Distance beyond which all objects can be brought into an acceptable focus
In optics and photography, hyperfocal distance is a distance from a lens beyond which all objects can be brought into an "acceptable" focus. As the hyperfocal distance is the focus distance giving the maximum depth of field, it is the most desirable distance to set the focus of a fixed-focus camera. The hyperfocal distance is entirely dependent upon what level of sharpness is considered to be acceptable.
The hyperfocal distance has a property called "consecutive depths of field", where a lens focused at an object whose distance from the lens is at the hyperfocal distance H will hold a depth of field from "H"/2 to infinity, if the lens is focused to "H"/2, the depth of field will be from "H"/3 to H; if the lens is then focused to "H"/3, the depth of field will be from "H"/4 to "H"/2, etc.
Thomas Sutton and George Dawson first wrote about hyperfocal distance (or "focal range") in 1867. Louis Derr in 1906 may have been the first to derive a formula for hyperfocal distance. Rudolf Kingslake wrote in 1951 about the two methods of measuring hyperfocal distance.
Some cameras have their hyperfocal distance marked on the focus dial. For example, on the Minox LX focusing dial there is a red dot between and infinity; when the lens is set at the red dot, that is, focused at the hyperfocal distance, the depth of field stretches from to infinity. Some lenses have markings indicating the hyperfocal range for specific f-stops, also called a "depth-of-field scale".
Two methods.
There are two common methods of defining and measuring "hyperfocal distance", leading to values that differ only slightly. The distinction between the two meanings is rarely made, since they 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 the closest distance at which a lens can be focused while keeping objects at infinity acceptably sharp. 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 hyperfocal distance is the distance beyond which all objects are acceptably sharp, for a lens focused at infinity.
Acceptable sharpness.
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.).
Formula.
For the first definition,
formula_1
where
For any practical f-number, the added focal length is insignificant in comparison with the first term, so that
formula_2
This formula is exact for the second definition, if H 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 H 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.
Derivation using geometric optics.
The following derivations refer to the accompanying figures. For clarity, half the aperture and circle of confusion are indicated.
Definition 1.
An object at distance H forms a sharp image at distance x (blue line). Here, objects at infinity have images with a circle of confusion indicated by the brown ellipse where the upper red ray through the focal point intersects the blue line.
First using similar triangles hatched in green,
formula_3
Then using similar triangles dotted in purple,
formula_4
as found above.
Definition 2.
Objects at infinity form sharp images at the focal length f (blue line). Here, an object at H forms an image with a circle of confusion indicated by the brown ellipse where the lower red ray converging to its sharp image intersects the blue line.
Using similar triangles shaded in yellow,
formula_5
Example.
Depths of field of 3 ideal lenses of focal lengths, "f"1, "f"2 and "f"3, and f-numbers "N"1, "N"2, and "N"3 when focused at objects at different distances. "H"1, "H"2, and "H"# denote their respective hyperfocal distances (using "Definition 1" in that article) with a circle of confusion of diameter. The darker bars show how that, for fixed subject distance, the depth of field is increased by using a shorter focal length or smaller aperture. The second topmost bar of each set illustrates the configuration for a fixed focus camera with the focus permanently set at the hyperfocal distance to maximise the depth of field. 
As an example, for a lens at ;templatestyles src="F//styles.css" / ;f/8 using a circle of confusion of , which is a value typically used in photography, the hyperfocal distance according to "Definition 1" is
formula_6
If the lens is focused at a distance of , then everything from half that distance () to infinity will be acceptably sharp in our photograph. With the formula for the "Definition 2", the result is , a difference of 0.5%.
Consecutive depths of field.
The hyperfocal distance has a curious property: while a lens focused at H will hold a depth of field from "H"/2 to infinity, if the lens is focused to "H"/2, the depth of field will extend from "H"/3 to H; if the lens is then focused to "H"/3, the depth of field will extend from "H"/4 to "H"/2. This continues on through all successive neighboring terms in the harmonic series (1/"x") values of the hyperfocal distance. That is, focusing at "H"/"n" will cause the depth of field to extend from "H"/("n" + 1) to "H"/("n" − 1).
C. Welborne Piper 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".
History.
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.
Sutton and Dawson 1867.
Thomas Sutton and George Dawson define "focal range" for what we now call "hyperfocal distance":
 ;templatestyles src="Template:Blockquote/styles.css" / ;Focal Range. In every lens there is, corresponding to a given apertal ratio (that is, the ratio of the diameter of the stop to the focal length), a certain distance of a near object from it, between which and infinity all objects are in equally good focus. For instance, in a single view lens of 6 inch focus, with a 1/4 in. stop (apertal ratio one-twenty-fourth), all objects situated at distances lying between 20 feet from the lens and an infinite distance from it (a fixed star, for instance) are in equally good focus. Twenty feet is therefore called the "focal range" of the lens when this stop is used. The focal range is consequently the distance of the nearest object, which will be in good focus when the ground glass is adjusted for an extremely distant object. In the same lens, the focal range will depend upon the size of the diaphragm used, while in different lenses having the same apertal ratio the focal ranges will be greater as the focal length of the lens is increased.
The terms 'apertal ratio' and 'focal range' have not come into general use, but it is very desirable that they should, in order to prevent ambiguity and circumlocution when treating of the properties of photographic lenses. 'Focal range' is a good term, because it expresses the range within which it is necessary to adjust the focus of the lens to objects at different distances from it – in other words, the range within which focusing becomes necessary.
Their focal range is about 1000 times their aperture diameter, so it makes sense as a hyperfocal distance with CoC value of ;templatestyles src="F//styles.css" / ;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.
Abney 1881.
Sir William de Wivelesley Abney says:
 ;templatestyles src="Template:Blockquote/styles.css" / ;The annexed formula will approximately give the nearest point p which will appear in focus when the distance is accurately focussed, supposing the admissible disc of confusion to be :
formula_7
when
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:
 ;templatestyles src="Template:Blockquote/styles.css" / ;It can be shown that an enlargement from a small negative is better than a picture of the same size taken direct as regards sharpness of detail. ... Care must be taken to distinguish between the advantages to be gained in enlargement by the use of a smaller lens, with the disadvantages that ensue from the deterioration in the relative values of light and shade.
Taylor 1892.
John Traill Taylor recalls this word formula for a sort of hyperfocal distance:
 ;templatestyles src="Template:Blockquote/styles.css" / ;We have seen it laid down as an approximative rule by some writers on optics (Thomas Sutton, if we remember aright), that if the diameter of the stop be a fortieth part of the focus of the lens, the depth of focus will range between infinity and a distance equal to four times as many feet as there are inches in the focus of the lens.
This formula implies a stricter CoC criterion than we typically use today.
Hodges 1895.
John Hodges discusses depth of field without formulas but with some of these relationships:
 ;templatestyles src="Template:Blockquote/styles.css" / ;There is a point, however, beyond which everything will be in pictorially good definition, but the longer the focus of the lens used, the further will the point beyond which everything is in sharp focus be removed from the camera. Mathematically speaking, the amount of depth possessed by a lens varies inversely as the square of its focus.
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.
Piper 1901.
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:
 ;templatestyles src="Template:Blockquote/styles.css" / ;
It is unclear what distinction he means. Adjacent to Table I in his appendix, he further notes:
 ;templatestyles src="Template:Blockquote/styles.css" / ;If we focus on infinity, the constant is the focal distance of the nearest object in focus. If we focus on an extra-focal distance equal to the constant, we obtain a maximum depth of field from approximately half the constant distance up to infinity. The constant is then the hyper-focal distance.
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.
Derr 1906.
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 p for hyperfocal distance, D for aperture diameter, d for the diameter that a circle of confusion shall not exceed, and f for focal length, he derives:
formula_8
As the aperture diameter, D is the ratio of the focal length f to the numerical aperture N ("D" = "f"/"N"); and the diameter of the circle of confusion, "c" = "d", this gives the equation for the first definition above.
formula_9
Johnson 1909.
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:
 ;templatestyles