Documentation for the HP48Gx calculator - Large Format ...

A large form at Photographer's vade m ecum ¤
HP48GX program
docum entation
Robert E.W
heeler
Jan 1998
¤ c°1997-1999 by Robert E.W heeler,a lrights reserved.
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Contents
1 Introduction 4
1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Fractionalstops . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Num ericalaccuracy . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Setting param eters 7
3 The viewfram e 8
3.1 Program Nam e: VIEW V,From v . . . . . . . . . . . . . . . . . . 9
3.2 Program Nam e: VIEW U,From u . . . . . . . . . . . . . . . . . . 10
3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4 Determ ining lens tilt 11
4.1 TILTB,By back focusing . . . . . . . . . . . . . . . . . . . . . . 12
4.2 TILTD,Tilt from distance andangle . . . . . . . . . . . . . . . . 13
4.3 Tilt from geom etric param eters . . . . . . . . . . . . . . . . . . . 14
4.3.1 TILTJ,Tilt from J . . . . . . . . . . . . . . . . . . . . . . 14
4.3.2 TILTG,Tilt from ° . . . . . . . . . . . . . . . . . . . . . 15
4.4 Tilting the back . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
5 Determ ining DOF 16
5.1 DOFU,As a function of u . . . . . . . . . . . . . . . . . . . . . . 16
5.2 DOF$,From ray angle,for tiltedlenses . . . . . . . . . . . . . . 17
5.3 DOFV,From v . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
5.4 DOFH,From subject height . . . . . . . . . . . . . . . . . . . . . 20
5.5 DOFM,From m agnī cation . . . . . . . . . . . . . . . . . . . . . 20
5.6 DOFD,From defocus . . . . . . . . . . . . . . . . . . . . . . . . 20
5.7 DDOF,From depth of ¯eld . . . . . . . . . . . . . . . . . . . . . 21
5.8 HYP,Hyperfocaldistance . . . . . . . . . . . . . . . . . . . . . . 21
5.9 FUZZ,Blurredim ages . . . . . . . . . . . . . . . . . . . . . . . . 21
6 Finding the subject's height 23
7 Finding the F-num ber,N 24
7.1 NDU,N from ±=2 andu . . . . . . . . . . . . . . . . . . . . . . . 24
7.2 NNF,N from near andfar values . . . . . . . . . . . . . . . . . . 25
7.3 NDOF,N from DOF andu . . . . . . . . . . . . . . . . . . . . . 26
8 Finding the focallength 26
9 Finding the m agnī cation 27
10 UV 28
11 Be lows Factor 28
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12 Misce laneous program s 29
12.1 AOV,from m agnī cation andfocallength . . . . . . . . . . . . . 29
12.2 MOTIO,shutter speedto stop m otion . . . . . . . . . . . . . . . 30
12.3 STOPD,stop di®erence for light m ovem ent . . . . . . . . . . . . 30
12.4 TRIAN,triangulation . . . . . . . . . . . . . . . . . . . . . . . . 30
12.5 RESL,resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
12.6 RESU,at the DOF lim its . . . . . . . . . . . . . . . . . . . . . . 31
13 Technicalities 32
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1 Introduction
W hen a university education wasm ostly a study ofGreek andLatin,schoolboys
carriedabout sm a lbooks containing crib notes,gram m aticalm em oranda,and
other helpfulbitsof inform ation. Itisnotsurprising thatthey referredto them
by a Latin tag,vade m ecum ,which translates as \go with m e." I,andI think
m ostlarge form atphotographers,do som ething sim ilar. In m y bag isa notebook
¯ ledwith variousthings,such as¯lter factors,reciprocity tim es,shiftlim itsfor
various lenses,etc.
In addition,I carry one other thing,which I consider m y realvade m ecum . It
isan HP48GX pocketcalculator,andin itI have program m edthose calculations
that are usefulfor m y sort of photography. Yesterday,for exam ple,I took two
pictures while on a walk. One requireda tilted lens, andthe other involved a
nice calculation of DOF to include an object in the foreground. Neither took
m ore than ¯ve m inutes, and a third photograph was abandoned before the
cam era was unpacked,because prelim inary calculations indicatedthat it could
not be taken in the way I envisioned | I'lprobably go back and rethink the
com position som etim e.
W hen I startedin large form at photography,I was unable to ¯ndadequate
instructions for m any ¯ddling problem s in the books I consulted. I found exhaustive
discussion of light and photographic m aterials, but little help on the
practicalproblem s. Lens tilt,for exam ple,seem edas m uch a m ystery to m any
authorsof photography booksasitdidto m e | atleastif they knew itsincantation,they
chose not to revealit.
As near as I could m ake out,tilting the lens was som ething to be done by
cut-and-try | focuson som ething,tiltthe lensa bit,focuson som ething,adjust
the tilt,etc. untila lparts of the subjectare in focus 1 . I triedthis a few tim es
with m iddling success, but found it hardly a satisfactory procedure: if for no
other reason,than thatittakesa long tim e. I am sure the inform ation existsin
the technicalliterature som ewhere,but in exasperation I sat down andderived
the equations from ¯rst principles. By that I m ean I went back to a theorem
of projective geom etry due to Desargueswhich underliesthe rulesartistsuse to
produce perspective drawings,andderivedthe lens equation relating the focus
distance,the lens-¯lm distance,andthe lensfocallength. From thiseverything
else fo lows.
The program s described here, calculate the lens tilt angle in several ways.
The easiest is by focusing on two points in the subject plane,which is the idea
that underlies the Sinar, and Linho® built-in calculations. Alm ost as easy, is
to use the distance and angle of a pair of points in the subject plane. There
are two additionalm ethodswhich m ay appealto som e who fancy thatthey can
judge angles by eye.
Of course,a photographer hasother problem s: depth of ¯eld,DOF,for one.
The standardform ulasthatappear in bookswork we lenough for interm ediate
anddistant objects,but are considerably in error for m acro photography. This
1 A discussion of this procedure in exquisite detailm ay be foundin Bond (1998).
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isstrange,since the correctform ulasare notdi±cult. Perhapsthe authorshave
stuck to form ulas that they thought wouldbe easy to calculate. In any case,if
one is going to use a calculator, which I hope you wil, then there is no need
to use the wrong form ula. In addition, the practicing photographer needs to
m ake judgm entsaboutDOF for tiltedlenses,andassessthe degree of blur for a
backgroundor foregroundobject. Six program sare providedfor these problem s,
di®ering in the type of input required.
Various questionsarise in practice about focal length: (1) given near and
far objects, which focal length wil cause these to be the near and far DOF
lim its? (2) Given the m agnī cation andthe distance to the subject,whatisthe
proper focallength to choose. Sim ilar questions occur in relation to the proper
F-num ber. Andwhat about the be lows factor? There are other questions stil
{ what shutter speedwilstop m otion,or what is the angle of view? Program s
are providedfor a l.
Perhaps the m ost im portant thing,is the viewfram e: a sim ple fram e m ade
from what you wil with a m easuring cord attached. I am not aware of any
serious discussion of this m ost usefuldevice. It not only concentrates the photographers
attention by fram ing the view, but enables the appropriate lens to
be chosen,andwith the two program sincluded,a lowsthe photographer to ¯nd
the DOF { a lwithout unpacking the cam era.
The program s m ay be entered into a calculator by hand, or they m ay be
downloaded from a PC via a serial cable. Instructions for downloading are
given in an associated docum ent. The m athem atical details of m any of these
calculations do notseem to be available in the literature,andso a co lection of
notes m ay also be downloaded.
I assum e thatthe reader isa large form atphotographer,andthatthe problem
sdiscussedwilbe those thathave been thoughtabout. I wouldn'tdiscourage
som eone who isnew to thisform atfrom reading thism aterialandusing the program
s,butI think itunlikely thatthey wilfu ly appreciate the program suntil
they have experienced som e of them ¯rsthand. There are a num ber of good 2
books on large form at photography,andI wouldhope that the neophyte would
consult them ¯rst. Two that I can recom m end are: Stroebel et.al. (1986),
Stroebel (1993) For the m ore technica ly inclined, Ray (1994) and Jacobson
et.al. (1988) are also good.
1.1 Notation
I have chosen to use a few standardsym bols both in the program s andin this
text,rather than repeatdescriptive phrasesatevery point. They are ilustrated
in Figure 1, and listed in Table 1. The ¯gure shows three planes: the subject
plane,the lens plane,andthe focalplane. In addition,the Near andFar DOF
planes are shown aboutthe subjectplane,asare their cognates about the focal
plane. Note that the Near and Far planes are m easured from the lens plane.
2 Although on topic,I cannotrecom m endMerklinger (1992)or (1993) because sim ple ideas
are m ade overly com plicated.
5

The distances of these planes from the subject plane are denoted FrontDOF
andBackDOF.The Near andFar planesare notsym m etricalaboutthe subject
plane,buttheir cognatesaboutthe focalplane are sym m etrical,for a lpractical
purposes. Thus one m ay use ±=2 to represent the distance from either to the
focalplane. The lens tilt angle,µ,wilbe discussedlater.
Subject
Plane
Far
Lens
Plane
Focal
Plane
Near
Lens
BackDOF
FrontDOF
u
F
v
δ
Figure 1: Para lelplanes
u Distance from subject focus point to lens
v Distance from lens to ¯lm
F Focallength
N
The F-num ber
± Defocus ´ Depth of focus
µ Lens tilt angle
Near Near DOF lim it
Far Far DOF lim it
FrontDOF = (u ¡Near): i.e. DOF in front of focus
BackDOF = (F ar¡u): i.e. DOF in back of focus
Table 1: Frequently usedsym bols
In addition,units of m easurem ent are indicatedby appending a unit designator
starting with an underline \ " to values. Thus5 m m eans5 m eters,3 m m
m eans 3 m ilim eters,10 ± m eans 10 degrees,etc.
1.2 Fractionalstops
The N valuesare spaceduniform ly on the shutter,andone can setinterm ediate
values, such as a value half way between 16 and 22. The corresponding N is
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19.6,which requiressom e calculation to discover,andwouldbe a nuisance were
the program s to dem andits input. Therefore,the program s a low the input of
interm ediate values using the form \16 5",where both the quotes andthe \ "
are required. \16 5" m ay be read as the shutter setting half way between 16
and 22. In actuality, the program translates the quoted string into a num ber
(19.6) which it stores in the variable N; and when it displays N, it displays a
quotedstring if needed. Fu lF-num bers m ay be input either as num bers 3 ,like
16,or as strings with quotes like \16."
1.3 Num ericalaccuracy
The am ount of light adm itted through the aperture (ilum inance) is of great
im portance, as early photographersquickly learned. It m ay be contro led by
the duration of the exposure andby the lens aperture. About 1880 (Kingslake
1989) it becam e custom ary to designate a lens by its F-num ber. Iris shutters
were also introducedabout 1880 (ibid),andm arkings on the lenses were likely
to have been in term s of F-num bers. Sequences of stops and F-num bers were
proposed at the turn of the century, but the present standard sequence of F-
num bers (1,1.4,2,2.8,4,:::) is fairly recent.
There is not a lot thatcan be saidin justī cation of thisstandardsequence
for a practical photographer. It requires an act of m em ory to use, and it increasesasthe
ilum ination decreases,which iscontrary to whatone wants. One
way to think of itis asa m easure of the lum inance of the object,in which case
a sm a l opening and large F-num ber represents a bright object, but the odd
sequence is stilsom ething to rem em ber.
Twice the base 2 logarithm of the F-num ber sequence is (0, 1, 2, 3, :::),
which would m ake things easier if only cam eras were m arked this way 4 . In
this sequence, the steps represent a halving of the light intensity, so that 3
represents half the light intensity of 2. For interm ediate values, light m eters
report the decim alfraction from this scale: thus one m ay read 1.45 on a light
m eter,representing a tic half way between the f/1.4andf/2 tics on a cam era 5 .
Shutterson autom atic cam erascan be capable of contro ling aperture to a tenth
of a step,butothersm ay notbe so accurate. A likely practicallim itisa setting
to the nearest 1/3 stop.
The notation adopted herein is 1.4 5. The \ " is required. The program s
read and write num bers in this form , but interna ly they are translated into
F-num bers,so that 1.4 5 is translatedinto N = 1:68.
2 Setting param eters
Program
Nam e: P
3 Actua ly inputnum bers are rounded to the nearest F-num ber before assignm ent to the
variable N,so inputting 20 results in N being set to 16.
4 The EV system was origina ly developedfor this purpose,but never becam e popular.
5 Such an interm ediate tic doesnotrepresentthe halfway intensity of 75%,butrather about
71%.
7

Four param eters are needed for m ost calculations. They are focal length,
F, the F-num ber, N, the form at and optiona ly C, the diam eter of the circle
of confusion. It is incum bent on the user to m ake sure that they are always
current. Som e program s use them , others do not, but it is a waste of tim e to
try to rem em ber which is which,so always m ake sure they are current.
They m ay be set by running the P program . This can always be done by
entering 6 P or pressing the corresponding m enu key ifitisvisible. Thisfunction
throws up a display showing the current values,andthen shows a m enu along
the bottom ,as in the fo lowing screen capture:
To change the F or N values,enter a new value in the calculator andpressF
or N.To change the ¯lm size,pressFILM andchoose one of the options: they
are 35m m ,645,6x7,6x9,4x5,and8x10. In generalC shouldnot be changed.
A ¯lm size change wilsetC to a standardvalue,which m ay then be changedif
desired.W hen the param eters are correct,press the EXIT key. The needfor
pressing EXIT can be confusing since thisisthe only program requiring it. A l
other program sperform their calculations,display them ,andexitautom atica ly.
I lustrations in the rest of this docum ent assum e the param eters are set as
in the above screen capture { F at 210 m m ,N at 32,andFILM at 45.
3 The viewfram e
Even before the cam era is set up, a decision about the view to be captured
should be m ade. This is aided considerably by the use of a sim ple viewfram e
heldso asto fram e the scene. Alm ostanything with the rightsizedhole wildo
| I use a bent coat hanger.The inside dim ensions shouldbe the sam e as the
size of the im age on the ¯lm . In addition som e m eans should be provided for
m easuring the distance from the viewfram e to the eye. I attach a °exible tape
m arkedin m ilim eters to m y viewfram e. Figure (2) shows som e item s from m y
gadget bag,am ong them m ay be seen m y bent coat hanger viewfram e.
Use the viewfram e to fram e the view,andnote either of two distances: (1)
the distance, v, from the viewfram e to one eye | the other should be closed,
since a cam era has only one eye; or (2) the distance, u, to the point of focus.
The fo lowing two program s wilprovide depth of ¯eldandother inform ation.
6 Press ®,then P,then ENTER
8

Figure 2: Gadgets
3.1 Program Nam e: VIEW V,From v
The program asks for two param eters. The input is on the left,the output on
the right.
Here v is the distance in m ilim eters from the lens to the ¯lm plane. The
F-num ber,N,m ay be input since one som etim es needs to com pare the e®ects
of di®erentN. If N isinputas0 or blank (i.e. as"")the currentvalue of N,set
by the P program ,wilbe used. This current value is unchangedby VIEW V.
On the output, the u value is the distance from the lens to the in-focus
subject. The FrontDOF and BackDOF values are the distances from u to the
near andfar DOF lim its. In thisilustration,the DOF is19.57 m (3:52+16:05).
9

3.2 Program Nam e: VIEW U,From u
The program asks for two param eters. The input is on the left,the output on
the right.
The only di®erence from the previous program ,is that u,the focus distance,is
input. The F-num ber,N,m ay be input since one som etim es needs to com pare
the e®ects of di®erent N. If N is input as 0 or blank (i.e. as "") the current
value of N,setby the P program ,wilbe used. Thiscurrentvalue isunchanged
by VIEW U.
The u value is repeated from the input, and the v value calculated. The
FrontDOF and BackDOF values are the distances from u of the near and far
DOF lim its.
3.3 Discussion
A viewfram e im provesthe visualization of a com position. Itaidsin deciding the
com position's orientation (landscape or portrait), and aids in the elim ination
of extraneous detailwhich the eye m ight otherwise not notice. There are other
uses.
Itm ay be usedto ¯ndthe appropriate focallength. Suppose a com position
fram es nicely in the viewfram e when the subject is about 4 m away, andthat
the viewfram e to eye distance,v,isabout220 m m . Any lenswith lessthan 220
m m focallength m ay be chosen andthen extendedto 220 m m for focusing. A
210 m m m ight be appropriate. Of course one couldalso choose a 150 m m ,but
the extension to 220 m (150 m m + 70 m m ) wouldresult in a be lows factor of
about 2. In this case, VIEW V using v = 220 m indicates the FrontDOF and
BackDOF values are 1.12 m and2.17 m ,assum ing F = 210 andN = 32.
In m aking a decision aboutfocallength,there isno needto be overly precise
about v. A few m ilim eters either way wilnot change the com position m uch.
The im portant thing is to decide on a focal length with approxim ately the
coverage desired and leave the ¯ne details to that tim e when the cam era is
focused.
Using a viewfram e in this way is quite easy. W hat happens if one needs
m ore depth of ¯eld, such as a foreground object? I raise the point, because
som e wilattem pt to adjust the DOF by changing their lens. There is nothing
wrong with this if it m eets the artistic needs, butoften the resulting change
distortsthe com position. Extraneousdetailwhich wasoutside the fram e isnow
included,or the com position loosensup andbecom esuninteresting. To keep the
10

sam e com position when changing the lens,the cam era m ustbe m oved. Thiswil
keep the m agnī cation constant. Unfortunately,the DOF doesnotchange very
m uch when m agnī cation is ¯xed. You m ight like to use the DOFM program
which gives DOF as a function of m agnī cation to establish the truth of this
statem ent 7 .
To increase DOF in a substantialway,the F-num ber,N,m ust be changed.
In this case doubling N from 32 to 64, produces FrontDOF = 1:8 m and
BackD OF = 8:21 m . The com position rem ains unchanged.
4 Determ ining lens tilt
W hen the subject,lens,and¯lm planes are para lel,focusing on any one part
of the subject focuses on a lparts. This is ilustratedin Figure (1).
Subject Plane
FrontDOF
Lens
Plane
Focal
Plane
u
u'
Lens
v
BackDOF
δ
F
Figure 3: Tiltedplanes
This is not true when the subject plane lies atan angle to the other planes.
Focusing on one part of the tilted subject plane m ay cause other parts to be
outof focus. To bring a lpartsof the subjectinto focus,the lensm ustbe tilted
so that a l three planes m eet at a line. This is Scheim p°ug's rule. Figure (3)
ilustrates this rule. The para lel planes in Figure(1) also obey Scheim p°ug's
rule if one agrees,as is usual,that para lelplanes m eet at in¯nity.
If the subject plane is tilted, then it wil m eet the focalplane som ewhere.
The problem isto ¯ndthe tiltangle of the lenssuch thata plane through itwil
also m eet where the other two planes m eet. Many experienced photographers
decide on the tilt angle by cut-and-try m ixed with considerable experience. If
the ground glass were brightly ilum inated, I would not ¯nd m uch fault with
7 The m agnī cation for the above ilustration is about 0.021,which can be foundfrom the
MA GV program .
11

such a procedure: but it is not,it is alm ost always dim requiring a dark cloth
to block light,and for som e lens's with sm a lm axim um apertures,the ground
glass can be grainy m aking nice judgm ents di±cult. I prefer to calculate the
angle from observations,andthe fo lowing program s do this. The BACK TILT
param eter which appears in the input m enus of the fo lowing program s wilbe
discussedin Section (4.4).
It is im portant to note, that although \tilt" is used in this section, the
inform ation applies equa ly to \swing."
4.1 TILTB,By back focusing
The easiest way to determ ine lens tilt is by focusing on two subjects which
im age near the top andbottom of the groundglass. Figure (4) shows this for
two back positions, one when point A is in focus,and one when point C is in
focus. The distance along the rail (in m ilim eters) between the focus points,
together with the distance between the im ages (in m ilim eters) on the ground
glasscan be usedto determ ine lenstilt. In ¯gure (4) the Rail¢ is the distance
from ato b,andthe Glass¢ isthe distance from bto c. The calculation of the
lenstiltangle in degrees isgiven approxim ately by W heeler'srule of 60. To wit
Á ¼ 60(b¡a)=(c ¡ b). The rule is not appropriate for close up work, but the
TILTB program always gives the correct value.
Subjet Plane
Tilted
Lens Plane
Untilted
Lens Plane
Focal Plane
A
Lens
u'
φ
Back
with A in
focus
c
Back
with C in
focus
C
a
b
Figure 4: Tilt Diagram
The program inputison the leftandthe program output on the right. The
inputusesthe distance u 0 shown in Figure(4)which isthe horizontaldistance to
the subjectplane. The tiltcalculation doesnotm ake use of u 0 ,butitisusedin
12

calculating the slope ° of the subject plane,which param eter is savedfor later
use by DOF $.
The program outputsthe lenstiltangle in degrees. The angle ispositive for
forwardtilts,andnegative for backward. Once the lensistilted,andrefocused,
a lpoints in the lens plane wilbe in focus.
In refocusing after the lens tilt,those with center tilt cam eras wil¯ndthe
focus point som ewhere between the two previous points,while those with base
tilt cam eras wil ¯nd it necessary to m ove the rear standard a considerable
distance forward. Base tilt cam eras m ove the lens in addition to rotating it,
andthe rear standardm ust be brought forwardto adjust for this m ovem ent.
4.2 TILTD,Tilt from distance andangle
Subjet Plane
A
High u'
Tilted
Lens Plane
High
angle
φ
Untilted
Lens
Plane
Focal Plane
u'
C
Low angle
Low u'
J
γ
Figure 5: Distance andangle
Another way to determ ine lenstilt is by specifying the anglesanddistances
to two objects in the desiredsubject plane. Figure (5) shows two points A and
C on the subject plane. The distances and angles of these points are input
13

to TILTD. The program assum es that angles above the horizon are positive,
and negative below the horizon. One can buy palm size devices to m easure
anglesabove andbelow the horizon from shopsthatse lsurveyors'sequipm ent.
Distance can be guessed. The calculations are not very sensitive to the far
distance,but the near distance shouldbe as accurate as possible. If a com pass
is available, then the program TRIAN m ay be used to triangulate from two
bearings | the com pass m ust read in degrees or ¯ner. The program input is
on the left,the output on the right.
4.3 Tilt from geom etric param eters
The nexttwo program sask for inputsderivedfrom the geom etry ofthe problem .
These inputs can often be guessed with reasonable accuracy, and m ay prove
easier to use in certain circum stances. Figure (5) shows the planes and rays
involvedfor lens tilt calculations. The distance J is the verticaldistance from
the lens center to the subject plane,and° is the angle that the subject plane
m akes with the lens plane.
4.3.1 TILTJ,Tilt from J
The values of J andu 0 are guessed. The output is shown at the right.
Merklinger (1992)and(1993)recom m endsthism ethodusing J alone,which
isa lthat is requiredto calculate the tiltangle µ. TiLTJ asksfor u 0 in order to
calculate ° which is neededfor DOF$.
14

4.3.2 TILTG,Tilt from °
Guessing the subject plane angle ° andu 0 . The output is shown at the right.
4.4 Tilting the back
Subjet Plane
Tilted
Lens Plane
φ
Untilted
Lens
Plane
θ
Untilted
Focal Plane
Negative
angle
Tilted
Focal Plane
u'
J
γ
Figure 6: Tiltedback
Tilting the back changes perspective by altering the way lines converge to
vanishing points. Som etim es this change is desired. Scheim p°ug's rule stil
applies when the back is tilted,as Figure (6) shows. The appropriate lens tilt
angle wil be calculated by each of the above program s when the Back Tilt
param eter is input. The input param eters shouldbe obtainedwith an untilted
back precisely as has been done above. The only change in the program input,
is the setting of Back Tilt to a non-zero value. Consider the TILTB exam ple
15

above:
W ith the back tiltedforward10 ± ,andthe lenstilted11:2 ± the subjectplane
wil be in focus, although previously para lel vertical lines wil diverge. It is
assum edthat positive back tilt angles im ply a forwardback tilt,and negative
angles a backwardtilt,just as they do for the lens tilt.
Back tiltscan be usedeven when the subjectplane isvertical. The program s
wiloutputthe necessary lens tilt degrees. For TILTB,a verticalsubject plane
is indicatedby setting the Rail¢ to 0. The Glass ¢ m ust have a non-negative
value. For TILTD,the distances andangles m ust specify a verticalplane. For
TILTJ and TILTG inputting 0 for J and 0 for °, the subject plane angle,
indicates a verticalsubject plane.
5 Determ ining DOF
DOF is an im portant topic,andthe photographer needs to assess it in several
ways. The usualway is to calculate DOF as a function of u. But one can also
use v, subject height, m agnī cation, or the defocus ±. Program s are provided
for each of these input values, as we l as one that calculates the DOF along
an arbitrary ray when the lens is tilted. Please refer to Figures (1) and (3).
In addition, there is a program to calculate the hyperfocal distance, one that
assessesthe blur of objectsatvariousdistances,andtwo thattranslate DOF to
andfrom ±.
A lDOF program s report the sam e inform ation,so this output wilbe describedm
ore fulsom ely in the next subsection, (5.1) than in the other subsections.
5.1 DOFU,A s a function of u
The input is u,andthe initialoutput is shown on the right.
16

The Be lowsfactor andthe param eter u appear above the top of the output
list,andare not initia ly visible. To see them one m ust scro lthe display,as is
done here:
A lDOF program s output u. For DOFU,it is input. For other DOF program
s it is calculated. The FrontDOF and BackDOF are distances about u.
Thus the near point of the depth of ¯eldis u ¡F rontDOF,andthe far point
is u + BackD OF .
The defocus distance ± is the distance on the rail corresponding to DOF,
see Figure(1). For practicalpurposes,thisdistance is sym m etricalaboutv,the
distance on the railcorresponding to the subject distance u. Thus,the points
v ¡ ±=2 and v + ±=2 are defocus lim its, corresponding to near and far DOF
lim its. One can use the defocus lim its to locate the near andfar DOF planes.
Sim ply m ove the standardback or forwardby ±=2 andobserve those objects in
sharpest focus | these correspondto objects on one of the DOF planes.
The adjustedF-num ber,Adj N,shown isthe be lowscorrectedF-num ber. In
this case,the originalF-num ber,N,was 32. A lowing for the be lows extension
produces a value of 22.9,which is not practica ly di®erent from 32.
Before releasing the shutter, it is always a good idea to perform a
DO F calculation in order to check the bellowse®ecton the F-num ber
{ surprises do occur.
The DOF lim its shown by the DOF program s are calculated assum ing the
inputN.If the adjustedN di®ers practica ly from thisN,andif the adjustedN
is actua ly usedto set the shutter,it wilbe necessary to recalculate the DOF
lim its by inputting the adjusted N to DOFU. Of course a doubly adjusted N
wilbe output,but this shouldbe ignored.
5.2 DOF$,From ray angle,for tiltedlenses
In order to setparam etersthatwilbe usedby DOF$,itisnecessary to run one
of the tiltprogram s. Other program sm ay intervene. DOF$usesthe param eters
from the last run TILT program ,andno other program s tinker with the saved
TILT param eters.
The DOF$program acceptsa single input,an angle,andoutputsthe FrontDOF
andBackDOF values along a ray at this angle to the horizontal. Figure (7) illustrates
the situation. Assum ing that TILTB has been run with the fo lowing
17

Subject Plane
Lens
Plane
Focal
Plane
BackDOF
FrontDOF
Lens
BackDOF
FrontDOF
u'
Figure 7: DOF for tiltedlens
param eters
then inputting the angle zero into DOF$,ason the left,producesthe outputon
the right.
Scro ling up wouldshow that u = 10,which was the value input in TILTB.
This calculation represents DOF for horizontal distances through a tilted
lens. It can be com paredwith the output of DOFU for an untiltedlens:
18

.
There seem sto be little di®erence,butthisisnotthe case when one looksalong
rays at an angle.
The output resulting from an angle of 10 ± is shown below on the left, and
¡10 ± on the right. The corresponding u's are 12.78 m and8.42 m .
The e®ect of lens tilt on DOF is substantial.
It is im portant to note that the defocus, ± m ay be used to ¯nd the near
andfar DOF planes for tiltedlenses,just as it can be for untiltedlenses. The
defocus lim itsare v ¡±=2 andv + ±=2,andthe distance v is the position of the
rear standardwhen the subject plane is in focus. By m oving the rear standard
forwardor back ±=2 m ilim etersthe DOF planesare those which are in sharpest
focus.
5.3 DOFV,From v
The distance between the lens andthe ¯lm plane is v. It m ay be usedto ¯nd
DOF.Inputandoutputfor the DOFV program are shown below. Asa sidelight,
note thatthe adjustedN value islittle di®erentfrom the nom inalN of 32. This
is because the extension 215 ¡ 210 = 5 m m is very sm a l relative to F = 210
m m . For close up work,however,the adjustedN wilbe considerably di®erent.
Suppose one were interested in a 1 to 1 im age, then the 210 m m lens would
have to be focusedat 440 m m ,andthe adjustedN wouldbecom e \11 9," and
the DOF wouldshrink to about 11 m m .
19

5.4 DOFH,From subject height
By subject height is m eant the verticalextent of the subject which wil¯ lthe
long side 8 of the im age on the ¯lm . In the present case with u = 10 m , the
height is 5.78 m . The input andoutput are shown for this value.
5.5 DOFM,From m agnī cation
Magnī cation isthe ratio v=u. In the presentcase itis210=10000 = 0:021 with
both v and u in m ilim eters. Magnī cation is sm a l for distant objects, but
becom es large for close up photography. A m agnī cation of 1 occurs when the
im age andsubject heights are equal.
The outputwouldhave been slightly di®erenthad0.02 been usedfor m agni-
¯cation. W henever the output of two program s is unexpectedly di®erent,look
to the input values for an explanation. The calculations are done to m any
m ore places than are shown, and using too few places in the input can cause
discrepancies.
5.6 DOFD,From defocus
The defocus,±,is the distance along the railbetween the cognates of the near
and far DOF: i.e. the di®erence between the locations of the two standards
when the lens is focusedon each of the two DOF lim its. This is ilustratedin
Figure (1). For practical purposes, it is sym m etrical about v, the position of
the rear standard when the subject is in focus. Using ±=2 = 3:27 m m to be
8 The program chooses a ¯lm height of 124m m for 4x5.
20

consistent with the previousexam ples,the inputandoutputscreensappear as:
5.7 DDOF,From depth of ¯eld
This program accepts the DOF. To be consistent with the previous exam ples,
the DOF is set to 28.7 m .
5.8 HYP,Hyperfocaldistance
This sim ply returns the hyperfocaldistance for the globalparam eters.
The ±=2 andadjustedN are di®erentfrom the previousprogram sbecause u
has changedfrom 10 m to the hyperfocaldistance of 13.991 m .
5.9 FUZZ,Blurredim ages
Bokah is the Japanese term for out of focus or blurred objects. There is good
bokah andbadbokah,butthis is not the place to discuss its nature,andis only
broughtup here to introduce the factthatblurredim agescan form usefulparts
of a picture. Som etim es one has intrusive objects in the fram e that needto be
blurred, and som etim es it is just better to have a fuzzy area in a picture to
support in a sense the m ain subject.
21

Focal Plane
Subject
Plane
Far
Lens
Plane
Near
Lens
Object fuzz
Back fuzz
Front fuzz
Figure 8: Fuzz at three distances
In any case, the FUZZ output shows the degree of blurring for objects at
specī c distances. FUZZ always calculates blurring values for the DOF lim its,
andin addition itwilcalculate them for a user inputdistance. Consistentwith
the above exam ples, suppose u = 10 m and let us suppose we need to know
aboutan objectat50 m . Figure (8) showsthe size of objectsatthree distances
which im age on the ¯lm atexactly twice the diam eter of the circle of confusion.
The input andoutput are:
The fuzz values reportedin the output,are the sizes of objects at the given
distanceswhich wilim age astwice the diam eter of the circle of confusion 9 Thus
objectsatthe Near DOF ofsize 2.7 m m wilbarely be distinguishable. Sim ilarly,
objects of size 16.1 m m at the Far DOF lim it wil not be distinguishable. At
9 To ¯ndthe size of an object which im ages k tim es the diam eter of the circle of confusion,
one should m ultiply the size, s, outputby the program , by k ¡ 1. For k = 2 the m ultiplier
k ¡1 is1,andthisissasoutputby the program . The size of the objectim aging atfour tim es
the diam eter of the circle of confusion would be (k ¡ 1)s = 3s. Using 4s instead, however,
does little harm .
22

50m ,a 26.3 m m object wilnot be distinguishable. This m eans that inch high
lettering on an intrusive bilboard som e 50 m distant, wil not be readable in
a print. The lettering would have to be at four or m ore tim es this size to be
readable,andeven then wouldbe very fuzzy.
This is especia ly useful for m acro photography. Suppose you are photographing
a °ower at one-to-one m agnī cation and there is an unavoidable
object in the background. How visible wilthis object be? For a 210m m lens,
one-to-one m agnī cation puts the subject plane 420m m in front of the lens.
Suppose the objectionable object is twice this far, say 800m m away from the
lens. The FUZZ input andoutput are:
If the object is sm a ler than 5.9 m m , then it wil not be visible. If it is
larger, say 25m m , then it wil be very fuzzy because its top edge wil not be
distinguishable from a line 1=4of thisdistance down. The size of the objectwil
thus correspondto about four tim es the circle of confusion on the ¯lm ,clearly
a negligible am ount.
6 Finding the subject's height
By subject height is m eant the heightof the subject thatjust¯ ls the long side
of the ¯lm im age. The ratio of the long side of the ¯lm to the height is the
m agnīcation 10 . The height m ay be calculatedfrom any of severalparam eters.
Five program sare provided. They andtheir param etersare shown in Table (2).
Program Param eter
HU u
HV v
HM m agnī cation
HDOF FrontDOF
HD ±=2
Table 2: Height program
param eters
A l input screens are sim ilar, requiring a single param eter. Only the HU
program wil be ilustrated. The output screen shows the subject height in
10 Subject and¯lm diagonals are often usedinstead of the height.
23

m eters. The input andoutput screens are:
7 Finding the F-num ber,N
There are three program s, NNF, NDU, and NDOF, which calculate the F-
num ber,N.
7.1 NDU,N from ±=2 andu
The inputsare ±=2 andu,asshown on the inputscreen atthe left. The output
is shown on the right,andif it is scro led,one ¯nds u = 10 m m .
The be lows correction shown at the bottom of the output depends only on
the be lows extension andis thus the sam e for any N.It is given in stops,and
m ay be subtractedfrom whatever N one selects. Thus\8 1" becom es\8 0" and
\32 3" becom es \32 2."
Figure (9) ilustrates the situation for the case when the lens is focused at
in¯nity. In this ¯gure, the optim um N line m ay be visualized as the crest of
a m ountain with the land sloping downward away from it. The \10 l/m m N"
curve relatesN to ± such thatthe on-¯lm resolution is10 l/m m . The \40 l/m m
N" and\20 l/m m N" curvesdo the sam e for 40 l/m m and20 l/m m respectively.
The extent of the ± scale is appropriate for the 4x5 form at. As a contrast,the
dotted box at the lower left represents the ± scale appropriate for the 35 m m
form at. The reason that m anufacturers choose N = 16 or N = 22 as the
m axim um F-num ber for 35 m m shouldbe clear from this ¯gure.
Should one use N or optim um N? It a l depends on what is wanted. The
output N corresponds to 10 l/m m ,which is appropriate for an 8x10 print. The
optim um N wil of course support larger prints, but there seem s little reason
for choosing the optim um unless such large prints are the goal, and even then
as m ay be seen in Figure (9) the optim um N wil be less than 20 l/m m for
24

40 l/mm N
N
64
45
32
22
16
11
8
20 l/mm N
10 l/mm N
Optimum N
0 1 2 4 6 8 10
δmm
Figure 9: Optim um
andresolution curves
±'s greater than about four. The program
resolutions for particular com binations.
RESL m ay be usedto calculate the
7.2 NNF,N from near andfar values
NNF acceptsNear andFar values for which it¯nds the N that wilm ake them
Near andFar DOF lim its.
The input and output screens for NNF are as fo lows. Do not confuse the
\Near" and\Far" values with \FrontDOF" and\BackDOF" values.
Scro ling up the output screen shows u = 10:24: The calculatedN is about
25

32. In addition the N which produces m axim um resolution is shown as the
optim um N.
7.3 NDOF,N from DOF andu
NDOF accepts DOF and u, and ¯nds N. The input screen is on the left, the
output on the right.
8 Finding the focallength
Focal length can be chosen in a variety of ways. The best way is to use a
view¯nder to obtain a proper fram ing of the scene; however, other ways are
possible. One m ay choose two distances,andm ap them into the DOF lim itsby
a proper choice of focallength. Sim ilarly, one m ay ¯x som e other param eter,
such as DOF or u,andcalculate the focallength. Five program s are provided.
They andtheir input param eters are show in Table (3).
Program Param eters
FNF Near Far
FUD ±=2 u
FDOF DOF m agnīcation
FMU u m agnīcation
FMV v m agnīcation
Table 3: Focallength program s
The program FNF wil be ilustrated. The input and output screens are
shown below. The F returnedisthe F requiredto m ake 6 m and35 m the DOF
lim its.
26

9 Finding the m agnī cation
Magnī cation is the ratio of the ¯lm size to the subject size 11 ,thus for a 2 m
ta lsubject im agedon a 4x5 ¯lm with long side 124 m m , the m agnī cation is
124=2000 = 0:062. Magnī cation controls the appearance of a picture, in that
the subjectsize wil¯ lthe sam e area ofthe ¯lm ifthe m agnī cation isconstant.
A picture taken with a 210 m m lenswil fram e the sam e com position as one
taken with a 600 m m lens if the m agnī cation for the two is the sam e. For
exam ple,in the case of a 2 m ta lsubject taken with a 210 m m lens on a 4x5
cam era,the subject m ust be 3.6 m away. For a 600 m m lens,the subject m ust
be 10.2 m away. The UVM program m ay be usedto con¯rm this.
Two pictures with the sam e m agnī cation taken with di®erent lenses m ay
or m ay not appear identical { there can be a di®erence in resolution. The
di®erence wilbe quite sm a l,as m ay be seen by checking the DOFM program
which produces the values in Table (4). The F-num ber in this table is N = 32
for both lenses.
Lens Distance FrontDOF BackDOF
210 m m 3.6 m 0.71 m 1.17 m
600 m m 10.6 m 0.81 m 0.97 m
Table 4: Constant m agnī cation for 2 m
subject on a 4x5
Of course changing N wil produce considerable di®erences in resolution,
since DOF changes dram atica ly as N changes, but the point that has been
m ade is that lens changes have little e®ect when m agnīcation is constant.
There are four program s for m agnī cation,as shown in Table (5).
Program Param eter
MA GU u
MA GV v
MA GH height
MA GD ±=2
Table 5: Magnī cation program
param eters
11 Any ¯lm dim ension m ay be chosen. Each dim ension wilproduce a slightly di®erentvalue.
The long side of the im age is chosen here. Magnī cation is also equalto v=u.
27

The input andoutput for MA GU is shown below:
10 UV
The param eters u and v satisfy an equation ca led the lens equation, and one
m ay be com puted from the other. In addition, either m ay be obtained from
other param eters. Four program s are given here. The UV program translates
u into v or v into u. The others produce u andv from m agnī cation, subject
height,and±=2. The program s andtheir param eters are given in Table (6).
Program Param eter
UV u or v
UVH height
UVM m agnī cation
UVD ±=2
Table 6: u andv program
param eters
The input andoutput for UV with v as input are shown below. Note that
the input m ust always be in m eters.
11 Be lows Factor
The F-num ber that is readfrom a light m eter strictly applies to the situation
when the lens is focusedat in¯nity. W hen the focus point is closer to the the
lens,the lensm ustbe extendedandthisdecreases the am ountof lightreaching
the ¯lm by the inverse square law. Thatisif the lens is m ovedtwice asfar out,
the exposure wil be one quarter of the original. To com pensate for this, one
28

shouldm ultiply the shutter speedby a factor. The factor isca ledthe \be lows
factor." The program sin thissection give the be lowsfactor asa function either
of the m agnī cation or of the lens extension. They also translate the be lows
factor into stops so that one m ay adjust the F-num ber instead of the shutter
speedif desired.
The program s andtheir param eters are given in Table (7).
Program
BFM
BFEXT
Param eter
m agnī cation
extension
Table 7: be lows factor program
The input andoutput for the BFM program
param eters
are shown below.
12 Misce laneous program s
This section docum ents a num ber of m isce laneous program s such as AOV for
¯nding the angle of view,andMOTIO for calculating the shutter speedrequired
to stop m otion.
12.1 A OV,from m agnī cation andfocallength
The input andoutput are as shown.
The last line of the output gives the focallength of a 35 m m
angle of view shown.
lens with the
29

12.2 MOTIO,shutter speedto stop m otion
The screen on the left shows input for a 30 m ph hour object 100 m eters away.
The direction of m otion is assum edto be at right angles to the lens axis. The
outputon the rightindicatesthata shutter speedof 1/282 secondswilstop this
m otion. One m ay notalways be able to shootatthe indicatedspeed,andsom e
idea of the degree of blurring is useful. If the desiredshutter speed,in seconds
is input, then the degree of blurr is shown on the output screen. The output
gives the distance the im age of the subjectm oves in units of c,the diam eter of
the circle of confusion. A tim e of 1/2 secondwasinputon the screen on the left
and the output screen shows that the im age of a subject wilcover a distance
of 141 tim es c during the 1/2 secondexposure. It wilbe very blury. As a rule
a subject that m oves only two or three tim es c m ay be acceptable.
12.3 STOPD,stop di®erence for light m ovem ent
If the lights in a studio are m oved, the F-num ber wil change. This program
givesthe stop di®erence requiredto m ake the adjustm ent. On the left,the lights
are at 4 m and they are to be m oved to 8 m . The output on the right shows
that the F-num ber m ust be increasedby 2 stops.
12.4 TRIA N,triangulation
The distance to an object m ay be foundby triangulation. The input screen on
the leftrequiresthe base distance andtwo com passreadings. I genera ly choose
a two m eter base,since I carry a spring retraction pocketm easure anditiseasy
for m e to hook the end on the cam era and m ove out one m eter on each side.
30

My com pass is shown am ong the gadgets in Figure (2).
12.5 RESL,resolution
This is a theoretical calculation of no practical use in the ¯eld, but included
for com pleteness. W ith it one m ay calculate the resolutions shown in Figure
(9). On the left one sees the input screen with ±=2 entered as 4. The output
screen on the right shows the resolution at the optim um for ±=2 = 4. Below
thisare the resolutionsfrom two di®erentsourceswhich com bine to produce the
¯nalresolution. The ¯rst of these is the di®raction resolution,and the second
is the defocus resolution at a ±=2 distance from the plane of exact focus. The
two resolutions are com bined using root m ean square on their inverses { this
is strictly an em piricalcom bination since there is no sim ple theoreticalway to
com bine them . It is interesting to note that the resolution due to di®raction
is actua ly the largest resolution shown,which shouldgive pause to those who
ascribe poor quality to di®raction { in this case,the principalcause is extrem e
defocus.
12.6 RESU,at the DOF lim its
Although di®raction isoften thoughtof asan im portantsource of im age degradation,
it is not in fact. This calculation shows the di®raction and defocus
resolutionsaswasdone in the previoussubsection,butthistim e the resolutions
are in term sofpracticalcam era settings. The calculationsare m ade atthe DOF
lim its,which represent a worst case { resolutions for any distances closer to u
than these DOF lim its wilbe higher 12 .It m ay be seen that for such practical
settings,di®raction isalwaysso m uch larger than defocusresolution,thatithas
12 This m ay be checkedusing the previous program by decreasing d=2. The value of d=2 at
the DO F lim its m ay be obtainedfrom any of the DO F program s.
31

little e®ect on the com binedresolution. The m axim um N engravedon a lens is
chosen so that di®raction has an e®ect only in extrem e cases. An extrem e case
wouldbe one with N at the m axim um andu just slightly larger than the focal
length.
13 Technicalities
Param etersusedby the program sare shown in Table (8). They m ay be changed
if desired, but rem em ber that they wil be reset when one of the indicated
program s is run.
Param eter Set by Explanation
F± P Focallength
N± P F-num ber
C± P diam eter of the circle of confusion
DM± P Film long side in m m
FS± P Film size 35,645,67,69,45,810
°± Any tilt program Slope of subject plane
J± Any tilt program Vert distance from lens to subject plane
µ± Any tilt program Lens tilt angle
®± Any tilt program Back tilt angle
Table 8: Param eters
32

BIBLIOGR A PHY
1. Bond,H.(1998). Photo Techniques,May/June
2. Merklinger,Harold,M.(1992)The insandoutsof focus. Ottawa,Canada.
3. Merklinger,Harold,M.(1993)Focusing the view cam era. Ottawa,Canada.
4. Jacobson, R.E.; Ray, S.F.; and Attridge, G.G. (1988). The m anual of
photography. Eighth Ed. FocalPress,Boston.
5. bib:K ingslake) Kingslake, R. (1989) A history of the photographic lens.
Academ ic Press,Boston.
6. Ray,Sidney F.(1994) Photographic optics. FocalPress,Boston.
7. Sears,F.W .(1958) Optics,Addison W esley,Reading MA .
8. Stroebel,Leslie; Com pton,John; Current,Ira; andZakia,Richard. (1986)
Photographic m aterials andprocesses. FocalPress,Boston.
9. Stroebel,Leslie. (1993). View cam era technique. FocalPress,Boston.
33

Directory Program Nam e Page Checksum Description
P 7 #4318d Set and display param eters
VIEW
Viewfram e inform ation
VIEW V 9 #11856d u,FrontDOF,BackDOF from v
VIEW U 10 #9427d u,FrontDOF,BackDOF from u
TILT
Calculates tilt angle
TILTB 12 #43025d By back focusing
TILTD 13 #52396d From distances and angles
TILTJ 14 #62975 From u and J
TILTG 15 #63438d From subject plane slope
DOF
Depth of ¯eld
DOFU 16 #22354d From u
DOF$ 17 #16249d From ray angle,for tiltedlens
DOFV 19 #23592d From v
DOFH 20 #40427d From height of the subject
DOFM 20 #17139d From m agnī cation
DOFD 20 #18561d From ±=2
DDOF 21 #19545d ±=2 from DOF
HYP 21 #40002d The hyperfocal distance
FUZZ 21 #40176d Blurredsubject size
HIGH
Subject height
HU 23 #23625d From u
HV 23 #28437d From v
HM 23 #13283d From m agnī cation
HDOF 23 #20163d From DOF
HD 23 #52578d From ±=2
N
Calculates F-num bers
NNF 25 #40611 From near andfar DOF lim its
NDU 24 #62842 From ±=2 andu
NDOF 26 #1200d From DOF andu
FOCA L
Focallength
FNF 26 #58387d From near andfar DOF lim its
FUD 26 #27295d From ±=2 andu
FDOF 26 #31383d From DOF andu
FMU 26 #51068d From DOF andm agnī cation
FMV 26 #43577d From v and m agnī cation
MA G
Magnī cation
MA GU 27 #64906d From u
MA GV 27 #42492d From v
MA GH 27 #65288d From subject height
MA GD 27 #27241d From ±=2
UVDIR
u andv
UV 28 #57029d u to v or v to u
UVH 28 #33989d u andv from subject height
UVM 28 #16880d u andv from m agnī cation
UVD 28 #64295d u andv from ±=2
BFA CT
Bellows factors
BFM 28 #32060 From m agnī cation and N
BFEXT 28 #21325d From extension andN
MISC
Various utilities
A OV 29 #9063d A ngle of view from m agnī cation and focallength
MOTIO 30 #2351d Shutter spreadto stop m otion
STOPD 30 #31310d Stop di®erence for two light distances
TRIA N 30 #32858d Triangulates to ¯nd subject distance
RESL 31 #17469d Theoreticalresolution
RESU ?? #47548 Resolution at the DOF lim its
34