**Circle of Confusion** *n* (abbr.: COC) A group of photographers desperately trying to understand --> DOF.

This article is intended as a continuation of a

thread in the

*Beginner's Questions* subforum which asked about field of view and magnification but quickly was carried away with questions about DOF and COC. Warning: This article contains a lot of math. If you don't like formulas then please read on elsewhere.

My contentious hypothesis is this: When taking an image from the same point of view with different format cameras, using equivalent focal lengths (i. e. same angle of view) and equivalent apertures (i. e. same depth-of-field) then the images still won't be identical. Instead, the transition from perceived sharpness within DOF range to blur outside DOF range will be different---that is, smoother for the larger-format image. At close distance, this effect will be more pronounced that at or near infinity.

To discuss this hypothesis, I'd like to introduce a few formula symbols:

**f** = focal length

**g** = subject-to-lens distance

**b** = lens-to-image distance

**d** = focus distance

**k** = aperture number

**z** = diameter of the circle of confusion

**q** = quotient of the linear sizes of two frame formats

The simple thin-lens equation thus becomes:

[1]

**1/f = 1/g + 1/b**The focus distance is the distance from subject to image:

[2]

**d = g + b**Please note that [2] ignores the fact that real lenses are not "thin"; it assumes the nodal planes' distance to be zero.

Now we need two functions g = G(f, d) and b = B(f, d) which calculate g and b from f and d so that f, d, g, and b will meet both [1] and [2].

[3]

**g = G(f, d) = d/2 + sqrt(d*d/4 - d*f)**[4]

**b = B(f, d) = d/2 - sqrt(d*d/4 - d*f)**As you can see, for d > 4*f, these are the two real solutions to a quadratic equation. They hold for magnifications below 1:1; for larger magnifications, simply swap b and g. Please note that lenses cannot create images at focus distances shorter than 4*f.

In the thread mentioned above, some contributors have digged out a formula from the Internet which calculates the diameter z of the circle of confusion of an object at a distance d for a lens with focal length f, focused to the distance d[span style=\'font-size:8pt;line-height:100%\']o[/span]:

[5]

**z = (f * f * |d - d[span style=\'font-size:8pt;line-height:100%\']o[/span]|) / (k * d * (d[span style=\'font-size:8pt;line-height:100%\']o[/span] - f))**Common wisdom suggests that if one frame format is larger than another by a linear factor q then you'd get the larger format's equivalent focal length by multiplying the smaller format's focal length by q:

[6]

**f[span style=\'font-size:8pt;line-height:100%\']_lf[/span] = q * f[span style=\'font-size:8pt;line-height:100%\']_sf[/span]**(Here, "_sf" is supposed to mean small format; "_lf" means large format.) Common wisdom also suggests the same method to calculate equivalent aperture numbers:

[7]

**k[span style=\'font-size:8pt;line-height:100%\']_lf[/span] = q * k[span style=\'font-size:8pt;line-height:100%\']_sf[/span]**However [5] suggests a slightly different formula for calculating equivalent aperture numbers. Obviously, [7] only holds for focus distances near infinity; at shorter distances it will become increasingly inaccurate. So the modified formula for equivalent aperture numbers, depending on focus distance d, becomes:

[8]

**k[span style=\'font-size:8pt;line-height:100%\']_lf[/span] = q * [(d - f[span style=\'font-size:8pt;line-height:100%\']_sf[/span]) / (d - f[span style=\'font-size:8pt;line-height:100%\']_lf[/span])] * k[span style=\'font-size:8pt;line-height:100%\']_sf[/span]**For d approaching infinity, the factor in brackets [] will approach 1, and thus [8] will approach [7].

Using [6] to calculate equivalent focal lengths, and [8] to calculate equivalent aperture numbers, [5] suggests that the COC curves of all frame formats will match exactly across all possible object distances d, thus rendering exactly the same image. Some contributors in the above-mentioned thread used this as an argument against my hypothesis stated above.

However [5] is not accurate; it holds for long focus distances only. At shorter focus distances it will become increasingly inaccurate. It ignores the fact that at shorter focus distances, the lens-to-image distance is not equal to the focal length anymore. By the way, the same flaw affects [6] because angle of view is not determined by focal length f but by lens-to-image distance b! In other words, the equivalent focal length depends on focus distance; the well-known rule to multiply with the form factor holds at infinite focus distance only.

So we need a new, improved formula to calculate the larger format's focal length f[span style=\'font-size:8pt;line-height:100%\']_lf[/span] equivalent to the smaller format's focal length f[span style=\'font-size:8pt;line-height:100%\']_sf[/span] for the form factor q at focus distance d. Using [4], we get:

[9]

**f[span style=\'font-size:8pt;line-height:100%\']_lf[/span] = q*q * f[span style=\'font-size:8pt;line-height:100%\']_sf[/span] - (q*q - q) * B(f[span style=\'font-size:8pt;line-height:100%\']_sf[/span], d)**Compare this to [6] and note how complicated things are getting when taking the effects of shorter focus distances into account properly! Further note that [9] will be acurate only if you know your lens' actual focal length at focusing distance d---for lenses employing internal, front-part, or rear-part focusing, this cannot be taken for granted.

Finally, to improve upon [5], I developed the following formula, to calculate the diameter of the circle of confusion z for an object at distance d for a lens with focal length f, focused to d[span style=\'font-size:8pt;line-height:100%\']o[/span] (for B, refer to [4]):

[10]

**z = (f / k) * |1 - (B(f, d[span style=\'font-size:8pt;line-height:100%\']o[/span]) / B(f, d))|**When using [10] rather than [5], and when using [9] to calculate equivalent focal lengths, then we'll find that for different frame formats the run of the COC curve across all possible object distances d will be different, and more different at shorter focus distances d[span style=\'font-size:8pt;line-height:100%\']o[/span]. Surprisingly, the growth of the COC diameter outside the DOF range is

*slower* for the larger format! While this seems counter-intuitve, it does make sense because a COC growing at a slower rate in the blurred area beyond DOF will make for a smoother transition from perceived sharpness to blur---just as my hypothesis suggests. To evaluate [5] and [10], I used this

function plotter.

So [10] provides strong evidence that my hypothesis actually is true.

-- Olaf

P.S. Unfortunately, I have not found an exact formula to calculate equivalent aperture numbers. While [8] is better than [6], it still is based on the over-simplified formula [5] and thus not really accurate. To find equivalent aperture numbers, so far I resorted to trial-and-error, trying various values until DOF matches, using the function plotter.