In my previous post I described how easy it is to calculate apertures to get equivalent depth of field on different formats. I presented there all the equations needed for DoF calculation, but left the actual “paperwork” to you.

In this post I’ll do these calculations for those of you who did not do the homework ;)

Our goal is to get the same amount of depth of field for two setups. Object distance is also the same, so we can simply work with hyperfocal distances.

$H\approx <!--\; \approx \; -->{\displaystyle \frac{f_F^2}{N_F{c}_F}}\approx <!--\; \approx \; -->{\displaystyle \frac{f_C^2}{N_C{c}_C}}$

Where the index $F$ denotes full frame and the $C$ index denotes crop sensor cameras.

We also know how the required full frame and crop factor focal lengths relate ($X$ denotes the crop factor), so:

${\displaystyle \frac{X^2{f}_C^2}{N_F{c}_F}}\approx <!--\; \approx \; -->{\displaystyle \frac{f_C^2}{N_C{c}_C}}$

Now let’s see how the circle of confusion changes with the format. Having the exact same print dimensions, magnification will be higher for smaller formats.

$m_C=X{m}_F$

Viewing distance and your eye’s resolution are also the same, so:

$c_C={\displaystyle \frac{\tan <!--\; \; -->(\frac{\mathrm{\pi <!--\; \pi \; -->}}{180R_e})D_v}{X{m}_F}}$

$X{c}_C={\displaystyle \frac{\tan <!--\; \; -->(\frac{\mathrm{\pi <!--\; \pi \; -->}}{180R_e})D_v}{m_F}}=c_F$

To summarize what we have:

${\displaystyle \frac{X^2{f}_C^2}{N_{F}X{c}_C}}\approx <!--\; \approx \; -->{\displaystyle \frac{f_C^2}{N_C{c}_C}}$

Which after simplification leads to:

${\displaystyle \frac{X}{N_F}}\approx <!--\; \approx \; -->{\displaystyle \frac{1}{N_C}}$

That is, we arrive to the result:

$X{N}_C\approx <!--\; \approx \; -->N_F\text{.}$