[abonnement gratuit]
the author
Emmanuel Bigler is a professor in optics
and microtechnology at ENSMM,
Besançon, France, an engineering college (École Nationale Supérieure
d'Ingénieurs) in mechanical engineering and microtechnology . He got
his Ph.D. degree from Institut d'Optique, Orsay (France). E. Bigler
uses an ArcaSwiss 6X9 FC view camera.
ENSMM, 26 chemin de l'Épitaphe 25030 Besançon cedex, France
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Depth of field and Scheimpflug's rule : a minimalist geometrical approach
(Version Française)
Emmanuel BIGLER
ENSMM, 26 chemin de l'Epitaphe, F25030 Besancon cedex,
FRANCE, email : bigler@ens2m.fr
Résumé : We show here how pure geometrical
considerations with an absolute minimum of algebra will yield the
solution for the position of slanted planes defining the limits of
acceptable sharpness (an approximation valid for distant objects) for
DepthofField (DOF) combined with Scheimpflug's
rule. The problem of DepthofFocus is revisited using a similar
approach. General formulae for DepthOfField (DOF) are given in
appendix, valid in the closeup range. The significance of the circle of
least confusion, on which all DOF computations are based, even in the
case of a tilted view camera lens and the choice of possible numerical
values are also explained in detail in the appendix.
Introduction
We address here the question that immediately follows the application of
Scheimpflug's rule: when a camera is properly focused for a pair of
object/image conjugated slanted planes (satisfying "Scheimpflug's
rules of 3 intersecting planes''), what is actually the volume in object
space that will be rendered sharp, given a certain criterion of acceptable
sharpness in the (slanted) image/film plane?
Again the reader interested in a comprehensive, rigorous, mathematical study
based on the geometrical approach of the circle of confusion should refer to
Bob Wheeler's work [1]
or the comprehensive review by Martin Tai [2].
A nice graphical explanation is presented by Leslie Stroebel
in his reference book [5], but no details
are given. We recompute in the appendix Stroebel's
DOF curves and show how they are related to the classical DOF theory. The
challenge here is to try and reduce the question to the absolute minimum of
maths required to derive a practical rule.
It has been found that, with a minimum of simplifications and sensible
approximations, the solution can be understood as the image formation
through the photographic lens fitted with an additional positive or negative
"closeup'' lens of focal length ± H, where H is the
hyperfocal distance. This analogy yields an immediate solution to the
problem of depth of field for distant objects, the same solution as
documented in Harold M. Merklinger's work [3],
[4], which appears simply as an
approximation of the rigorous model, valid for far distant objects.
1 Derivation of the position of
slanted limit planes of acceptable sharpness
1.1 Starting with reasonable
approximations
Consider a situation where we are dealing with a pair of corresponding
slanted object and image planes according to Scheimpflug's
rule (fig. 4), and let us first assume a few
reasonable approximations:

first we neglect the fact that the projection of a
circular lens aperture on film, for a single, out of focus point object,
will actually be an ellipse and not a circle. This is
well explained by Bob Wheeler [1]
who shows, after a complete rigorous calculation, that this
approximation is very reasonable in most practical conditions.

second we consider only far distant objects; in other
words we are interested to know the position of limit surfaces
of sharpness far from the camera, i.e. distances s or p
much greater than the focal length f. We'll show that those
surfaces in the limit case are actually planes, the more
rigorous shape of these surfaces for all objecttocamera distances can
be found in Bob Wheeler's paper, in Leslie
Stroebel's book [5], and here in
the appendix.

finally we'll represent the lens as a single positive
lens element; in other words we neglect the distance between the
principal planes of the lens, which will not significantly change the
results for far distant objects, provided that we consider a
quasisymmetrical camera lens (with the notable exception of telephoto
lenses, this is how most view camera lenses are designed).
1.2 A "hidden treasury'' in classical
depthoffield formulae !
Let us restart, as a minimum of required algebra, with the wellknow
expressions for classical depth of field distances, in fact the ones used in
practice and mentioned in numerous books, formulae on which are based the
DOF engravings on classical manually focused lenses.
Figure 1 : Depthoffield distances
s_{1},
p_{1} and s_{2},
p_{2} for a given circle of confusion
c
Consider an object AB perpendicular to the optical
axis, let p_{1} and p_{2}
the positions (measured from the lens plane in O) of the planes of
acceptable sharpness around a given position of the object p.
It should be noted (see fig. 1) that the ray tracing for
a couple of points outside the optical axis like D_{1}
and D'_{1} yields in the image plane
A'B' an outoffocus image of circular shape (not an
ellipse, as it could be considered at a first); this outoffocus image
is exactly the same as the circular spot originated from P_{1};
this is simply the classical property of the conical projection of a
circular aperture between two parallel planes. The outoffocus spot near D' is centred on the median ray
D D_{1} O D' D'_{1}
that crosses the lens at its optical centre. This point will be important in
the discussion about transversal magnification factors for outoffocus
images.
Assuming a given value for c, the diameter of the circle of
confusion, p_{1} and p_{2}
are identical whether we consider AA' onaxis or DD' offaxis
and is given, for far distant objects, by
In eq.(1), H is the hyperfocal distance for
a given numerical aperture N and diameter of the circle of confusion
c, defined as usual as
The previous expressions of eq. (1) are valid only for
fardistant objects; readers interested by more exact expressions, valid
also for closeup situations, will find them below in the appendix.
Now let us combine eq.(1) with the wellknown
objectimage equation (known in France as Descartes
formulae), written here with positive values of p and p'
(the photographic case)
Combining equation (1) into (3) yields
the interesting formula (4)
which is nothing but the objectimage equation for the image
A'
located at a distance p' of the lens centre O, but as seen
through an optical system of inverse focal length (1/f+1/H)
for the near plane p_{1}
(point P_{1}) and (1/f1/H)
for the distant plane p_{2}
(point P_{2}). The
expression (1/f+1/H) is simply the inverse 1/f_{1}
of the focal length of a compound system made of the original lens, fitted
with a positive closeup lens of focal length H.
When you "glue'' two thin single lens elements together into a
thin
compound with no air space, their convergences (inverse of the
focal length) should simply be added. Thus in a symmetric way (1/f1/H)
is nothing but the inverse 1/f_{2} of
the focal length of a compound system made of the original lens, fitted with
a negative "closeup'' lens of focal length H.
1.3 A positive or negative "closeup''
lens to visualise DOF at full aperture?
1.3.1 a classical DOF rule revisited
Before we proceed to the Scheimpflug case, let us
examine the practical consequence of the additional closeup lens approach
in the simple case of parallel object and image planes. We'll show how this
additional lens element will allow us to revisit some wellknow DOF rules
(figure 2).
Figure 2 : When the an additional
"closeup'' lens of positive or negative power +H or H allows us to
find wellknow DOF rules
It is known to photographers that when a lens is focused on
the hyperfocal distance H, all objects located between H/2 and
infinity will be rendered approximately sharp on film, i.e. sharp within the
DOF tolerance. Consider a lens focused on the hyperfocal distance and let us
add a positive closeup lens element of focal length +H. The ray
tracing on figure 2 shows that the object plane
located at a distance H/2 is now imaged sharp on film if the lens to
film distance is unchanged. In a symmetric way, the same photographic lens
fitted with a negative "closeup'' additional lens element of focal length H
will focus a sharp image on film for objects located at infinity. The same
considerations actually apply whatever the object to lens distance might be,
in this case formulae (4) are simply a more general rule
valid for any object to lens distance p; the result is eventually the
same, i.e. the positions of acceptable sharpness p_{1}
and p_{2} are located where the film
"would see sharp'' through the camera lens fitted with a positive or a
negative "closeup'' lens of focal length +H or H.
1.3.2 DOF visualisation at full
aperture??
It would be nice to be able to use this trick in practice to check for depth
of field without stopping the lens down to a small aperture. In large format
photography, f/16 to f/64 are common, and the brightness of the image is
poor; it is difficult to evaluate DOF visually in these conditions. The
closeup lens trick would, in theory, allow to visualise the positions of
limit planes of acceptable sharpness at full aperture simply by swapping a +H
or H supplementary lens by hand in front of the camera lens with the
fstop kept wide open.
There is no reason why this could not work from an optical point of view.
Unfortunately classical closeup lenses are always positive and, to
the best of my knowledge, my favourite opticist round the corner will not
have in stock eyeglasses with a focal length longer than 2 metres (a power
smaller than 0.5 dioptre). The shop will probably have all kinds of positive
and negative lenses in stock, but none, even on special order, will exhibit,
say, a focal length of 10 metres (1/10 dioptre) because this is quite
useless for correcting eyesight.
In large format photography, the hyperfocal distance is always greater than
2 metres. For example a large format lens, with a focal length of 150 mm,
for which we consider appropriate a circle of confusion of 100 microns has
an hyperfocal distance smaller that 2 metres only when closed down at a
fstop smaller than f/112. An impossible aperture, and moreover for such
"pinhole'' kind of values, diffraction effects make the classical DOF model
questionable.
Let us however see if this could work in 35mm photography. Shift and tilt
lenses do exist for 35mm SLR cameras. Consider a moderate wideangle of 35
mm focal length and assume that the circle of confusion is chosen equal to
the conventional and widely used value of 33 microns. The hyperfocal
distance is equal to 2 metres at f/18: this is a more realistic value. Those
who use shift and tilt lenses, 35mm focal length on a 35mm SLR can actually
use a +0.5 or 0.5 dioptre supplementary lens to get an idea of the DOF
planes at f/16f/22 without actually stopping down the lens. And we'll show
below that this will be useful also for moderate tilt angles.
For large format photographers, actually the majority of users of tilts and
shifts, the trick of a positive or negative "closeup'' lens will only be a
very simple geometrical help to determine where the slanted planes of
acceptable sharpness in object space are located, as explained now.
1.4 Where Mr. Scheimpflug helps us
again and gives the solution
When the film plane is tilted, the ray tracing is similar to the one on
figs.1 and 2, but the object
plane is slanted (figures 3 and 4).
We show now that even in this case, we can also consider the camera lens
fitted with a positive or negative closeup lens to determine the object
planes of acceptable sharpness.
1.4.1 a last argumentation without
analytical calculations...
Now a subtle question that arises is: we now have the formula connecting the
longitudinal position of outoffocus pseudoimages (actually:
elliptical patches, close to a circle, when the tilt angle is small) in the
slanted film plane with the corresponding longitudinal position of a point
source in the object space. But what is the transversal
magnification factor? To find this we need an additional diagram (figure 3).
Figure 3 : The transversal magnification
factor for an outoffocus pseudoimage, p'/p_{2},
is the same as for a true image through a compound lens
Due to basic properties of a geometrical projection of
centre O, if we neglect the "ellipticity'' of the DOF spot,
the centre of the outoffocus image, A''_{2},
is aligned with the median ray A_{2} O A'_{2}.
Hence, the transversal magnification factor for an outoffocus image A''_{2}
is the same as a for a true image when A''_{2}
is formed "sharp'' through a compound lens fabricated by adding a thin
supplementary lens to the camera lens. This transversal magnification factor
(see fig. 3) is equal to p'/p_{2},
the same value would be obtained for A_{2}''
as a true image through the compound lens.
So both in longitudinal and
transversal position the
correspondence between the object space and the image space for outoffocus
images is exactly the same as if viewed "sharp'' through the compound lens.
Applying basic rules of true objectimage formation we already know
that the image of a slanted plane is another slanted plane, we do not need
any analytical proof to derive what follows.
1.4.2 ...and Mr. Scheimpflug gives us
the solution without any calculation!!
As a consequence, without any further calculations we apply
Scheimpflug's rule to the compound optical system and
we
conclude that the limit surfaces of acceptable sharpness for distant objects
are the slanted conjugate planes of the film plane with respect to a
compound, thin lens centred in O, with a focal length equal to f_{1}
(for p_{1}) or f_{2}
(for p_{2}), and that all
those planes G_{1} P_{1}
and G_{2} P_{2}
intersect together in S with the slanted object plane AS and
the slanted Scheimpflugconjugated image plane S
A' as on fig.4.
Figure 4 : Position of slanted planes of
acceptable sharpness, for distant objects, according to this simplest
model: fit the original lens with "closeup'' lenses of focal length (±
H)
To actually define where those planes are located, we simply
have to impose that they should cut the optical axis at a distance p_{1}
(point P_{1}) or p_{2}
(point P_{2}), respectively. Then,
simple geometric considerations on homothetic triangles P_{2}
A B_{2} vs. P_{2}
O S as well as B_{1}
A P_{1} vs. S O
P_{1}
combined with eq.1 yield the interesting and most simple
final result: with h=OS, both distances h_{1}
and h_{2} are equal to
Now consider a plane G_{1}
G G_{2}
perpendicular to the optical axis and located at the hyperfocal distance H
from the lens plane (fig.4). Considering homothetic
triangles G_{1} G S vs.
B_{1} A S and G
G_{2} S
vs. A B_{2} S, we
eventually get
a nice result given by Harold M. Merklinger in
ref.[4]. Note that for far distant objects,
the image point A' on the optical axis is located very close to the
image focal point F'; thus the distance h, hard to estimate in
practice, can be computed from the "camera triangle'' O
S A'
from the focal length f and the estimated tilt angle OSA' as:
tan (O S A') ~ f/h. For small tilt angles tan(O
S A') ~
sin(O S A'), which eventually yields the same result as
in reference [4], where the diagram is
drawn with a reference line perpendicular to the film plane (hence
a sine instead of a tangent) instead of the lens plane like
here.
2 Application to the problem of
DepthofFocus
Another classical photographic problem is the determination of DepthofFocus. The question is:
for a given, fixed, object plane,
what is the mechanical tolerance on film position in order to get a good
image, within certain acceptable tolerances? The following (figure 5)
yields the solution, at least to start with the case of an object plane
perpendicular to the optical axis and an image plane parallel to the object
plane.
Figure 5 : A ray tracing similar to the
ones used in the DepthofField problem yields the solution of the
DepthofFocus problem
If p' denotes the film position for an ideally sharp
image of an object plane at a distance p, the two acceptable limit
film plane positions p'_{1} and p'_{2}
are given by
p'_{1}
= p'(1+ 

) ; p'_{2} =
p'(1 

) (7) 
In order to keep the derivation as simple as possible and keep the
equivalence with a true optical image formation valid, we need an additional
but reasonable approximation, namely that the hyperfocal distance H
is much greater that the focal length f. This is what happens in most
cases and is argumented in the appendix. Within this approximation, it is
found (see details in the appendix), not so surprisingly, that the limit
positions p_{1} and p_{2}
as defined above for the DepthofField problem are approximately the
optical conjugates of the positions p'_{1}
and p'_{2} of the
DepthofFocus problem through the photographic lens of focal length f,
as given by Descartes formula
Combining this eq.(8) with the transversal
magnification formula p_{1}'/p
or p_{2}'/p, still the same for
pseudoimages (the centre of outoffocus light spots) as for real images,
we find that in the general case of a slanted object plane, for far
distant objects (so that equation (1) is valid), the limit positions for the image planes in the DepthofFocus problem are
given by two slanted planes, those slanted image planes of acceptable
sharpness being the optical conjugates (through the lens of focal
length f) of the slanted object planes in the DepthofField problem.
Hence, applying Scheimpflug's rule, we conclude
again that those slanted planes intersect together at the same point S
(see fig. 6)
Figure 6 : For a slanted object plane
AC
located far from the lens, and when the hyperfocal distance is much greater
than the focal length, the image planes of acceptable sharpness SP_{1}'
and SP_{2}' in the DepthofFocus
problem are the optical conjugates of the slanted DepthofField object
planes SP_{1}
and SP_{2} through the camera lens
Appendix : depth of field formulae also valid for closeup,
reasonable limits for the choice of the circle of confusion c
DepthofField formulae valid for closeup
>From Newton's objectimage formulae s×
s'=f× f = f^{2}
it is not too difficult (although rather lengthy) for an object AB
perpendicular to the optical axis to derive more general formulae giving the
position p_{1} and p_{2}
of the planes of acceptable sharpness around a given position of the object
p
(as measured from the lens plane, see fig.1).
Those exact formulae (9) and (10)
are used in a htmljavascript [7] and a
downloadable spreadsheet [8] on Henri
Peyre's French web site. Another derivation, strictly equivalent, is
proposed by Nicholas V. Sushkin [6]
offering an inline graph.
There is however a restriction: those formulae will be also valid for a
thick compound lens where the pupil planes are located not too
far from the nodal planes identical to principal planes in
air. This is the case obviously for a single lens element or a cemented
doublet, but also for quasisymmetric view camera lenses; however for
asymmetric lenses or more generally speaking for a lens where pupil planes
are far from nodal planes, an extreme case being, for example, socalled telecentric lenses, this classical depthoffield approach is no longer
valid. Another ray tracing diagram has to be taken into account; of course
depthoffield will increase when stopping down such a lens, but this will
not be quantitatively
described by equations (9) or (10).
Assuming a given value for c, the diameter of the circle of
confusion, a derivation not shown here yields the following (and
surprisingly simple) result, which is presented in a slightly different but
strictly equivalent form by Nicholas V. Sushkin on his web site [6]


= 

+ 

(1 

) ; 

= 

 

(1 

) (9) 
these formulae can be also written as
where (see fig.1) f is the focal length of the
lens (here considered as a single, positive, thin lens element) p the
position (measured from the lens plane O) of the object plane AB,
assumed to be perpendicular to the optical axis.
Then p_{1} is the position of the
near
limit plane of sharpness and p_{2} the
position of the far limit plane of sharpness. It should be noted in
eqs.(9), that all distances p, p_{1},
and p_{2} are (positive) distances
measured with respect to the lens plane plane O. Here, for a
thin positive lens, O is identical to the principal planes. No
problem with a thick compound lens if pupillar planes are not too far from
principal planes, restarting from the single thin lens element you just
have to "separate'' "virtually'' the object side from the image side by a
distance equal to the (positive of negative) spacing between principal
planes.
Definition of the "true'' hyperfocal distance
Let us first point out that there is a subtle difference in what appears as
the "true'' hyperfocal distance when exact formulae are used. If one tries
in (9) or (10) to find the proper
distance p for which p_{2} goes
to infinity, the value of H+f is found instead of H in
the conventional approach. In this case, the near limit of acceptable
sharpness will be H_{true}=(H+f)/2.
In practice as soon as H is much greater than 5 f, the
difference is not meaningful. It could be possible to rewrite equations (9)
and (10) as a function of H_{true},
but we eventually prefer to denote by hyperfocal distance the
wellaccepted value H=f.f/N.c since it
naturally comes out of the computation, and as it is referred to in many
classical photographic books.
With this assumption on pupillar planes, the formulae given in eq.(9)
are derived from Newton's formulae within the
only, nonrestrictive, reasonable approximation that the circle of confusion
c
(in the range of 20 to 150 microns) is smaller than the diameter of the exit
pupil f/N. Taking c <0.5 f/N
sounds reasonable. For example with f=100mm, the aperture diameter a=f/N
should be smaller than f/100 to be smaller than one millimetre,
whereas conventional values for c never exceed 0.5mm.
It is also possible to think again about the significance of the hyperfocal
distance H by reintroducing the value a of the lens aperture
diameter, a=f/N. The following expression is obtained:
H/f = a/c, in other words the ratio between the hyperfocal distance
H
and the focal length f is equal to the ratio between the lens
aperture diameter a and the circle of confusion c. In most
practical cases, H is much greater than f. Considering a limit
case where H could be close to f, although acceptable from a
geometrical point of view, would yield values for c that are too big
to be acceptable: for example if c can be as big as a/2,
equivalent to H=2.f, the resultant image quality will be
terrible.
Let's put in some numerical data to support this idea. Consider a standard
focal length f equal (by conventional definition of a standard lens)
to the diagonal of the image format; assume that the format is square to
simplify. The image size will be equal to 0.7f by 0.7f (diagonal size = 1.4
times the horizontal or vertical size of the square). If we assume that c=a/2=f/(2.N),
the number of equivalent image dots will be only 2× 0.7× N=1.4× N
both horizontally or vertically. Even at f/90, N=90 this yields a
total number of image points smaller than 20,000 (128 × 128)!!! Even if this
"unsharp'' outoffocus image concerns only a small fraction of the whole
image, such a terrible image quality is clearly unacceptable.
Now that we have shown that it is necessary to limit the upper value for
c
for image quality reasons, this upper limit being somewhat arbitrary, lets
us demonstrate that there is also an absolute, unquestionable, minimum value
for c due to diffraction effects.
This pure geometrical DOF approach is valid as long as diffraction effects
are neglected. Considering a value equal to N microns (1.22 × N
× l, with l = 0.8 µm
in the worst case) for a diffraction spot in the image plane, the other
reasonable condition is c(in microns) < N microns. For example
in medium 6x6cm format with c=50µm, f/32 is a reasonable fstop
whereas f/64 is irrelevant to the present purely geometrical approach. In
4"x5" format taking c=150µm, f/128 will be the smallest
nondiffractive aperture for depthoffield computations.
In macro work at 1:1 ratio (2f2f), DOF does not depend on
the focal length
With all abovementioned assumptions, equation (9) is
valid even for short distances p as in macro work, with p>f
of course to get a real image. This will be in fact irrelevant to our
purpose to find a simple expression and graphical interpretation for
fardistant objects, but is of practical use in macro and
microphotography. For example when p=2f at 1:1 magnification
ratio, the total depth of field is given by p_{2}
 p_{1}
= 4.N.c, and is totally independent from the focal length, a
wellknow result.
A numerical computation in agreement with Stroebel's
diagrams
Unfortunately there is probably nothing really simple in terms of
understanding geometrically depthoffield zones for closeup when
the film plane is tilted at a high angle with respect to the optical axis.
>From eq.(9) we easily derive a limit form valid for far
distant objects, i.e. p much greater than f i.e. p
» f. In this case we can write that f/p « 1 and 1f/p
~ 1, H_{true}= H+f
~ f . This yields the wellknown expressions of eq.(2).
To go a little further, a numerical computation and graphical computer plot
(fig.7) is required. However it is interesting to
find the origin of the diagram presented in Stroebel's
excellent reference book [5], stating that
limit planes of acceptable sharpness intersect all in the same pivot
point P_{f}
located not in the lens plane (like in our approximate model here) but also
in the slanted object plane AS, and one focal length ahead
of the "regular'' Scheimpflug's pivot point S (figure 7).
Without any calculation when p decreases down to the limit value p=f,
it is easy to see from eq.(9) that both values p_{1}
and p_{2} become equal to f this
defining the pivot point Pf.
Figure 7 : A better determination of
slanted object planes of acceptable sharpness, with a pivot point
located one focal length ahead of the lens, according to Stroebel (ref.[5])
and recalculated numerically from eqs. (9)
From the computed diagram, here plotted the particular value
of c=f/1000 (c=f/1750 is mentioned sometimes and
is more stringent), the simplified approach of the "plus or minus H''
closeup lens (yielding slanted planes at large distances p) still
holds remarkably well at f/16, even in the macro range. However at f/64 in
the closeup range the exact calculation will be required, at least for
those inclined to the highest degree of precision, the approximate model
being still an excellent starting point to manually refine the focus for
slanted Scheimpflug's planes. This has been
computed with a very simple gnuplot [9]
freeware script, and will be gladly mailed to all interested readers.
2.1 DepthofFocus formulae
Starting from equation (7) defining depthoffocus
limits without approximation, and combining with the exact DOF formulae (9)
and (10) yields a complicated expression
which would be useless except that its limit form when
H » f
is nothing but equation (8), the additional term
inside the bracket vanishing as f^{2}/H^{2}.
In most photographic situations, with a circle of confusion smaller than f/1000, the correcting factor is also of a magnitude smaller than
1/1000. Equation (11) is actually very close to
Descartes formula (8) connecting
p_{1}
to p_{1}' and p_{2}
to p_{2}', as long as the basic DOF
equation (1) is valid, namely when p » f a
common photographic situation except in macro work.
Equation (7) yields the expression for the
total
depth of focus p_{1}'  p_{2}'
equal to 2p' f/H. Subtituing H by its value
f^{2}/(Nc) yields the expression
2Nc p'/f. In the case of the 1:1 magnification ratio
(2f2f), p'=2f, the same value for depthoffocus or
depthoffield is found i.e. 4Nc, which makes sense considering the
perfect objectimage symmetry at 1:1 ratio.
In the practical case of far distant objects,
p' will be very close
to one focal length f; in this case the total depthoffocus is found
close to 2Nc. Surprisingly this result does not depend on the choice
of focal length f but only the conventional value for c. In
other words, for a given film format or a given camera (35mm, medium format,
large format) if the same circle of confusion is chosen for all lenses
covering a given format with the same camera body, the conclusion, under
those assumptions, is that the choice of focal length has no influence on
the total depth of focus for far distant objects.
However, in order to peacefully conclude on a potentially controversial
subject, the conventional value chosen for c increases somewhat
proportionally to the standard focal length when changing from 35 mm to
medium and large formats; in a sense it can also be said that depthoffocus
is larger in large format than in small format. How this "large format
advantage'' actually helps getting better images in large format for given
mechanical manufacturing tolerances or film flatness cannot be simply
inferred without deeper investigations.
Acknowledgements
I am very grateful to Yves Colombe for his explanations about subtle pupil
effects in a nonsymmetric or telecentric lens. In the more general case,
the projection of the exit pupil actually defines outoffocus
"pseudoimages'' of object points. In the general case, those
"pseudoimages'' do not obey the classical depthoffield formulae nor, of
course, basic objecttotrueimage relations. Simon Clément has pointed to
me the fact the the "true'' hyperfocal distance becomes H+f
when the exact DOF formulae are in use. The subtle question of the
transversal magnification for outoffocus pseudoimages has been clarified
by a passionate debate on one of the US Internet discussion groups on
photography, the key point being mentioned by Andrey Vorobyov [10].
Références
[1] 
Bob Wheeler, "Notes on
view camera'',
http://www.bobwheeler.com/photo/ViewCam.pdf 
[2] 
Martin Tai, "Scheimpflug
, Hinge and DOF'',
http://www.accessv.com/~martntai/ public_html/Leicafile/lfdof/tilt1.html 
[3] 
Harold M. Merklinger, "View Camera Focus''
http://www.trenholm.org/hmmerk/VuCamTxt.pdf 
[4] 
Harold M. Merklinger,
http://www.trenholm.org/hmmerk/HMbooks5.html 
[5] 
Leslie D. Stroebel, "View
Camera Technique'', 7th Ed., ISBN 0240803450, (Focal Press, 1999)
page 156 
[6] 
Nicholas V. Sushkin,
"Depth of Field Calculation'',
http://www.dof.pcraft.com/dof.cgi 
[7] 
Henri Peyre's web site,
in French, "A Javascript to compute DOF limits'',
http://www.galeriephoto.com/profondeur_de_champ_calcul.html 
[8] 
Henri Peyre, "A
spreadsheet application to compute DOF'', in French,
http://www.galeriephoto.com/ profondeur_de_champ_avec_excel.html 
[9] 
"gnuplot, a freeware plotting program for many
computer platforms'',
http://www.ucc.ie/gnuplot/gnuplot.html,
http://sourceforge.net/projects/gnuplot 
[10] 
Discussion group on large format photography,
photo.net, july 2002 :
http://www.photo.net/bboard/qandafetchmsg?msg_id=003Rdn 
Emmanuel Bigler 16 novembre 2002
