The first definition: the hyperfocal distance is the closest distance at which a lens can be focused while keeping objects at infinity acceptably sharp; that is, the focus distance with the maximum depth of field. When the lens is focused at this distance, all objects at distances from half of the hyperfocal distance out to infinity will be acceptably sharp.
The second definition: the hyperfocal distance is the distance beyond which all objects are acceptably sharp, for a lens focused at infinity.
The distinction between the two meanings is rarely made, since they are interchangeable and have almost identical values. The value computed according to the first definition exceeds that from the second by just one focal length.
The hyperfocal distance is entirely dependent upon what level of sharpness is considered to be acceptable. The criterion for the desired acceptable sharpness is specified through the circle of confusion (COC) diameter limit. This criterion is the largest acceptable spot size diameter that an infinitesimal point is allowed to spread out to on the imaging medium (film, digital sensor, etc.).
For the first definition,
For any practical f-number, the focal length is insignificant in comparison with the first term, so that
This formula is exact for the second definition, if is measured from a thin lens, or from the front principal plane of a complex lens; it is also exact for the first definition if is measured from a point that is one focal length in front of the front principal plane. For practical purposes, there is little difference between the first and second definitions.
As an example, let's compute the hyperfocal distance for a 50 mm lens at using a circle of confusion of 0.03 mm (which is a value typically used in 35 mm photography):
If we focus the lens at a distance of 5.2 m, then everything from half that distance (2.6 m) to infinity will be acceptably sharp in our photograph. With the more exact formula for the first definition, the result is not much different.
Piper 1901 calls this phenomenon "consecutive depths of field" and shows how to test the idea easily. This is also among the earliest of publications to use the word hyperfocal.
The concepts of the two definitions of hyperfocal distance have a long history, tied up with the terminology for depth of field, depth of focus, circle of confusion, etc. Here are some selected early quotations and interpretations on the topic.
Thomas Sutton and George Dawson define focal range for what we now call hyperfocal distance:
Their focal range is about 1000 times their aperture diameter, so it makes sense as a hyperfocal distance with COC value of f/1000, or image format diagonal times 1/1000 assuming the lens is a “normal” lens. What is not clear, however, is whether the focal range they cite was computed, or empirical.
Sir William de Wivelesley Abney says:
That is, a is the reciprocal of what we now call the f-number, and the answer is evidently in meters. His 0.41 should obviously be 0.40. Based on his formulae, and on the notion that the aperture ratio should be kept fixed in comparisons across formats, Abney says:
J. Traill Taylor recalls this word formula for a sort of hyperfocal distance:
This formula implies a stricter COC criterion than we typically use today.
John Hodges discusses depth of field without formulas but with some of these relationships:
This "mathematically" observed relationship implies that he had a formula at hand, and a parameterization with the f-number or “intensity ratio” in it. To get an inverse-square relation to focal length, you have to assume that the COC limit is fixed and the aperture diameter scales with the focal length, giving a constant f-number.
C. Welborne Piper may be the first to have published a clear distinction between Depth of Field in the modern sense and Depth of Definition in the focal plane, and implies that Depth of Focus and Depth of Distance are sometimes used for the former (in modern usage, Depth of Focus is usually reserved for the latter). He uses the term Depth Constant for H, and measures it from the front principal focus (i. e., he counts one focal length less than the distance from the lens to get the simpler formula), and even introduces the modern term:
It is unclear what distinction he means. Adjacent to Table I in his appendix, he further notes:
At this point we do not have evidence of the term hyperfocal before Piper, nor the hyphenated hyper-focal which he also used, but he obviously did not claim to coin this descriptor himself.
Louis Derr may be the first to clearly specify the first definition, which is considered to be the strictly correct one in modern times, and to derive the formula corresponding to it. Using for hyperfocal distance, for aperture diameter, for the diameter that a circle of confusion shall not exceed, and for focal length, he derives :
George Lindsay Johnson uses the term Depth of Field for what Abney called Depth of Focus, and Depth of Focus in the modern sense (possibly for the first time), as the allowable distance error in the focal plane. His definitions include hyperfocal distance:
His drawing makes it clear that his e is the radius of the circle of confusion. He has clearly anticipated the need to tie it to format size or enlargement, but has not given a general scheme for choosing it.
Johnson's use of former and latter seem to be swapped; perhaps former was here meant to refer to the immediately preceding section title Depth of Focus, and latter to the current section title Depth of Field. Except for an obvious factor-of-2 error in using the ratio of stop diameter to COC radius, this definition is the same as Abney's hyperfocal distance.
The term hyperfocal distance also appears in Cassell's Cyclopaedia of 1911, The Sinclair Handbook of Photography of 1913, and Bayley's The Complete Photographer of 1914.
Rudolf Kingslake is explicit about the two meanings:
Kingslake uses the simplest formulae for DOF near and far distances, which has the effect of making the two different definitions of hyperfocal distance give identical values.
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