Hyperfocal distance

File:Minox LX hyperfocal.JPG

Minox LX camera with hyperfocal red dot

File:Nikon 28mm lens at hyperfocus.jpg

Nikon 28mm Template:F/ lens with markings for the depth of field. The lens is set at the hyperfocal distance for Template:F/. The orange mark corresponding to Template:F/ is at the infinity mark (Template:Math). Focus is acceptable from under Template:Val to infinity.

File:Minolta 100-300 at hyperfocal distance.jpg

Minolta 100–300 mm zoom lens. The depth of field, and thus hyperfocal distance, changes with the focal length as well as the f-stop. This lens is set to the hyperfocal distance for Template:F/ at a focal length of Template:Val.

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.<ref name="Kingslake1951">Template:Cite book</ref> 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 Template:Mvar will hold a depth of field from Template:Math to infinity, if the lens is focused to Template:Math, the depth of field will be from Template:Math to Template:Mvar; if the lens is then focused to Template:Math, the depth of field will be from Template:Math to Template:Math, etc.

Thomas Sutton and George Dawson first wrote about hyperfocal distance (or "focal range") in 1867.<ref name=":0">Template:Cite book</ref> 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 Template:Val and infinity; when the lens is set at the red dot, that is, focused at the hyperfocal distance, the depth of field stretches from Template:Val to infinity. Some lenses have markings indicating the hyperfocal range for specific f-stops, also called a depth-of-field scale.<ref>Template:Cite book</ref>

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.

Definition 1

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.

Definition 2

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,

<math display="block">H = \frac{f^2}{N c} + f</math>

where

• Template:Mvar is the hyperfocal distance;
• Template:Mvar is the focal length of the lens;
• Template:Mvar is f-number (Template:Mvar for aperture diameter Template:Mvar); and
• Template:Mvar is the circle of confusion limit.

For any practical f-number, the added focal length is insignificant in comparison with the first term, so that

<math display="block">H \approx \frac{f^2}{N c}\,.</math>

This formula is exact for the second definition, if Template:Mvar 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 Template:Mvar 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

File:Hyperfocal distance definitions.svg

Accompanying figures

The following derivations refer to the accompanying figures. For clarity, half the aperture and circle of confusion are indicated.<ref>Template:Cite book</ref>

Definition 1

An object at distance Template:Mvar forms a sharp image at distance Template:Mvar (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, <math display="block">\begin{array}{crcl}

          & \dfrac{x-f}{c/2} & = & \dfrac{f}{D/2}  \\

\therefore & x-f & = & \dfrac{cf}{D} \\ \therefore & x & = & f+\dfrac{cf}{D} \end{array}</math>

Then using similar triangles dotted in purple, <math display="block">\begin{array}{crclcl}

          & \dfrac{H}{D/2} & = & \dfrac{x}{c/2}                                              \\

\therefore & H & = & \dfrac{Dx}{c} & = & \dfrac{D}{c}\Big(f+\dfrac{cf}{D}\Big) \\

          &                & = & \dfrac{Df}{c}+f & = & \dfrac{f^2}{Nc}+f

\end{array}</math>

as found above.

Definition 2

Objects at infinity form sharp images at the focal length Template:Mvar (blue line). Here, an object at Template:Mvar 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, <math display="block">\begin{array}{crclcl}

          & \dfrac{H}{D/2} & = & \dfrac{f}{c/2}                      \\

\therefore & H & = & \dfrac{Df}{c} & = & \dfrac{f^2}{Nc} \end{array}</math>

Example

As an example, for a Template:Val lens at Template:F/ using a circle of confusion of Template:Val, which is a value typically used in Template:Val photography, the hyperfocal distance according to Definition 1 is

<math display="block">H = \frac{(50)^2}{(8)(0.03)} + (50) = 10467 \mbox{ mm}</math>

If the lens is focused at a distance of Template:Val, then everything from half that distance (Template:Val) to infinity will be acceptably sharp in our photograph. With the formula for the Definition 2, the result is Template:Val, a difference of 0.5%.

Consecutive depths of field

The hyperfocal distance has a curious property: while a lens focused at Template:Mvar will hold a depth of field from Template:Math to infinity, if the lens is focused to Template:Math, the depth of field will extend from Template:Math to Template:Mvar; if the lens is then focused to Template:Math, the depth of field will extend from Template:Math to Template:Math. This continues on through all successive neighboring terms in the harmonic series (Template:Math) values of the hyperfocal distance. That is, focusing at Template:Math will cause the depth of field to extend from Template:Math to Template:Math.

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.<ref name="Piper1901" />

History

File:Derr Hyperfocal 1906.png

This early use of the term hyperfocal distance, Derr 1906, is by no means the earliest explanation of the concept.

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:<ref name=":0" />

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.{{#if:|{{#if:|}}

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Their focal range is about 1000 times their aperture diameter, so it makes sense as a hyperfocal distance with CoC value of Template:F/, 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:<ref>Template:Cite book</ref>

The annexed formula will approximately give the nearest point Template:Mvar which will appear in focus when the distance is accurately focussed, supposing the admissible disc of confusion to be Template:Val:

<math display="block">p = 0.41 \cdot f^2 \cdot a</math> when

• Template:Math the focal length of the lens in cm
• Template:Math the ratio of the aperture to the focal length{{#if:|{{#if:|}}

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That is, Template:Mvar 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:

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.{{#if:|{{#if:|}}

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Taylor 1892

John Traill Taylor recalls this word formula for a sort of hyperfocal distance:<ref>Template:Cite book</ref>

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.{{#if:|{{#if:|}}

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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:<ref>Template:Cite book</ref>

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.{{#if:|{{#if:|}}

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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).<ref name="Piper1901">Template:Cite book</ref> He uses the term Depth Constant for Template:Mvar, 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:

This is the maximum depth of field possible, and Template:Math may be styled the distance of maximum depth of field. If we measure this distance extra-focally it is equal to Template:Mvar, and is sometimes called the hyperfocal distance. The depth constant and the hyperfocal distance are quite distinct, though of the same value.{{#if:|{{#if:|}}

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It is unclear what distinction he means. Adjacent to Table I in his appendix, he further notes:

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.{{#if:|{{#if:|}}

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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,<ref>Template:Cite book</ref> which is considered to be the strictly correct one in modern times, and to derive the formula corresponding to it. Using Template:Mvar for hyperfocal distance, Template:Mvar for aperture diameter, Template:Mvar for the diameter that a circle of confusion shall not exceed, and Template:Mvar for focal length, he derives:<ref>Template:Cite book</ref>

<math display="block>p = \frac{(D + d) f}{d}\,.</math>

As the aperture diameter, Template:Mvar is the ratio of the focal length Template:Mvar to the numerical aperture Template:Mvar (Template:Math); and the diameter of the circle of confusion, Template:Math, this gives the equation for the first definition above.

<math display="block">p = \frac{\left(\tfrac{f}{N} + c\right) f}{c} = \frac{f^2}{N c} + f</math>

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),<ref>Template:Cite book</ref> as the allowable distance error in the focal plane. His definitions include hyperfocal distance:

Depth of Focus is a convenient, but not strictly accurate term, used to describe the amount of racking movement (forwards or backwards) which can be given to the screen without the image becoming sensibly blurred, i.e. without any blurring in the image exceeding 1/100 in., or in the case of negatives to be enlarged or scientific work, the 1/10 or 1/100 mm. Then the breadth of a point of light, which, of course, causes blurring on both sides, i.e. Template:Nowrap (or Template:Nowrap).{{#if:|{{#if:|}}

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His drawing makes it clear that his Template:Mvar 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.

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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.

Others, early twentieth century

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.

Kingslake 1951

Rudolf Kingslake is explicit about the two meanings:<ref name="Kingslake1951" />

if the camera is focused on a distance Template:Mvar equal to 1000 times the diameter of the lens aperture, then the far depth Template:Math becomes infinite. This critical object distance "Template:Mvar" is known as the Hyperfocal Distance. For a camera focused on this distance, Template:Math and Template:Math, and we see that the range of distances acceptably in focus will run from just half the hyperfocal distance to infinity. The hyperfocal distance is, therefore, the most desirable distance on which to pre-set the focus of a fixed-focus camera. It is worth noting, too, that if a camera is focused on Template:Math, the closest acceptable object is at Template:Math (by equation 21). This is a second important meaning of the hyperfocal distance.{{#if:|{{#if:|}}

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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.

See also

• Circle of confusion
• Deep focus

External links

• http://www.dofmaster.com/dofjs.html to calculate hyperfocal distance and depth of field