 35mm, 645 and 67. So that one can compare between formats.
I once did the following graph that compares FF (35mm) to APS-C (using the sensor diagonal).

Name:  AoV-FF-APS-C.png
Views: 9993
Size:  9.7 KB

The graph nicely shows that the difference in the AOV for the two formats becomes less and less significant towards longer focal lengths.

What 645 format are you interested in? The original film dimensions or the new 645D crop format?


Last edited by Class A; 04-30-2010 at 07:03 AM.
04-30-2010, 07:30 AM   #9
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QuoteOriginally posted by falconeye Quote
What about a double-logarithmic plot, like this?
Once again we think alike! I was hoping to do a log plot myself but my charting library is too limited. Though I can twist the equation into log form myself, it will not display the axis ticks -- rather necessary to read the thing.

Must find a better library someday.

P.S. Axis going to 1000 degrees... now that is optimism!
04-30-2010, 08:21 AM   #10
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QuoteOriginally posted by rparmar Quote
P.S. Axis going to 1000 degrees... now that is optimism!
180 and some smart software genius thought it may be ok to round this to 1000 Cannot be changed in my package but I have another package which can do it if it were that important ...


But something else ...

In a log-linear plot where the focal length axis is logarithmic but the angle scale is still linear, we get an S-shaped curve. And the inflexion point universally is at f=d/2 excatly. I didn't know that and found it an interesting discovery.

So, for a 23.6mm image width, the "most interesting" focal length is 11.8mm, let's say 12mm So, the mathematically inclined may not call it UWA yet
04-30-2010, 11:23 PM   #11
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QuoteOriginally posted by falconeye Quote
And the inflexion point universally is at f=d/2 exactly.
That's when the AOV becomes 90.

QuoteOriginally posted by falconeye Quote
So, the mathematically inclined may not call it UWA yet
That's right. 90 is not quite obtuse yet.

Curious that the AOV for a "normal focal length" is 53.13. I know it is 2*ATAN(1/2) but it seems like an odd number to have that significance.
04-30-2010, 11:40 PM   #12
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It would be interesting to see the AOV juxtaposed with full frame 645 and the 1.3 reduced format the pentax 645D is most likely to be using.
05-01-2010, 12:25 AM   #13
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QuoteOriginally posted by Digitalis Quote
It would be interesting to see the AOV juxtaposed with full frame 645 and the 1.3 reduced format the pentax 645D is most likely to be using.
According to the information I found the original 6x4.5 format is 56mm x 41.5mm. The 645D uses a 44mm x 33mm sensor. This results in a crop factor of 1.27 and the following graph:
Name:  645.png
Views: 2150
Size:  10.0 KB

Hope that helps.
05-01-2010, 01:13 AM   #14
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Another observation:
The difference between the AOVs for two different formats (for a given focal length) first rises with the focal length and then falls again. The maximum AOV difference for two formats, where d1/d2=c (c == crop factor) is at the focal length that yields the AOV 2*ATAN(SQRT(C)) for the larger format with diagonal d1 and the AOV 2*ATAN(1/SQRT(C)) for the smaller format with diagonal d2.
The mean of these two AOVs is 90.

Last edited by Class A; 05-01-2010 at 01:26 AM.
05-01-2010, 02:12 AM   #15
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QuoteOriginally posted by Class A Quote
Another observation:
The difference between the AOVs for two different formats (for a given focal length) first rises with the focal length and then falls again. The maximum AOV difference for two formats, where d1/d2=c (c == crop factor) is at the focal length that yields the AOV 2*ATAN(SQRT(C)) for the larger format with diagonal d1 and the AOV 2*ATAN(1/SQRT(C)) for the smaller format with diagonal d2.
The mean of these two AOVs is 90.
In a linear-log plot (cf. my post above), this is the exact middle between the two inflexion points
Also, in such a plot, the multitude of curves for different sensor sizes are just shifted left<->right against one another.

QuoteOriginally posted by Class A Quote
Curious that the AOV for a "normal focal length" is 53.13. I know it is 2*ATAN(1/2) but it seems like an odd number to have that significance.
I always thought it was arbitrarily defined to be 50mm focal length for the 35mm format. At least, this is what viewfinder magnification refers to.

This would be 46.79 diagonal AoV and 39.60 horizontal AoV.

Your definition would require d=f or f=43.27mm.

I know that the Pentax Limited series does indeed propose 43mm as the normal focal length. But does anybody know why everybody else including Pentax non-limited proposes 50mm? What is the history behind this?
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