Zeiss formula

In photographic optics, the so-called Zeiss formula is sometimes said to be used for computing a circle of confusion (CoC) criterion for depth of field (DOF) calculations. The formula is $c\; =\; d/1730$, where $d$ is the diagonal measure of a camera format, film, sensor, or print, and $c$ the maximum acceptable diameter of the circle of confusion.

The Zeiss formula is apocryphal, in the sense that it has grown to be a well-known named concept by propagation through the internet, even though it has no official origin, little connection to Carl Zeiss Company, and no recognition or usage in the photographic industry outside the web community.

Several hundred web pages state:

Using the 'Zeiss formula' the circle of confusion is calculated as $d/1730$ where $d$ is the diagonal measure of the film in millimeters.

As of January, 2006, the Wikipedia circle of confusion article even dignifies it with "industry-standard" in front of "Zeiss formula." Through extensive research (on Google and the Internet Archive), the origin of this formula, including the unnecessary appendage "in millimeters", is now known.

The phrase has spread from the online help pages of "f/Calc", an interactive DOF calculator that was written in 1998 by Warren Young. The original help pages said:

f/Calc uses the commonly-accepted CoC value of 0.033mm for 35mm film, but some companies like Zeiss use a more demanding value of 0.025mm when making the depth of field marks on their lens barrels. That number is calculated as 1/1730 of the diagonal of the frame. You can use the same formula for other film formats.

By March 2001, before f/Calc was moved to its new site, its new CoC.htm page says

f/Calc calculates the CoC using the 'Zeiss formula': "d/1730", where "d" is the diagonal measure of the film, in millimeters.

It appears that Young thereby just coined the term "Zeiss formula" and the superfluous "in millimeters" phrase that everyone else copied.

But where did Young get the idea that Zeiss uses 0.025 mm or 1/1730 of the format diagonal? From David M. Jacobson’s popular online "Photographic Lenses FAQ" and "Photographic Lenses Tutorial".Fact|date=March 2007 The FAQ of December 1996 says:

Q8. What is meant by circle of confusion?

A. When a lens is defocused, an object point gets rendered as a small circle, called the circle of confusion. (Ignoring diffraction.) If the circle of confusion is small enough, the image will look sharp. There is no one circle ‘small enough’ for all circumstances, but rather it depends on how much the image will be enlarged, the quality of the rest of the system, and even the subject. Nevertheless, for 35 mm work "c"=.03 mm is generally agreed on as the diameter of the acceptable circle of confusion. Another rule of thumb is $c=1/1730$ of the diagonal of the frame, which comes to .025 mm for 35 mm film. (Zeiss and Sinar are known to be consistent with this rule.)

The December 1995 Tutorial gives a little background on how Jacobson came up with those numbers:

Although there is no one diameter that marks the boundary between fuzzy and clear, .03 mm is generally used in 35mm work as the diameter of the acceptable circle of confusion. (I arrived at this by observing the depth of field scales or charts on/with a number of lenses from Nikon, Pentax, Sigma, and Zeiss. All but the Zeiss lens came out around .03mm. The Zeiss lens appeared to be based on .025 mm.)

In a personal communication, Jacobson confirms that this estimate was based on his Rollei B 35 "with 40mm f/3.5 Carl Zeiss Triotar lens", and that he got a similar value from measurements of a friend’s Sinar camera, and that these led to his 1/1730 estimate and his comments in the FAQ. Jacobson's estimate can be exactly confirmed from the photo of his lens above if the hyperfocal distance at f/16 is estimated as 4 meters (13 feet).

An online copy of the Rollei B35 instruction manual (external link below) confirms

The inner set of dots applies to the aperture f/8 and the outer set of dots to f/16.

The manual also provides the example:

With the lens set to 10 feet and the aperture f/8, the depth of field extends from 6.5 feet to 20 feet.

A quick analysis and check with another DOF calculator indicates that this example is approximately consistent with $COC\; =\; 0.033mathrm\{mm\}$, or $d/1300$. The picture of the lens, however, is clearly not consistent with the example, since the distance from the 10 to the 20 exceeds the distance between the dots. Somewhere, Zeiss got inconsistent between their documentation and their lens markings.

In 1997, Carl Zeiss began publication of their "Camera Lens News" quarterly newsletter; the first issue had this to say about the COC limit used in depth of field scales:

A certain amount of blur is supposed to be tolerable. According to international standards the degree of blur tolerable is defined as 1/1000th of the camera format diagonal, as the normally satisfactory value. With 35 mm format and its 43 mm diagonal only 1/1500th is deemed tolerable, resulting in 43 mm/1500 » 0.030 mm = 30 µm of blur.

This article explains that

'Depth of field is insufficient' is the most common complaint to meet the Carl Zeiss service department today,

due to the improvements in lens sharpness and film sharpness since the standards were set. It is perhaps possible that they had earlier reacted to this increase in service calls by tightening up their COC limit for computing lens DOF markings from about 30 µm to about 25 µm, but they did not do so explicitly, nor did they ever state the divisor 1730. David Jacobson reverse engineered what Zeiss had done on one or two cameras, approximately, and put a number to it; later, Warren Young put a name to it, and the "Zeiss formula" caught on rapidly from there, among web amateurs. It is a perfectly reasonable formula, but does not have the historical basis that its name seems to claim.

ee also

*Circle of confusion
*Depth of field
*Hyperfocal distance

External links

* [http://www.faqs.org/faqs/rec-photo/lenses/tutorial/ Photographic Lenses Tutorial, by David Jacobson, 1995]
* [http://www.faqs.org/faqs/rec-photo/lenses/faq/ Photographic Lenses FAQ, by David Jacobson, 1996]
* [http://tangentsoft.net/fcalc/help/CoC.htm f/Calc Circle-of-confusion help page, by Warren Young, 2001]
* [http://www.zeiss.com/C12567A8003B58B9/allBySubject/15C75F926592E5C1C1256CED0054968D Zeiss "Camera Lens News" (No.1, 1997)]
* [http://www.rolleicamera.com/manb35.htm Rollei B35 manual at rolleicamera.com, 1995]