How to Calculate Crop Factor, F-Stops, and Equivalent Lenses in Google

Discussion in 'Open Gear Talk' started by Amin Sabet, Jul 29, 2012.

  1. Amin Sabet

    Amin Sabet Administrator

    Jul 3, 2010
    DPReview did something in their RX100 Preview that I haven't seen before from a major review site although a number of us have been doing it for quite a while now. They expressed both the focal length range as well as the relative aperture (f-number) range of various cameras in 135 format (35mm format) equivalent terms. For example, they list the 10-37mm f/1.8-4.9 lens on the Sony RX100 as equivalent in terms of angle of view and depth of field control to a 28-100mm f/4.9-13.4 lens for 135 format. Some also extend this idea of equivalence to include signal/noise, meaning that the RX100 would offer the same noise performance characteristics as a 135 format camera with a 28-100mm f/4.9-13.4 lens if that 135 format camera had the same sensor technology as the RX100.

    As time goes on, we have more and more formats to compare. For example, Fuji brought back 2/3" format with the X10, and the RX100 uses a 1" sensor, while the Canon S100 uses a 1/1.7" sensor and the Panasonic LX7 in 4:3 aspect ratio effectively uses a 1/1.8" sensor. Expressing focal length and relative aperture in terms of 135 format equivalents is very useful for comparing angle of view and depth of field possibilities across formats and gives a rough estimate of how well these systems will compare for handheld, low light photography if using similar sensor technology.

    format.


    The purpose of this post is to provide an easy method to calculate things like 135 format equivalent focal length and relative aperture using only the sensor dimensions, the information marked on the lens itself, and Google. If you are not interested in doing such calculations, then definitely skip the rest of this post!


    In order to calculate equivalent focal lengths and equivalent relative apertures, we need to know the crop factor for each format. Crop factor can be defined in either of the following two ways:

    1) Crop factor = the ratio of the diagonal dimensions of the respective formats

    2) Crop factor = the square root of the ratio of the areas of the respective formats.

    The above two definitions/methods will give the same results when calculating the crop factor for a format with a 3:2 aspect ratio but will differ when calculating for a format with an aspect ratio other than 3:2. For example, 4/3 has a 2.00X crop factor based on the ratio of diagonal format dimensions and a 1.96X crop factor based on the root of the sensor area ratio.

    Many of you realize that Google has a very convenient built-in calculator. For example, if you search for "2+2", the first search result will be "4". Below are a series of calculations which can be plugged directly into Google search* for deriving useful relationships between formats.

    *It should be noted that the ISO-standard for math notation says that "lg(x)" denotes the 10-logarithm of x, rather than the 2-logarithm. Google does it differently, by taking lg(x) as the 2-logarithm of x. The equations below will work for their intended purpose (doing calculations in Google), but mathematically they don't use commonly accepted notation.


    The following are a series of calculations based on the ratio of the diagonal dimensions of the two formats and are written in a form which can be copied directly to Google:


    Crop factor, where SW=sensor height (mm); SH=sensor width (mm):

    (sqrt((36^2)+(24^2)))/(sqrt((SW^2)+(SH^2)))


    Difference in stops between two formats, where SW1=sensor 1 width (mm); SH1=sensor 1 height (mm), SW2=sensor 2 width (mm); SH2=sensor 2 height (mm):

    2*lg((sqrt((SW1^2)+(SH1^2)))/(sqrt((SW2^2)+(SH2^2))))


    135 format equivalent focal length, where SW=sensor height (mm); SH=sensor width (mm); FL=focal length:

    ((sqrt((36^2)+(24^2)))/(sqrt((SW^2)+(SH^2))))*FL


    135 format equivalent relative aperture (equivalent f-number), where SW=sensor height (mm); SH=sensor width (mm); RA=relative aperture (f-number):

    ((sqrt((36^2)+(24^2)))/(sqrt((SW^2)+(SH^2))))*RA


    The following are a series of calculations based on the square root of the ratio of the areas of the two formats and are written in a form which can be copied directly to Google:


    Crop factor, where SW=sensor height (mm); SH=sensor width (mm):

    sqrt((36*24)/(SW*SH))


    Difference in stops between two formats, where SW1=sensor 1 width (mm); SH1=sensor 1 height (mm), SW2=sensor 2 width (mm); SH2=sensor 2 height (mm):

    2*lg(sqrt((SW1*SH1)/(SW2*SH2)))


    135 format equivalent focal length, where SW=sensor height (mm); SH=sensor width (mm); FL=focal length:

    (sqrt((36*24)/(SW*SH)))*FL


    135 format equivalent relative aperture (equivalent f-number), where SW=sensor height (mm); SH=sensor width (mm); RA=relative aperture (f-number):

    (sqrt((36*24)/(SW*SH)))*RA


    Using the above calculations in Google, it's easy to derive the following table

    2012-07-29_1734.


    Here are a few additional Googleable calculations which may be useful:


    Number of stops separating two f-numbers for a given format, where RA1=f-number 1; RA2=f-number 2:

    2*lg(RA1/RA2)


    F-number which is X stops slower than another given f-number (RA):

    ((sqrt(2))^(X)))*RA


    F-number which is X stops faster than another given f-number (RA):

    ((sqrt(2))^(-X)))*RA


    Special thanks to everyone in this thread (especially GideonW) for double checking my math and offering ways to improve this resource.
     
    • Like Like x 11
  2. serhan

    serhan SC All-Pro

    May 7, 2011
    NYC
    Lots of formulas with very good table:) I think you should post the table somewhere as a reference for future buyers.
     
    • Like Like x 1
  3. Andrewteee

    Andrewteee SC All-Pro

    Jul 8, 2010
    I think I'm getting a headache... Need to go take pictures...

    Amin, you're not analytical are you :wink:
     
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  4. gregorb

    gregorb New to SC

    6
    Sep 22, 2012
    Slovenia
    Gregor
    I thought you might be interested to know that I just included 35-mm equivalent aperture for all fixed lens cameras (2896 to be precise) on my site - Digital Camera Database.

    I'm calculating equivalent aperture from crop factor based on diagonal dimensions.

    Gregor
     
    • Like Like x 3
  5. bartjeej

    bartjeej SC Hall of Famer

    Nov 12, 2010
    bart
    yikes - that's a lot of work! Thanks for doing it for us!
     
  6. gregorb

    gregorb New to SC

    6
    Sep 22, 2012
    Slovenia
    Gregor
    It's actually done automatically. I just wrote a script that takes the aperture, extracts the numbers, multiplies them with crop factor and glues the whole thing together.
     
  7. bilzmale

    bilzmale Super Moderator Emeritus

    Jul 17, 2010
    Perth, Western Australia
    Bill Shinnick
    Thanks Amin and Gregor.
     
  8. Boid

    Boid SC All-Pro

    Dec 15, 2011
    Bangalore, India
    Rajiv
    I found this write-up about "equivalence" quite helpful -

    “Full Frame Equivalence” and Why It Doesn’t Matter @ Admiring Light

    Edit: This guy is totally and completely wrong about equivalence. When looked at in absolute quantities of light that reaches the sensor, the (35mm equivalent) f stop of say 2.8 in front of a crop sensor camera is a lie. My apologies, it took me a while to get it. This video does a fair job explaining it

    https://www.youtube.com/watch?v=DtDotqLx6nA&list=PLBE338967F8DB7F2A