Hyperfocal distance

Minox LX camera with hyperfocal red dot

Nikon 28mm <templatestyles src="F//styles.css" />f/2.8 lens with markings for the depth of field. The lens is set at the hyperfocal distance for <templatestyles src="F//styles.css" />f/22. The orange mark corresponding to <templatestyles src="F//styles.css" />f/22 is at the infinity mark (

\infty

Script error: No such module "Check for unknown parameters".). Focus is acceptable from under Script error: No such module "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 <templatestyles src="F//styles.css" />f/32 at a focal length of Script error: No such module "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.^[1] 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 $H /2$ Script error: No such module "Check for unknown parameters". to infinity, if the lens is focused to $H /2$ Script error: No such module "Check for unknown parameters"., the depth of field will be from $H /3$ Script error: No such module "Check for unknown parameters". to Template:Mvar; if the lens is then focused to $H /3$ Script error: No such module "Check for unknown parameters"., the depth of field will be from $H /4$ Script error: No such module "Check for unknown parameters". to $H /2$ Script error: No such module "Check for unknown parameters"., etc.

Thomas Sutton and George Dawson first wrote about hyperfocal distance (or "focal range") in 1867.^[2] 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 Script error: No such module "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 Script error: No such module "val". to infinity. Some lenses have markings indicating the hyperfocal range for specific f-stops, also called a depth-of-field scale.^[3]

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,

$H = \frac{f^{2}}{N c} + f$

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

$H \approx \frac{f^{2}}{N c} .$

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.^[4]

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, $\begin{array}{crcl} \frac{x - f}{c / 2} & = & \frac{f}{D / 2} \\ ∴ & x - f & = & \frac{c f}{D} \\ ∴ & x & = & f + \frac{c f}{D} \end{array}$

Then using similar triangles dotted in purple, $\begin{array}{crclcl} \frac{H}{D / 2} & = & \frac{x}{c / 2} \\ ∴ & H & = & \frac{D x}{c} & = & \frac{D}{c} (f + \frac{c f}{D}) \\ = & \frac{D f}{c} + f & = & \frac{f^{2}}{N c} + f \end{array}$

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, $\begin{array}{crclcl} \frac{H}{D / 2} & = & \frac{f}{c / 2} \\ ∴ & H & = & \frac{D f}{c} & = & \frac{f^{2}}{N c} \end{array}$

Example

<timeline> ImageSize = width:512 height:512 PlotArea = left:88 right:0 top:0 bottom:20 Period = from:0 till:13799 TimeAxis = orientation:horizontal AlignBars = late Colors =

id:grid_major value:gray(0.5)
id:grid_minor value:gray(0.8)

ScaleMajor = unit:year increment:2000 start:0 gridcolor:grid_major ScaleMinor = unit:year increment:500 start:0 gridcolor:grid_minor BarData =

bar:title0 text:f₁:50 mm; N₁:8
bar:data00 text:focus at infinity
bar:data10 text:focus at H₁
bar:data20 text:focus at H₁/2
bar:data30 text:focus at H₁/3
bar:data40 text:focus at H₁/4
bar:title1 text:f₂:50 mm; N₂:16
bar:data01 text:focus at infinity
bar:data11 text:focus at H₂
bar:data21 text:focus at H₂/2
bar:data31 text:focus at H₂/3
bar:data41 text:focus at H₁/4
bar:title2 text:f₃:25 mm; N₃:8
bar:data02 text:focus at infinity
bar:data12 text:focus at H₃
bar:data22 text:focus at H₃/2
bar:data32 text:focus at H₃/3
bar:data42 text:focus at H₁/4

PlotData =

fontsize:M
bar:data00 from:10417 till:end color:pink
bar:data00 at:13500 text:∞
bar:data00 at:10467 text:\nH₁−f₁=10417 mm
bar:data10 from:5233 till:end color:pink
bar:data10 at:13500 text:\n∞
bar:data10 at:10467 text:H₁ = 10467 mm mark:(line,white)
bar:data10 at:5233 text:\nH₁/2
bar:data20 from:3489 till:10467 color:pink
bar:data20 at:10467 text:\nH₁
bar:data20 at:5233 text:H₁/2 mark:(line,white)
bar:data20 at:3489 text:\nH₁/3
bar:data30 from:2617 till:5233 color:pink
bar:data30 at:5233 text:\nH₁/2
bar:data30 at:3489 text:H₁/3 mark:(line,white)
bar:data30 at:2617 text:\nH₁/4
bar:data40 from:2093 till:3489 color:coral
bar:data40 at:3489 text:\nH₁/3
bar:data40 at:2617 text:H₁/4 mark:(line,white)
bar:data40 at:2093 text:\nH₁/5

bar:data01 from:5208 till:end color:limegreen
bar:data01 at:13500 text:∞
bar:data01 at:5258 text:\nH₂ − f₂ = 5208 mm
bar:data11 from:2629 till:end color:limegreen
bar:data11 at:13500 text:\n∞
bar:data11 at:5258 text:H₂ = 5258 mm mark:(line,white)
bar:data11 at:2629 text:\nH₂/2
bar:data21 from:1753 till:5258 color:yellowgreen
bar:data21 at:5258 text:\nH₂
bar:data21 at:2629 text:H₂/2 mark:(line,white)
bar:data21 at:1753 text:\nH₂/3
bar:data31 from:1315 till:2629 color:limegreen
bar:data31 at:2629 text:\nH₂/2
bar:data31 at:1753 text:H₂/3 mark:(line,white)
bar:data31 at:1315 text:\nH₂/4
bar:data41 from:1052 till:1753 color:limegreen
bar:data41 at:1753 text:\nH₂/3
bar:data41 at:1315 text:H₂/4 mark:(line,white)
bar:data41 at:500 text:\nH₂/5

bar:data02 from:2629 till:end color:skyblue
bar:data02 at:13500 text:∞
bar:data02 at:2604 text:\nH₃ − f₃ = 2604 mm
bar:data12 from:1315 till:end color:powderblue2
bar:data12 at:13500 text:\n∞
bar:data12 at:2629 text:H₃ = 2629 mm mark:(line,white)
bar:data12 at:1315 text:\nH₃/2
bar:data22 from:876 till:2629 color:skyblue
bar:data22 at:2629 text:\nH₃
bar:data22 at:1315 text:H₃/2 mark:(line,white)
bar:data22 at:876 text:\nH₃/3
bar:data32 from:657 till:1315 color:skyblue
bar:data32 at:1315 text:\nH₃/2
bar:data32 at:876 text:H₃/3 mark:(line,white)
bar:data32 at:0 text:\nH₃/4
bar:data42 from:526 till:876 color:skyblue
bar:data42 at:876 text:\nH₃/3
bar:data42 at:657 text:H₂/4 mark:(line,white)
bar:data42 at:0 text:\nH₃/5

LineData =

at:2629 width:1 color:brightblue layer:back

TextData =

pos:(170,500) text:OBJECT AT A
pos:(170,490) text:DISTANCE OF 2629 mm
pos:(10,10) text:Distance of
pos:(10,0) text:object in mm
fontsize:L
pos:(475,0) text:→ ∞

</timeline>

Depths of field of 3 ideal lenses of focal lengths,

f 1

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f 2

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f 3

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N 1

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N 2

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N 3

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H 1

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H 2

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H #

Script error: No such module "Check for unknown parameters". denote their respective hyperfocal distances (using Definition 1 in that article) with a circle of confusion of Script error: No such module "val". 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.

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As an example, for a Script error: No such module "val". lens at <templatestyles src="F//styles.css" />f/8 using a circle of confusion of Script error: No such module "val"., which is a value typically used in Script error: No such module "val". photography, the hyperfocal distance according to Definition 1 is

$H = \frac{(50)^{2}}{(8) (0.03)} + (50) = 10467 mm$

If the lens is focused at a distance of Script error: No such module "val"., then everything from half that distance (Script error: No such module "val".) to infinity will be acceptably sharp in our photograph. With the formula for the Definition 2, the result is Script error: No such module "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 $H /2$ Script error: No such module "Check for unknown parameters". to infinity, if the lens is focused to $H /2$ Script error: No such module "Check for unknown parameters"., the depth of field will extend from $H /3$ Script error: No such module "Check for unknown parameters". to Template:Mvar; if the lens is then focused to $H /3$ Script error: No such module "Check for unknown parameters"., the depth of field will extend from $H /4$ Script error: No such module "Check for unknown parameters". to $H /2$ Script error: No such module "Check for unknown parameters".. This continues on through all successive neighboring terms in the harmonic series ( $1/ x$ Script error: No such module "Check for unknown parameters".) values of the hyperfocal distance. That is, focusing at $H / n$ Script error: No such module "Check for unknown parameters". will cause the depth of field to extend from $H /(n + 1)$ Script error: No such module "Check for unknown parameters". to $H /(n - 1)$ Script error: No such module "Check for unknown parameters"..

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.^[5]

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:^[2]

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.

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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 <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:^[6]

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 Script error: No such module "val".:
$p = 0.41 \cdot f^{2} \cdot a$ when

$f =$ Script error: No such module "Check for unknown parameters". the focal length of the lens in cm

$a =$ Script error: No such module "Check for unknown parameters". the ratio of the aperture to the focal length

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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.

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Taylor 1892

John Traill Taylor recalls this word formula for a sort of hyperfocal distance:^[7]

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.

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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:^[8]

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.

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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).^[5] 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 $H + f$ Script error: No such module "Check for unknown parameters". 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.

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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.

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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,^[9] 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:^[10]

$p = \frac{(D + d) f}{d} .$

As the aperture diameter, Template:Mvar is the ratio of the focal length Template:Mvar to the numerical aperture Template:Mvar ( $D = f / N$ Script error: No such module "Check for unknown parameters".); and the diameter of the circle of confusion, $c = d$ Script error: No such module "Check for unknown parameters"., this gives the equation for the first definition above.

$p = \frac{(\frac{f}{N} + c) f}{c} = \frac{f^{2}}{N c} + f$

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),^[11] 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. {{{1}}} (or {{{1}}}).

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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.

Depth of Field is precisely the same as depth of focus, only in the former case the depth is measured by the movement of the plate, the object being fixed, while in the latter case the depth is measured by the distance through which the object can be moved without the circle of confusion exceeding 2Template:Mvar.
Thus if a lens which is focused for infinity still gives a sharp image for an object at 6 yards, its depth of field is from infinity to 6 yards, every object beyond 6 yards being in focus.
This distance (6 yards) is termed the hyperfocal distance of the lens, and any allowable confusion disc depends on the focal length of the lens and on the stop used.
If the limit of confusion of half of the disc (i.e. Template:Mvar) be taken as 1/100 in., then the hyperfocal distance
$H = \frac{F d}{e},$

Template:Mvar being the diameter of the stop, ...

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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:^[1]

if the camera is focused on a distance Template:Mvar equal to 1000 times the diameter of the lens aperture, then the far depth $D 1$ Script error: No such module "Check for unknown parameters". becomes infinite. This critical object distance "Template:Mvar" is known as the Hyperfocal Distance. For a camera focused on this distance, $D 1 = \infty$ Script error: No such module "Check for unknown parameters". and $D 2 = h /2$ Script error: No such module "Check for unknown parameters"., 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 $s = \infty$ Script error: No such module "Check for unknown parameters"., the closest acceptable object is at $L 2 = sh /(h + s) = h /(h / s + 1) = h$ Script error: No such module "Check for unknown parameters". (by equation 21). This is a second important meaning of the hyperfocal distance.

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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.

References

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External links

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

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Hyperfocal distance

Contents

Two methods

Acceptable sharpness

Formula

Derivation using geometric optics

Definition 1

Definition 2

Example

Consecutive depths of field

History

Sutton and Dawson 1867

Abney 1881

Taylor 1892

Hodges 1895

Piper 1901

Derr 1906

Johnson 1909

Others, early twentieth century

Kingslake 1951

See also

References

External links

Navigation menu

Hyperfocal distance

Two methods

Acceptable sharpness

Formula

Derivation using geometric optics

Definition 1

Definition 2

Example

Consecutive depths of field

History

Sutton and Dawson 1867

Abney 1881

Taylor 1892

Hodges 1895

Piper 1901

Derr 1906

Johnson 1909

Others, early twentieth century

Kingslake 1951

See also

References

External links

Navigation menu

Search