Main Focusing the View Camera : A Scientific Way to Focus the View Camera and Estimate Depth of Field
Focusing the View Camera : A Scientific Way to Focus the View Camera and Estimate Depth of FieldHarold M. Merklinger
This book is intended for all users of the view camera, whether they be veteran professionals or new apprentices. The book will help all such photographers to understand the optical principles behind the camera's remarkable flexibility, and to manipulate the lens tilt and swing with confidence.The Hinge Rule will allow the photographer to set the required lens tilt and swing perhaps even before the camera is mounted on the tripod. This book contains many tables that enable the photographer to use the Hinge Rule easily.
FOCUSING the VIEW CAMERA A Scientific Way to focus the View Camera and Estimate Depth of Field J by Harold M. Merklinger FOCUSING the VIEW CAMERA A Scientific Way to focus the View Camera and Estimate Depth of Field by Harold M. Merklinger Published by the author This version exists in electronic (PDF) format only. ii Published by the author: Harold M. Merklinger P. O. Box 494 Dartmouth, Nova Scotia Canada, B2Y 3Y8. v. 1.0 1 March 1993. 2nd Printing 29 March 1996. 3rd Printing 27 August 1998. Internet Edition (v. 1.6.1) 8 Jan 2007 ISBN 0-9695025-2-4 © All rights reserved. No part of this book may be reproduced or translated without the express written permission of the author. ‘Printed’ in electronic format, by the author, using Adobe Acrobat. Dedicated to view camera users everywhere. FOCUSING THE VIEW CAMERA iii CONTENTS Page Preface ...............................................................................................................iv CHAPTER 1: Introduction ............................................................................1 CHAPTER 2: Getting Started .......................................................................3 CHAPTER 3: Definitions .............................................................................11 The Lens ...................................................................................................11 The Film and the Image Space .................................................................19 The Plane of Sharp Focus and the Object Space .....................................23 Other Terms and Distances ......................................................................27 CHAPTER 4: View Camera Optical Principles .........................................29 Tilt and Swing ..........................................................................................36 Discussion ................................................................................................37 CHAPTER 5: Perspective and Distortion ..................................................39 CHAPTER 6: Depth of Field .......................................................................49 Image-Based Depth of Field ....................................................................49 Object-Based Depth of Field ...................................................................53 Discussion ................................................................................................55 CHAPTER 7: A Simpler Method ................................................................57 CHAPTER 8: An Example ...........................................................................61 CHAPTER 9: Tutorial ..................................................................................67 Considerations ..........................................................................................67 A Solution ................................................................................................71 Additional Comments ..............................................................................74 Other Ways ..............................................................................................75 CHAPTER 10: Summary ...............................................................................79 Main Message ..........................................................................................79 Table of Hyperfocal Distances .................................................................80 Bibliography ............................................................................................82 CHAPTER 11: Tables ....................................................................................85 Index to Tables .........................................................................................87 Effective focal length ...............................................................................88 iv Merklinger: FOCUSING THE VIEW CAMERA CHAPTER 11: Tables (continued) Hinge line tilt ...........................................................................................91 Effective tilt for both swing and tilt .........................................................90 Lens tilt angle for given focal length f and distance J .............................93 Distance J for given tilt ...........................................................................96 Back focus distance A ............................................................................100 Depth of Field ........................................................................................105 Lens-to plane of sharp focus distance D ................................................128 Hyperfocal Distance, H ..........................................................................134 Index to Tables (repeated) .....................................................................130 Blank Charts ..................................................................................................131 Inserts provided with printed version .........................................................133 The National Aviation Museum, Ottawa In this photograph, the plane of sharp focus was set to emerge from the ground at the lower edge of the picture, and rise slowly to pass half-way through the hangar. PREFACE v Preface I have been active in photography since about 1950, when I was eight or nine years old. Over the past 56 years most of my photography has been done with 35 mm cameras. I have always wanted to pack lots of detail into my pictures, and that yearning has caused me to investigate medium and larger format cameras from time to time. I even owned an 8⫻10 camera back in the early 1960SÑwhen I couldnÕt afford to run it. For the most part, however, I have been able to achieve what I intended with 35 mm cameras and slow film, sometimes resorting to 645 or 6⫻6 format. My previous book, The INs and OUTs of FOCUS, explains how to achieve highly detailed images with almost any format. From time to time, the Graphic Century or Super Technika 23 has been called into action, primarily for the rising front feature. These cameras have a tilting front standard, but it only tips backwards. To achieve a forward lens tilt, one must either suspend the camera upside down, or drop the bed. The inconvenience of that design ensured that I remained largely ignorant of view camera principles for most of my photographic life. Although I knew vaguely of the Scheimpflug Principle, and even tried to use it on occasion, I never really applied it wellÑor trusted it, for that matter. Was this a reliable photographic tool? Or was this another example of rules like the depth of field Òone-third ruleÓ which offers useful advice but lacks rigor. One of the very first people to buy a copy of The INs and OUTs of FOCUS was Raymond Clark, President of ImageQuest Corporation in Colorado. Within a couple of weeks I received a letter from him asking how things work for view cameras when the lens is tilted. I had never attempted to work out the optics of tilted planes before, but here was a challenge. Relatively quickly, I was able to satisfy myself that the Scheimpflug Principle was exactÑfor thin, rectilinear, flat field lenses. If I could convince myself what the standard ÔcorrectÕ perspective was, I would also be able to describe depth of field for tilted planes. The depth of field went to zero at a strange point in space. One often sees drawings showing the depth of field vanishing where the lens plane, film plane and plane of sharp focus intersect. But in my solution the vanishing point was somewhere else. The correspondence between Raymond Clark and myself continued, and offered the encouragement I needed to continue the investigation. Eventually, I realized that there was not just one rule at play, but two. The Scheimpflug Principle contains only half of the needed information. A second rule contains the other half. This ruleÑwhich I have called the hinge rule in this bookÑ is so simple, it seems unlikely to me that its photographic significance has not vi Merklinger: FOCUSING THE VIEW CAMERA been documented in photography texts. Yes, Theodore Scheimpflug did know about itÑit is noted in his British PatentÑbut it was apparently of less importance to his problem (making maps from oblique aerial photographs) than the rule that bears his name. Let me be the first to point out that I am not an experienced view camera user. Yes, I have used semi-view cameras for decades, but finding the right amount of lens tilt was more effort than warranted for my purposes. Since uncovering the second rule, however, I have found focusing the view camera to be less than twice the work required to focus an ordinary camera. Applying the hinge rule removes all necessity for seemingly endless cycles of tilt and back-focus adjustment. The hinge rule is more useful than the Scheimpflug Rule. The few photographs accompanying this book were taken with an old Burke and James wooden 5⫻7 view camera equipped with a C. P. Goerz American Optical Co. 9.5 inch Gold Ring Dagor, and with Kodak Tri-X film (not T-Max). All of them were taken at f/22: an aperture 11 millimeters in diameter. I will not claim that these are great works of art. Nor do they constitute particularly tough tests for a view camera. They are, however, relatively typical of the photographs I like to take. What impresses me most about them is how quickly they were taken. On one outing, without even trying to work quickly, I realized in retrospect that I had taken four photographsÑall with different focus settings and in two locationsÑin the space of eight minutes. Without a knowledge of the hinge rule, I think it would have taken meÑa novice view camera userÑabout half an hour each. Yes, I do feel a bit hesitant, offering a book on a topic of which I am not a master. I can justify it only on the grounds that there is information here that I have been unable to find in any other book. But it is information that has made life very much easier for me, and I hope it will for you also. And, of course, I wish to thank Raymond Clark for prompting me to undertake this study. Without his encouraging letters, this story might not have been told. Discussions with Phil Davis, John Ward and Craig Bailey have also contributed significantly to this book. As this is written I still have a few copies of the original Focusing the View Camera in stock, but I also have a somewhat modified and expanded version in electronic form that I assembled about five years ago.. The expanded version incorporates the Addendum into the main body of the book and adds other material, much of it from Shutterbug magazine articles published after the original book was printed. Version 1.6Ñthe version you are reading nowÑis a slightly updated of that 5-year-old Version 1.5. An important addition is the new Chapter 5 on perspective. This is basically a re-packaged version of the article that appeared in the February PREFACE vii 1994 issue of Shutterbug. There is an important caveat, however. I was never happy with some of the technical terms I used in that article and have struggled to find new language. Specifically, in the Shutterbug article I used the term “bore sight’ to describe an important (vector) direction for view cameras. Those words better describe a different direction. The term derives from fire arms where the bore sight is simply established by by looking down the bore of the gun, from breech to muzzle. The camera equivalent is probably looking through the center of the lens with one’s eye positioned in the geometric center of the ground view camera glass screen. To actually do this would, of course require a hole in the ground glass and the removal of the lens components leaving only the diaphragm. At the risk of confusing the readers of that earlier article, I have changed the meaning of “bore sight” to mean precisely what I have just described. The original direction I used that term for, I have relabeled the “principal axis” of the camera. Please note that this is the principal axis of the camera, not the lens. This principal axis is established primarily by the orientation of the film, or electronic image sensor. It is the principal axis that determines the apparent perspective of the image. I am told that, in the usual language of perspective, what I am calling the principal axis is normally termed “the line of vision”. For a ‘normal’ camera, that is for a camera that has its lens axis perpendicular to the film and centered on the image, the principal axis of the camera, the axis of the lens and the bore sight all coincide. This coincidence of important axes simplifies the associated optical physics considerably. This coincidence is an assumption that underlies nearly all descriptions of photographic optics that I have seen. For view cameras, where it cannot be assumed that these three axes coincide, the mathematics unfortunately gets rather complex. Yet the fundamental principles are still relatively simple, and are within the grasp of all of us to understand. Essentially, the Scheimpflug rule and the hinge rule explain everything; add one extra consideration and depth-of-field is explained also. In order to assist people visualize the Scheimpflug and hinge rules, I generated a few animated computer “movie” files. These files can be found on my web site: http://www.trenholm.org/hmmerk/HMbook18.html. I would also like to acknowledge that in spite of my best intent, I have realized that I did make one or two unintended mathematical approximations (in both books) concerning depth of field and perhaps should correct some of the formulae and tables, but the errors introduced are so minor that these corrections are really of academic interest only. I worry that that to make the corrections would be to convey the wrong message: the whole concept of depth of field is an approximation. If we really want to be super-precise there are many more optical phenomena we should consider in addition. How precisely do we know the actual aperture, focal length and distance to the point of focus etc? The truth is there are lots of factors we don’t really know or need to know all that precisely. viii Merklinger: FOCUSING THE VIEW CAMERA I once had a lens instruction book that provided depth of field tables to six-figure accuracy. Can you imagine using your cell-phone to call up your model positioned a mile away from the camera and say “You and that mountain are not quite both in focus. Would you move a sixteenth of an inch further back, please!” That’s what six-figure accuracy implies. The transition from “in-focus” to “out-of-focus” is usually so gradual that we would often be unable to detect a significant change in the state of focus if the distance changed by 25%. In this example, the model could probably move a quarter of a mile without our being able to detect a change in effective image sharpness. A question some people are bound to ask is, “Does all this still apply for digital cameras?” The answer is an unqualified “yes”. Adjustable view cameras will continue to exist in the age of the electronic camera. We may all end up looking at electronic display screens rather than ground glass screens, but the adjustable ‘tilt and shift’ camera will still have its place for high quality photography. I expect that the most common formats—sizes of film or electronic sensors—will be smaller than the typical four by five inches, and typical lens focal lengths will decrease as a result, but the highly adjustable camera will endure. I expect clear advantages will arise out of improved sensor flatness—no more film sag or pop—and better viewing conditions: a brighter, right-way-around view even with stopped down lenses. There may even be a few built-in calculators to help us set-up the camera. There will, nevertheless, continue to be a place for “art’ and individual judgement is setting up the camera. Perhaps the most common question I am asked is something like this: “OK, you have shown me how to figure out the lens tilt I need if I know where to put the Plane of Sharp Focus, but how do I know where to put the Plane of Sharp Focus?” Sometimes there is a simple answer, but it depends upon many aspects of the situation and the intent of the photographer. I don’t pretend to tell you where to point the camera either, or when to trigger the shutter. That’s all part of the art of photography. There will be situations where tilt and shift are of no value whatever, and there will be others where there are multiple options for obtaining a satisfactory image. Making those decisions requires experience and judgement. I’m only trying to give you tools that I hope will make it easy for you to set up the camera once those decisions are made. Harold M. Merklinger 29 December, 2006. Chapter 1: INTRODUCTION 1 Chapter 1 INTRODUCTION What makes the view camera special is the ability to tilt, shift and swing both the lens and the film. These adjustments permit the camera to take very high quality photographs, that would not be possible any other way. The great flexibility of the camera requires, however, that the photographer understand something of the optical principles that allow the camera to achieve the desired result. The purpose of this book is to help users of view cameras ‘set up’ for their pictures—at least so far as focus and depth of field are concerned. The book starts with a quick overview of the method. In the ‘Getting Started’ chapter we’ll skip over many of the details. To fully understand the method, however, will take some careful attention. A number of definitions are necessary to ensure that you the user, and I the author, are speaking the same language. Once we understand the words, we move on to a description of the basic optical principles of view cameras. Not all of the terms I use will be familiar to you. A key part of the story is a rule which I have not seen described elsewhere. I call it the Hinge Rule. Like the Scheimpflug principle, the hinge rule states that three planes must intersect along a common line. Two of the three planes, however, have not received much attention in the past. The emphasis here will be on the ‘bottom line’. For the most part, I will not attempt to prove the physics or the mathematics here; I leave that for another book and another day. I am not a fan of tables; I have almost never consulted depth-of-field or other tables when using ordinary cameras. (I will use the term ‘ordinary’ camera to describe a camera that lacks back and lens movements other than focus. I will use the term ‘normal’ camera to describe one lacking movements other than focus and shift.) The tables in this book began life as tools to illustrate the optical principles applicable to view cameras. Somewhat to my surprise, I found that I consulted the tables more and more when actually taking pictures. They considerably reduced the time I spent setting up the camera. In time, view cameras will be fitted with scales and indicators which will eliminate the need for these tables. For now, however, the only other source of precise help is experience. With that apology out of the way, I admit that a significant portion of this book is made up of various tables and graphs. These tables and graphs are intended to help you determine the amount of lens tilt required for any given situation. They also enable you to estimate quantitatively, and in advance, where the limits of depth of field will lie. 2 Merklinger: FOCUSING THE VIEW CAMERA The tabled limits of depth of field are those based on an assumed standard of required image resolution. This is the traditional depth of field. In The INs and OUTs of FOCUS, I described another way to estimate depth of field—based on object field resolution. The ‘object field method’ does not require tables, but does take a new twist when used for view cameras. This second method is covered only briefly. An example is provided in Chapter 8 to help illustrate how some of the various tables can be used. I will also have to admit that I have used some trigonometry: mathematical functions like sine, cosine and tangent. This is a natural consequence of allowing parts of the camera to rotate relative to one-another. That is, they change their relative angular relationships. I had difficulty avoiding such mathematical expressions at first, but I found I could indeed solve most of the the relationships exactly without triginometry if all measurements are made either parallel to or perpendicular to the film plane. The resulting way of looking at the view camera may seem strange, but there is a logic to it, and it does work. If you do not understand triginometry, don’t worry. The principles are quite understandable without it and the whole reason for resorting to tables is to let you get away from having to work with that sort of mathematics. I include the mathematical results here primarily to permit those who like to program their own calculators or computers do so. In order to understand the tables fully, it will be necessary to understand the optical principles described in Chapter 4. But if the math is a problem, stop reading that chapter when you get to Equation (1). Better still, read on, but ignore the mathematics. There is a simple way to do just about everything, anyway. The information and tables in this booklet have allowed me to shorten the time necessary to set up a view camera considerably—by a factor of four or five. This probably just indicates that I’m not all that skillful. I nevertheless hope you too will find that by understanding the “hinge rule”, as well as the Scheimpflug rule, and by using these tables, you will be able to work more quickly and to capture the intended photograph more easily. Chapter 2: GETTING STARTED 3 Chapter 2 GETTING STARTED In what follows it will be assumed that the reader possesses some basic familiarity with the view camera. You know what is meant by tilting and swinging the camera back and the lens. You know that tilting the lens relative to the back—or the back relative to the lens—causes the plane of sharp focus, that surface on which the camera is accurately focused, to move out of parallel with the film plane. You may or may not be aware that the Scheimpflug rule states that the film plane, the lens plane and the plane of sharp focus intersect along a common line. If you don’t know this rule, that’s OK. Contrary to what some might say, I would argue that it is not absolutely necessary to understand the Scheimpfluh rule, anyway. Figure 1 shows a schematic (symbolic) diagram of a normal camera: one with the lens attached in such a way that the lens axis must stay perpendicular to the film. Figure 2 serves to indicate what happens when the lens axis (or the lens plane which is a surface perpendicular to the lens axis) is tilted. The film plane, the lens plane and the plane of sharp focus obey the Scheimpflug rule. You need not concern yourself about it; the laws of physics will make sure that it is obeyed. The general principle is simple: if we tilt one of the three planes relative to any one of the others, the third plane will get tilted too. In a normal camera, the camera is always focused on a plane that is parallel to the film. The view camera allows the photographer to focus on objects arranged on or near a plane that is not parallel to the film. This FILM PLANE LENS PLANE LENS AXIS PLANE OF SHARP FOCUS FIGURE 1: For a ‘normal’ camera, the film plane, lens plane and plane of sharp focus are parallel to one another. Merklinger: FOCUSING THE VIEW CAMERA 4 LENS PLANE FILM PLANE LENS AXIS PLANE OF SHARP FOCUS SCHEIMPFLUG LINE FIGURE 2: For a view camera, tilting the lens causes the plane of sharp focus to tilt also. The Scheimpflug rule requires that the three planes intersect along one line condition is achieved by tilting either the lens or the film relative to the other. That is, we can leave the lens where it is and tilt the back, or we can leave the back where it is and tilt the lens. Or, indeed, we can do a bit of both: tilt both the back and the lens, but not by the same amount in the same direction. The trouble comes in trying to figure out what to tilt and by how much in order to achieve the intended position for the plane of sharp focus. A further challenge arises when we want to focus on the intended plane of sharp focus and maintain correct perspective in the image. Maintaining correct perspective is perhaps the easier task. Standard perspective usually requires that the film plane remain vertical and more-or-less square to the line of sight of the camera. Sometimes we actually want false perspective in order to make the photograph appear as though it was taken from a place other than the camera’s true location. A classic example is taking a picture of a glass-covered water colour painting. If we place the camera squarely in front of the painting, we risk seeing the camera in the final image due to its reflection in the glass. The solution is to move the camera to one side and so view the painting at an angle. This eliminates the reflection. But we also want to make the image look as though the camera had been facing the painting squarely. We accomplish the desired perspective by having the film face the painting squarely—that is, keep the film and the painting parallel to one another—and let the arrangement of the back and lens effectively squint sideways at the painting. Figure 3 illustrates the resulting arrangement. If achieving the desired perspective were the only problem, we could get by with lens and back shifts (plus rise and fall) only. Let’s look now at a somewhat more complex situation. We are photographing a painting, but we want to include in the image, not only the painting, but some of the room it is in. Specifically, the large painting is hanging in a church on a wall some 30 feet from the camera. We also want to Chapter 2: GETTING STARTED PAINTING 5 PLANE OF SHARP FOCUS GLASS LENS PLANE FILM PLANE FIGURE 3: The view camera can ‘squint’ sideways, maintaining the proportions of the painting. The final image will look as though it had been taken straight on. Taking the picture as illustrated here avoids seeing a reflection of the camera in the glass. include a plaque on the church floor indicating where the artist is buried. We want a sharp image of the painting, but also a sharp image of the plate on the floor some 10 feet from the camera. To ensure both are sharp, we wish the plane of sharp focus to pass through the centers of both the painting and the plaque. Figure 4 illustrates a side view of the problem. To keep the painting rectangular, and the other features of the building in correct perspective, the 12 ft. US FOC RP SHA F O NE PLA Inscription on Floor Painting on Wall 30 ft. FIGURE 4: Here the task is to adjust the plane of sharp focus so that it passes roughly through the centers of the painting and the inscription. What amount of lens tilt will accomplish this? Merklinger: FOCUSING THE VIEW CAMERA 6 PLANE PARALLEL TO FILM, THROUGH LENS 12 ft. NE PLA J US FOC RP SHA F O Inscription on Floor Painting on Wall 30 ft. FIGURE 5: The amount of lens tilt required is set by the special distance J and the focal length of the lens. J in this case is the height of the lens above the plane of sharp focus. camera back must remain vertical and parallel to the painting. And we employ the necessary rise and/or fall to achieve the desired composition. How do we arrange for the plane of sharp focus to fall precisely where we want it to be? There’s another rule that arises from the laws of optics. I call it the hinge rule. The hinge rule will tell us the precise amount of lens tilt needed. The hinge rule is another rule very much like the Scheimpflug principle, but let’s skip the details for now. A consequence of the hinge rule is that the required amount of lens tilt is related to only two things: the focal length of the lens, and the distance the lens is from the plane of sharp focus measured in a very special way. We must measure how far the lens is from the plane of sharp focus along a plane through the lens but parallel to the film. In the example at hand, the concept is quite simple. The camera back is vertical. Therefore we measure this special distance in a vertical direction. The special distance is quite simply the height of the lens above the plane of sharp focus, as illustrated in Figure 5. I use the symbol J to denote this distance, and the symbol α to denote the amount of lens tilt needed, measured in degrees. The required amount of lens tilt is given mathematically by this expression: α = arcsin (f/J). The symbol, f, is of course the focal length of the lens. Don’t be scared off by the math; it’s really quite tame. The arcsine function can be found on many $15 ‘scientific’ calculators, but we can do even better. Included in this book on page 96 and repeated on a card at the back of this book is a table. The table has columns for lenses of various standard focal lengths. In a column at the left of the table are a number of distances. Find a distance close to that of your distance J, and look in that row for the angle listed under the focal length Chapter 2: GETTING STARTED 7 of your lens. In our example, J is equal to 8.5 ft. and the lens in use has a 75mm focal length. The required tilt angle is thus about 1.75°. For small tilt angles we can even dispense with the table. For lens tilts less than 15°, we can get an approximate value of the lens tilt from either of the following: if we measure f in inches and J in feet: α = 5f/J. If we measure f in millimeters and J in feet: α = f/5J. It’s still math, but its pretty simple math. So we set the lens tilt to 1.75° towards the intended plane of sharp focus. Not all view cameras have tilt scales. My own does not. I use a high school geometry protractor to set the tilt. I can’t set it to better than about half a degree, but that’s usually good enough. (The direction of lens tilt will have a bearing on the orientation of the plane of sharp focus. The plane of sharp focus will always be parallel to the lens tilt axis. If we imagine a plane parallel to the film but passing through the lens, that plane will intersect with the plane of sharp focus. If we mark that intersection, we will find it is a line, and it will always be parallel to the axis about which we tilted our lens. In common view camera language, if we use vertical tilt only, the tilt axis is horizontal. If we use swing only, the tilt axis is vertical. If we use both tilt and swing, the matter gets a bit complicated.) In essence, the hinge rule tells us that if we move the back of the camera to and fro (without changing its angle), closer to or farther from the lens, the plane of sharp focus must pivot on a line a distance J from the lens. In our example this pivot line is on the plane of sharp focus directly below the lens. I call that line the hinge line. I call it that because that line is like the pin in a hinge. The plane of sharp focus hinges on that line. As we move the back away from the lens, the plane of sharp focus will swing up in front of the camera. If we move the camera back closer to the lens, the plane of sharp focus will swing down, away from the lens. (It’s the Scheimpflug rule working in consort with the hinge rule that causes this rotation, by the way.) So, to achieve the desired focus in our example, we focus, using the ground glass, either on the center of the painting, or on the center of the plaque. If we have done things right, when one is in focus, the other will be too. That’s it; we’re done focusing. But what about depth of field? Well, here the view camera really has the advantage over normal cameras. Calculating view camera depth of field is dead simple. Plainly put, the depth of field at a distance one hyperfocal distance, H, in front of the camera is our friend J. Like the distance J itself, this depth of field is measured in a direction parallel to the film. Either side of Merklinger: FOCUSING THE VIEW CAMERA 8 the plane of sharp focus, the depth of field is J. In this sense depth of field is symmetrical, alwaysÑjust so long as we measure it parallel to the film. CanÕt remember what the hyperfocal distance is? ThereÕs an easy way to remember it. The criterion for image sharpness is often that the circle of confusion at the image should be no greater in diameter than some fraction of the lens focal length. The number often cited is 1/1500. Well, the hyperfocal distance is then 1500 lens aperture diameters. If our 75 mm lens is set to f/22, the hyperfocal distance will be 1500 times 75mm divided by 22. That is about 5100 millimeters or 16.8 feet. ThereÕs also a card showing hyperfocal distances for a variety of focal lengths and apertures included with FOCUSING the VIEW CAMERA. It can be demonstrated that the limits of depth of field are also planes, and that they too pass through the hinge line. Getting back to our example, we can now sketch in the limits of depth of field. We know the depth of field at one hyperfocal distance, and we know the limits pass through the hinge line. See Figure 6 for the result. Looks fine: essentially everything included in our photograph is within the limits of depth of field. The example just described is pretty straight forward. Unfortunately, the photographic situation will not always be quite so easy to analyze. The film will not always be vertical, and the plane of sharp focus will not always be near-horizontal. The basic principles to remember are as follows: The lens tilt, measured relative to the film plane, determines the distance from the lens to a line about which the plane of sharp focus pivots. N r ea Li m i f to De pt h of Fi el d 12 ft. 8.5 ft. Inscription on Floor US FOC ARP H S OF NE PLA Painting on Wall 8.5 ft. 8.5 ft. H = 16.8 ft. Far Limit of Depth of Field FIGURE 6: Depth of field for view cameras is easy. At a distance of one hyperfocal distance, H, the depth of field measured in a direction parallel to the film is simply J on either side of the plane of sharp focus. (Camera is not to scale.) Chapter 2: GETTING STARTED 9 That line, called the hinge line, will also be parallel to the lens tilt axis. Shortening the distance between lens and film plane causes the plane of sharp focus to rotate (about the hinge line) away from the front of the lens. Increasing the distance between the film plane and the lens causes the plane of sharp focus to rotate towards the front of the lens. Depth of field, measured parallel to the film, varies directly as the distance, J. Increasing the lens tilt decreases J and so also decreases depth of field. Depth of field measured this way is always symmetrical about the plane of sharp focus. Now you have the basics; the rest is just details. We’ll continue with a review of the definitions necessary to analyse the situation more carefully. Later in this book we’ll also discuss how the “object field depth-of-field method” (described in The INs and OUTs of FOCUS) applies to view camera situations. 10 Merklinger: FOCUSING THE VIEW CAMERA Point Gatineau: Where the Gatineau flows into the Ottawa In a photograph like this, there is no real necessity for tilts or swings of any kind. But I did, nevertheless, use about one-half degree of forward lens tilt in order to ensure that the plane of sharp focus would pass below the camera. My old wooden camera could easily be half-a-degree out in almost any swing or tilt. Were the plane of sharp focus to lie on the left, right or above the camera, the image would not be uniformly sharp. An enlarged prtion of this image, centered on the church, may be seen on page 84. Chapter 3: DEFINITIONS 11 Chapter 3 DEFINITIONS This chapter is intended to tell you exactly what I mean by various terms, such as “Parallel-to-Film Lens Plane”. If you are reading this book from start to finish, you need not read closely and remember all these definitions right now. But please do look for the terms you do not recognize, and give them a quick read. The purpose of grouping the definitions here is primarily to give you one place to find all the definitions; you won’t have to search through the whole booklet just to find a definition. Where distances are defined, it is very important to note how the distance is measured. For example, should we measure depth of field along a ray through the lens, or in a direction perpendicular to the plane of sharp focus? There are many ways to measure some of these distances, and how they are measured will affect the answers one obtains. Measuring in certain directions simplifies things, too. I have tried always to measure distances in a way that is photographically meaningful. There’s more than one way to do it, however. If you think it should be done differently, I would appreciate hearing from you. The definitions that follow are grouped by function. First, we’ll look at the definitions that apply to the lens, then those for the film, and so on. After most definitions is a reference to a figure which should help you understand the definition. The Lens Symbol Name lens axis Definition The lens axis is an imaginary straight line running through the centers of all the glass elements making up the lens. A lens is virtually always symmetric about its axis; it is in fact a ‘body of revolution’ about its axis. See Figure 7 (overleaf). Merklinger: FOCUSING THE VIEW CAMERA 12 LENS LENS AXIS FILM FIGURE 7 LENS PLANE FILM PLANE REAR NODAL POINT FILM THICK LENS LENS AXIS FIGURE 8 FRONT NODAL POINT FRONT LENS PLANE FILM PLANE REAR LENS PLANE INTER-NODAL DISTANCE Chapter 3: DEFINITIONS Symbol Name 13 Definition lens plane The lens plane is an imaginary plane passing through the optical center of the lens, and oriented perpendicular to the lens axis. If a lens is optically ‘thick’, there are actually two lens planes: a front lens plane through the front nodal point of the lens, and a rear lens plane through the rear nodal point. See Figures 7 and 8. front nodal point The front nodal point is the effective optical center of the lens as seen from the front of the lens, that is, the side away from the film. See Figure 8. front lens plane The front lens plane is an imaginary plane through the front nodal point, and perpendicular to the lens axis. See Figure 8. rear nodal point The rear nodal point of a lens is the optical center of the lens as seen from the back side (film side) of the lens. See Figure 8. rear lens plane The rear lens plane is an imaginary plane through the rear nodal point, and perpendicular to the lens axis. See Figure 8. inter-nodal distance The inter-nodal distance is the distance between the front nodal point and the rear nodal point, measured along the lens axis. It can occur that for some lens designs, the front nodal point is actually closer to the film than the rear nodal point. When this happens the inter-nodal distance has a negative value. See Figure 8. Merklinger: FOCUSING THE VIEW CAMERA 14 LENS PLANE LENS FILM OBJECT AT INFINITY FIGURE 9 f FILM PLANE FOCAL LENGTH, f EFFECTIVE FOCAL LENGTH, f' f' LENS LENS AXIS δ FIGURE 10 f LENS PLANE OFF-AXIS RAY FILM PLANE PLANE OF SHARP FOCUS (at infinity) FOCAL LENGTH, f LENS PLANE FOCAL PLANE FRONT FOCAL PLANE f IMAGES OF OBJECTS LENS AXIS FIGURE 11 f OBJECTS ARRANGED ON A PLANE IMAGE PLANE Chapter 3: DEFINITIONS Symbol Name 15 Definition thin lens A thin lens is one where the inter-nodal distance is very small in relation to its focal length. Many lenses for view cameras, especially those of symmetrical or near-symmetrical design, are optically thin even though the actual thickness of the glass is large. For the sake of simplicity, it will be assumed that lenses are thin, unless otherwise stated. f focal length The focal length of a lens is the distance from the rear nodal point of a lens to the sharp image of a very distant object located in front of the lens but on the lens axis. See Figure 9. δ off-axis angle If an object in front of the lens does not lie on the lens axis, we can describe its position in part through the off-axis angle, δ. δ is the angle between the lens axis and a line from the front nodal point to the object. See Figure 10. (δ is the Greek letter ‘Delta’.) f' effective focal length For a distant object not located on the lens axis, the effective focal length, f', of a lens is greater than its nominal focal length, f. That is, the distance from the rear nodal point of the lens to the sharp image of that object is f', and f' is greater than f. See Figure 10. focal plane The focal plane of a lens is an imaginary plane parallel to the lens plane (that is, perpendicular to the lens axis) and one focal length behind the rear nodal point of the lens (measured along the lens axis). See Figure 11. Merklinger: FOCUSING THE VIEW CAMERA 16 DIAPHRAGM d = f/8 FIGURE 12 f If d = f/8, N = 8. PERPENDICULAR TO FILM PLANE LENS α FILM LENS AXIS FIGURE 13 α LENS PLANE FILM PLANE Chapter 3: DEFINITIONS Symbol Name 17 Definition front focal plane The front focal plane is an imaginary plane perpendicular to the lens axis and one focal length in front of (away from the film) the front nodal point of the lens. A small object located anywhere on the front focal plane will be focused an infinite distance behind the lens. See Figure 11. image plane Objects arranged on an imaginary plane in front of the lens will be imaged on an imaginary plane, the image plane, behind the lens. The image plane is not necessarily perpendicular to the lens axis. See Figure 11. When the image plane and the film plane coincide, all objects are in focus. d lens diameter A lens normally contains a diaphragm or other “stop” which blocks some of the light that would otherwise pass through the lens. This stop is usually approximately round, and its diameter, as seen from the front of the lens, is called the lens diameter, or working lens diameter. The effective lens diameters as seen from the front, the rear and as measured at the diaphragm may all be different. But what usually matters is the diameter as seen from the front. See Figure 12. N f-number The lens diameter is often described by its size in relation to the focal length of the lens. A lens whose diameter is one-eighth of its focal length is said to be an f-8, often written f/8, lens. In this case the f-number or numerical aperture, N, is equal to 8. See Figure 12. α lens tilt The total effective lens tilt, α, is the angle between the lens axis and a line perpendicular to the film plane. See Figure 13. (α is the Greek letter ‘Alpha’.) Merklinger: FOCUSING THE VIEW CAMERA 18 LENS PLANE LENS LENS AXIS FILM PARALLEL-TO-FILM LENS PLANE (PTF PLANE) FILM PLANE A FIGURE 14 Chapter 3: DEFINITIONS Symbol Name 19 Definition H hyperfocal distance For an ‘ordinary’ or ‘normal’ camera, that is one having no lens tilt, the hyperfocal distance is the distance, measured parallel to the lens axis, from the lens to the inner limit of depth of field, when the lens is focused at infinity. The hyperfocal distance is not strictly a property of the lens or its focal length and numerical aperture. The hyperfocal distance also depends upon the assumed maximum permissible size of the circle of confusion at the film. When a lens is focused at its hyperfocal distance, the depth of field extends from one-half the hyperfocal distance to infinity. Q image quality factor If the maximum permitted diameter of the circle of confusion (a) is equal to the lens focal length, f, divided by Q, the hyperfocal distance is Q times the lens’ working diameter, d. parallel-to-film lens plane The parallel-to-film lens plane, or PTF plane, for short, is an imaginary plane through the front nodal point of the lens, and parallel to the film plane. See Figure 14. PTF plane The Film and the Image Space film plane It is assumed in this book that the film is located on an imaginary plane, called the film plane. The film plane may have any orientation and may lie any distance behind the lens. The film plane is the primary reference plane for the view camera. In general, all other angles are measured with respect to the film plane or with respect to a line perpendicular to the film plane. See Figure 14. Merklinger: FOCUSING THE VIEW CAMERA 20 FILM PLANE LENS a OBJECT AT INFINITY g c f LENS PLANE FIGURE 15 Chapter 3: DEFINITIONS Symbol A Name 21 Definition lens-to-film distance Also called the back focus distance, the lens-to-film distance is the distance from the film plane to the rear nodal point of the lens, measured in a direction perpendicular to the film plane. In The INs and OUTs of FOCUS, I used the symbol B to denote the lens-to-image distance. For most purposes in the present book, the symbols A and B may be used interchangeably, since the image is assumed to be sharply focused at the film. It is frequently useful to measure A in focal lengths. The distance A measured in focal lengths is denoted as A/f. See Figure 14. Parallel-to-film lens plane The parallel-to-film lens plane, or PTF plane, for short, is an imaginary plane through the front nodal point of the lens, and parallel to the film plane. See Figure 14. c diameter of the circle-of-confusion When the image of a very tiny spot does not lie precisely at the film plane, the image on the film will be a small circle of diameter, c. This circle is called the circle of confusion. See Figure 15. a largest permissible diameter for c For the traditional method for calculating depth of field, it is assumed that there is a largest diameter which may be tolerated for the circle-of-confusion. This largest permissible diameter is denoted as a. See Figure 15. g depth of focus For a stated value of the largest permissible diameter of the circle of confusion, a, and for a lens of numerical aperture, N, the depth of focus is simply equal to the product of N and a: g = Na. In order to be rendered with acceptable resolution, the sharp image must lie within a distance, g, either side of the film plane. See Figure 15. PTF plane Merklinger: FOCUSING THE VIEW CAMERA 22 LENS LENS AXIS D FIGURE 16 FILM PLANE PLANE OF SHARP FOCUS (PSF) LENS PLANE OBJECT R RAY FIGURE 17 FILM PLANE Z PTF PLANE LENS PLANE PLANE OF SHARP FOCUS LENS PLANE PARALLEL-TO-PSF PLANE (PTPSF) FIGURE 18 FILM PLANE PLANE OF SHARP FOCUS (PSF) Chapter 3: DEFINITIONS Symbol 23 Name Definition The Plane of Sharp Focus and the Object Space PSF plane of sharp focus The plane of sharp focus is that plane in front of the camera, every point of which is focused precisely on the film plane. Any small object located on the plane of sharp focus is in perfect focus. See Figure 16. D lens-to-PSF distance Distance D is that from the front nodal point to the plane of sharp focus, measured in a direction perpendicular to the plane of sharp focus. D is the shortest distance between the plane of sharp focus and the lens. For some purposes it is useful to measure D in units of one focal length. D/f will be used to denote the distance D measured in this way. See Figure 16. R lens-to-object distance The distance from the lens to some object lying on the plane of sharp focus is denoted as R when the distance is measured simply as the shortest direct line from the object to the front nodal point of the lens. See Figure 17. Z lens-to-object distance The distance from the lens to some object lying on the plane of sharp focus is denoted by Z when the direction of measurement is perpendicular to the film plane. See Figure 17. parallel-to-PSF lens plane The parallel-to-plane of sharp focus lens plane is an imaginary plane through the rear nodal point of the lens, parallel to the plane of sharp focus. We will use this term rarely. See Figure 18. PTPSF Merklinger: FOCUSING THE VIEW CAMERA 24 LENS PLANE NEAR LIMIT OF DEPTH OF FIELD L1 L2 FIGURE 19 FAR LIMIT OF DEPTH OF FIELD PLANE OF SHARP FOCUS (PSF) LENS PLANE NEAR LIMIT OF DEPTH OF FIELD K1 FIGURE 20 K2 FAR LIMIT OF DEPTH OF FIELD PLANE OF SHARP FOCUS (PSF) PTF PLANE m = L1/Z LENS PLANE l = L2/Z L1 Lpf L2 FIGURE 21 FILM PLANE Z PLANE OF SHARP FOCUS (PSF) Chapter 3: DEFINITIONS Symbol Name 25 Definition L depth of field Objects within some distance, L, of the plane of sharp focus will be imaged sharply enough to be considered “in focus”. The depth of field, L, is measured in a direction perpendicular to the plane of sharp focus. Since the depth of field is not necessarily equal on both sides of the plane of sharp focus, we may use L1 to denote the depth of field on the lens side of the plane of sharp focus, and L2 to denote depth of field on the far side. See Figure 19. Lpf depth of field A variation on ways to describe depth of field is to measure its extent in a direction parallel to the film plane. Such a measure of depth of field is denoted as Lpf. See Figure 21. K depth of field Objects within some distance, K, of the plane of sharp focus will be imaged sharply enough to be considered “in focus”. The depth of field denoted by K is measured along a ray from the lens to some specified point on the plane of sharp focus. As for the depth of field, L, we may use K1 and K2 to denote the depth of field on the lens side and the far side of the plane of sharp focus respectively. Note that the only difference between L and K is the direction in which the distance in measured. See Figure 20. m depth of field fraction The depth of field, L, may be expressed as a fraction of the distance Z. m is that fraction (or coefficient) applying on the lens side of the plane of sharp focus: L1 = mZ. See Figure 21. l depth of field fraction The depth of field, L, may be expressed as a fraction of the distance Z. l is that fraction (or coefficient) applying on the far side of the plane of sharp focus: L2 = lZ. See Figure 21. Merklinger: FOCUSING THE VIEW CAMERA 26 LENS PLANE S FIGURE 22 FILM PLANE PLANE OF SHARP FOCUS (PSF) LENS PLANE LENS FILM LENS AXIS FIGURE 23 SCHEIMPFLUG LINE α Φ PLANE OF SHARP FOCUS FILM PLANE FILM PLANE LENS PLANE FRONT FOCAL PLANE PLANE OF SHARP FOCUS J SCHEIMPFLUG LINE f PTF PLANE HINGE LINE FIGURE 24 Chapter 3: DEFINITIONS Symbol Name 27 Definition S spot size, or disk of confusion The spot size, S, is the diameter an object must be in order to be registered at full contrast on the film. If an object is smaller in size than S, it may still show up in the image, larger than it should be, and at reduced contrast. See Figure 22. The spot size, or “disk-ofconfusion” was discussed at length in The INs and OUTs of FOCUS. Φ plane of sharp focus angle The angle Φ is the angle of the plane of sharp focus, relative to the film plane. See Figure 23. (Φ is the Greek letter ‘Phi’.) θ hinge line tilt The angle θ is the tilt of the hinge line compared to the horizontal plane. θ is measured in the PTF plane. (θ is the Greek letter ‘Theta’.) Other Terms and Distances Scheimpflug line In order for a view camera image to be sharp, the rules of optics state that the film plane, the lens plane and the plane of sharp focus must intersect along a common line in space. That line is the Scheimpflug line. See Figure 23. Since the Scheimpflug line is seen on end, it is represented in the figure as a dot. hinge line The rules of optics state that for rectilinear, flat-field lenses, the PTF plane, the front focal plane and the plane of sharp focus must intersect along a common line. That line is the hinge line. The hinge line is always parallel to the Scheimpflug line. See Figure 24. Merklinger: FOCUSING THE VIEW CAMERA 28 PRINCIPAL AXIS LENS LENS AXIS FILM (or image sensor) FILM PLANE BORE SIGHT LENS PLANE PLANE OF SHARP FOCUS SCHEIMPFLUG LINE FIGURE 25 Symbol J Name Definition lens-to-hinge line distance The distance from the front nodal point of the lens to the hinge line (measured perpendicular to the hinge line) is denoted as J. When forward lens tilt is used, J is effectively the lens height above the plane of sharp focus. See Figure 24 (on page 24). principal axis A line through the lens center, and perpendicular to the film plane is called the principal axis. The angle between the principal axis and the lens axis is the lens tilt, a. See Figure 25 (above). bore sight A line from the geometric center of the image through the center of the lens is here called the bore sight. This is essentially the Ôlook directionÕ of the camera. See Figure 25 (above). Chapter 4: OPTICAL PRINCIPLES 29 Chapter 4 VIEW CAMERA OPTICAL PRINCIPLES The fundamental optics of ‘normal’ cameras are described by just one rule: the lens equation. ‘Normal’ cameras are those having the lens axis fixed in a direction perpendicular to the film plane. The lens equation relates the lens-to-film distance, A, the lens-to-plane of sharp focus distance, D, and the focal length, f, to one-another. Although the standard lens equation does not apply for off-axis rays, the matter is circumvented by measuring all the distances in a direction perpendicular to the film plane, or parallel to the lens axis—which is the same thing for ‘normal’ cameras. View camera design complicates things by allowing the photographer to tilt or swing the lens and the film plane independently. Distances measured parallel to the lens axis are no longer equivalent to those measured perpendicular to the film plane. To make matters even more complicated, swinging the lens and/or the film, also rotates the plane of sharp focus so that distance measured perpendicular to the plane of sharp focus now presents yet a third way to measure things. Fortunately, we can explain view camera optical principles with just two or three basic rules. Using these rules it is possible to focus the view camera systematically, and to understand depth of field for tilted planes. We’ll discuss the details of the view camera lens equation itself a little later. Happily, the view camera lens equation can be interpreted in the form of two relatively simple rules that require no understanding of the mathematics. These rules are the Scheimpflug Rule and what I call the Hinge Rule. Together they tell us everything that the view camera lens equation does. The Scheimpflug rule is well known to many view camera users. This principle states that for thin, flat-field, rectilinear lenses, the film plane, the lens plane and the plane of sharp focus must intersect along a common line. Figure 26 illustrates. The line where all three planes intersect, we’ll call the Scheimpflug line. Since the figure depicts a cross-section through the scene in a direction perpendicular to all three planes, the planes are drawn as straight lines. In three dimensions, the planes would extend out of the paper, at right angles to the page. The Scheimpflug line, which also extends out of the paper, is shown as a simple dot. No matter how the lens and film are tilted and/or swung, it will always be possible to find some direction from which to look at the camera so that it looks something like Figure 26. In this book we use the symbol α to denote the lens tilt relative to the film plane, and Φ to denote the resulting tilt of the plane of sharp focus relative to the film. For thick lenses, the Scheimpflug rule must be adjusted to account for the inter-nodal distance of the lens. As illustrated in Figure 27, any ray is Merklinger: FOCUSING THE VIEW CAMERA 30 assumed to pass from one nodal plane to the other in a direction parallel to the lens axis. The same is true for the extensions of the plane of sharp focus and for the film plane. FILM LENS LENS AXIS LENS PLANE α FILM PLANE PLANE OF SHARP FOCUS Φ LINE OF INTERSECTION (SCHEIMPFLUG LINE) FIGURE 26: The Scheimpflug Principle states that the Film Plane, the Lens Plane and the Plane of Sharp Focus must intersect along a common line. BACK LENS PLANE FRONT LENS PLANE LENS AXIS FI LM PL AN E LINE PARALLEL TO LENS AXIS CONNECTS TWO SCHEIMPFLUG LINES OFF-A XIS R AY OF NE PLA US FOC P R SHA INTER-NODAL DISTANCE FIGURE 27: For thick lenses, the Scheimpflug Principle must be adjusted. Any ray or plane which passes through the lens planes, moves from one lens plane to the other in a direction parallel to the lens axis. Chapter 4: OPTICAL PRINCIPLES 31 LENS PLANE f FRONT FOCAL PLANE LENS AXIS α FILM PLANE J PLANE OF SHARP FOCUS HINGE LINE PARALLEL-TO-FILM LENS PLANE (PTF Plane) SCHEIMPFLUG LINE FIGURE 28: The Hinge Rule states that the Front Focal Plane, the Parallel-to-Film Lens Plane and the Plane of Sharp Focus must also intersect along a common line. J is measured parallel to the PTF Plane. The Scheimpflug principle is a necessary condition for the lens equation to be satisfied, but it is not enough. With reference to Figure 26, suppose that the lens and lens plane are swung about the Scheimpflug line, keeping the distance from the lens to the Scheimpflug line constant. We’ll also keep the film plane and the plane of sharp focus fixed. If the Scheimpflug rule were enough, the image would stay in focus as we swing the lens. We know from experience that the image does not stay in focus. As the lens moves through the full range, we will find one, two or no places where the image is in focus. The needed additional information is contained in the hinge rule. This rule states that the parallel-to-film lens plane (PTF plane), the plane of sharp focus (PSF) and the front focal plane must intersect along a common line. This second important line, we’ll call the hinge line. This rule is depicted in Figure 28. The hinge rule and the Scheimpflug rule together solve the view camera lens equation for us—no matter where the object and the image lie. The hinge rule alone can help a lot if we know where we wish the plane of sharp focus to be, relative to the lens and film. If we know the desired orientation of the film plane, and how far the lens is from the plane of sharp focus (measured along the PTF plane), we know something else. We know the required lens tilt. We’ll call the lens-to-plane of sharp focus distance, measured along the PTF plane, J. When the film plane is oriented vertically, and the lens tilt axis is horizontal, J will represent the height of the lens above the plane of sharp focus. (Table III on pages 96 and 97 shows several examples the relationship between J and α for a number of common focal lengths. This table gives us Merklinger: FOCUSING THE VIEW CAMERA 32 the angle, α, needed for a variety of J distances for each focal length. Table IV on pages 98 and 99 show similar information in a slightly different form. In Table IV, the distance J is given for a range of tilt angles.) If J is measured in focal lengths, there is just one simple relationship between J and α: J 1 = . f sin α (1a) This may, alternatively, be written as: f α = arcsin ( ). J (1b) This expression may be further abbreviated, as shown on page 6 in the Getting Started chapter. If the film is racked to and fro, the Scheimpflug rule and the hinge rule, working together, cause the plane of sharp focus to rotate or pivot about the hinge line (see Figure 29). (For convenience in calculating J/f, Table V gives the value of J/f for the same range of J distances and focal lengths used in Table III.) In practice, it is generally easiest to first use Table III to determine the required lens tilt or swing, then focus using the camera back to put the plane of LENS PLANE FRONT FOCAL PLANE HINGE LINE PLANE OF SHARP FOCUS FILM PLANE PTF PLANE SCHEIMPFLUG LINE FIGURE 29: The Hinge Rule, together with the Scheimpflug Rule, requires that if the film is moved closer to the lens, the Plane of Sharp Focus must move away from the lens, rotating about the Hinge Line. Chapter 4: OPTICAL PRINCIPLES 33 sharp focus in the desired orientation. Should this not be possible, we can use the following equation to determine the required lens-to-film distance, A: A 1 1 = (sin α) + . f tan (Φ−α) tan α (2a) Again, if the lens-to-film distance is expressed in focal lengths, the same graph or table can be used for lenses of all focal lengths. (Table VI shows values of A/f for various combinations of normalized distances, J/f, and desired plane of sharp focus angles, Φ.) We can also rewrite Equation (2a) to solve for Φ: Φ = 90° + arctan f - A sin (α) 1 tan α . (2b) The relationships between α, Φ, and A/f are shown graphically in Figures 62 and 63 on pages 104 and 105. Returning to the matter of the lens equation, the basic lens equation—the one we usually read about—may be written as: 1 1 1 + = A D f (3) where A is the lens-to-film distance, D is the lens-to-plane of sharp focus distance, and f is the focal length of the lens. Note that, for ‘normal’ cameras, distances A and D are usually assumed to be measured in a direction parallel to A' f' PLANE OF SHARP FOCUS LENS LENS AXIS δ OFF-AXIS RAY f LENS PLANE D' FOCAL PLANE (for ∞ focus) FILM PLANE FIGURE 30: The object distance, D', and image distance A' are illustrated here for an oblique ray, making angle δ with the lens axis. Merklinger: FOCUSING THE VIEW CAMERA 34 the lens axis, or perpendicular to the film plane, or perpendicular to the plane of sharp focus. The lens equation for oblique rays—those not parallel to the lens axis—is given by the following equation: 1 1 1 f + = , f' = A' D ' f' cos δ (4) where δ is the off-axis angle, A' is the lens-to-film distance measured along the ray, D' is the lens-to-plane of sharp focus distance measured along the ray, and f' is the effective focal length of the lens. Figure 30 (on the previous page) illustrates. A consequence of the lens equation for oblique rays is that the effective focal length of a lens is not a fixed quantity. As a lens is tilted or swung in view camera use, its effective focal length changes. The focal length usually appears to increase as the lens is tilted. The change is negligible for small angles of tilt, but can be quite significant for angles of 25° and more. Table I on page 90 in the Tables section shows the results. The major consequence of this effect is that when a lens is tilted, the camera’s angle of view changes. One may discover that when lens tilt or swing is used, a focal length shorter than anticipated is needed to cover the intended angle of view. If the main subject is closer to the lens axis after the lens is tilted or swung than it was with no tilt or swing, however, the effect may seem to go in the opposite direction. The effective focal length may appear to shorten. Moving the lens axis away from the main subject increases effective focal length; moving the lens axis closer to the main subject shortens the effective focal length. The point to be LENS PLANE FRONT FOCAL PLANE FILM f LENS AXIS A FILM PLANE D α Φ PLANE OF SHARP FOCUS HINGE LINE PTF PLANE SCHEIMPFLUG LINE FIGURE 31: The distances and angles important for the view camera lens equation, Equation (5), are illustrated here. D is measured perpendicular to the Plane of Sharp Focus. Chapter 4: OPTICAL PRINCIPLES FOCAL PLANE 35 LENS PLANE FILM PLANE f PLANE PARALLEL TO PLANE OF SHARP FOCUS A D LENS AXIS α RECIPROCAL HINGE LINE FILM PLANE Φ−α Φ PLANE OF SHARP FOCUS SCHEIMPFLUG LINE FIGURE 32: The reciprocal hinge rule states that if the film is rotated about the reciprocal hinge line in the direction shown, the Scheimpflug line moves downward along the lens plane. The Plane of Sharp Focus must then move farther from the lens, but without changing its angle relative to the lens plane. The angles α and Φ change, but Φ−α does not. made here is that the expected angle of view may be different from that expected, based upon the marked focal length of the lens. Tilting or swinging a lens changes its effective focal length. The lens equation for view cameras can be written as: 1 1 sin α + sin (Φ-α) + = A D f sin Φ (5) where A is the lens-to-film distance, and D is the lens-to-plane of sharp focus distance. Both these distances are measured in a direction perpendicular to the respective planes of interest, as shown in Figure 31. The expression after the “=” sign is very nearly equal to 1/f for a surprising range of angles. If α and Φ are both less than 25°, for example, the error will be less than 10%. If α is less than 5°, Φ can be as large as 95° for a similar error. It may also be observed that we can interchange A and D, or α and Φ−α, with no net change to Equation (5). This implies that there is a “reciprocal hinge line” and a “reciprocal hinge rule”. The reciprocal hinge line lies at the common intersection of the film plane, the (rear) focal plane, and a plane through the rear nodal point of the lens, parallel to the plane of sharp focus. The reciprocal hinge rule then states that if the film plane is rotated about the reciprocal hinge line, the plane of sharp focus moves closer to or 36 Merklinger: FOCUSING THE VIEW CAMERA farther from the lens without changing its orientation. Figure 32 illustrates. I originally thought this might not be of much use. While reading a 1904 photography text by the British author Chapman Jones, however, I realized that it is essentially the reciprocal hinge rule that has allowed view camera users to use back tilt as a substitute for lens tilt. According to Chapman Jones, one should never attempt to adjust the camera using lens tilt. He claims that will just result in trouble. If one must set the lens axis out of perpendicular with the film, only back tilt should be considered—even though this may lead to unnatural perspective. If one keeps the lens-to-film distance constant as one tilts a lens, the plane of sharp focus moves in a complicated way that is not easy to understand. The plane of sharp focus changes both its range from the camera and its angular orientation as the lens tilt is adjusted. Furthermore, the apparent movement of the plane of sharp focus depends upon the lens-to-film distance that is set. Thus the effect of tilting the lens is difficult to anticipate. It is very difficult to learn how to judge the right amount of lens tilt by adjusting the lens tilt directly. I refer in Chapter 10 to it being like driving a car on ice. Adjusting the back tilt is a much ‘friendlier’—more predictable—operation. According to the reciprocal hinge rule, rotating the back about some fixed axis (on the film plane) merely regulates the distance of the plane of sharp focus from the camera without changing its angular orientation. The angular orientation is fixed by the relative positions of the lens and the axis about which the back is being tilted. The plane of sharp focus must remain parallel to the plane defined by the lens and the back tilt axis. The reciprocal hinge rule makes it easier to understand some of the arguments over whether base tilts or axis tilts are preferable for the camera back. The ideal, I guess, is to be able to position the back tilt axis so as to determine the desired orientation of the plane of sharp focus. The difficulty I see with using back tilts is how to maintain correct perspective. One solution I’ve heard proposed is to determine the required amount of tilt by tilting the back, but then transfer that amount of tilt to the lens (in the opposite direction) and straighten the back. This method is not rigorous. It often works well enough, but not always. The lens equation for view cameras, Equation (5), implies that the effective focal length is always less than the nominal focal length of the lens. This may seem to be at variance with the statement made earlier to the effect that the effective focal length is often longer than the marked focal length. What really matters is where the main subject is relative to the lens axis. If the main subject initially lies near the lens axis, but the lens is tilted or swung to sharpen some other object, the focal length will seem to increase. If, on the other hand, the main subject is initially well off the lens axis, and one tilts the lens axis towards the main subject, the focal length will seem to decrease. Chapter 4: OPTICAL PRINCIPLES 37 Tilt and Swing So far, and for that matter almost throughout this book, it is assumed that we are looking at the camera in such a way that only lens tilt need be of concern. Now, real view cameras have tilt and swing movements for the lens. Having to accommodate both swing and tilt complicates the mathematics considerably, and so in this book I’ll avoid that issue as much as possible. Two tables are included, however, to help out a bit. Tables IIa and IIb will eventually help us to deal with depth of field under circumstances where both tilt and swing are used. For small angles (less than 10 degrees) swing and tilt can be considered as independent of one another. For the most part, in this book, I use the word “tilt” very generally to mean whatever angle the lens axis makes to a normal to the film plane. In standard view camera language, swing refers to the rotation of the lens carrier about a vertical axis. Tilt refers to the rotation of the lens about a horizontal axis. For most cameras, the tilt axis moves with the swinging of the lens carrier. Thus the swing and tilt motions interact to some extent. If we use both tilt and swing, the total effective tilt—for the purposes of determining effective focal length and depth of field, for example—can be found from Table IIa. We simply read off the resultant value from the appropriate row and column. In this case, it does not matter whether the rows represent tilt and the columns represent swing, or vice versa. We just need to know the two numbers. The swing and tilt do interact to affect the orientation of the plane of sharp focus. And here it does matter which is which. A useful way to describe the result is to examine the tilt of the hinge line in the PTF plane. Imagine, for example, that we have a camera with its film vertical. We tilt the lens forward and adjust the back focus so that the Plane of Sharp Focus lies horizontal. The hinge line is also horizontal. But, if we also swing the lens to the left, the plane of sharp focus rotates such that it rises on the left of the camera (if we are looking forward) and falls on the right. The hinge line must also rise on the left and sink on the right. The angle by which the hinge line rotates is given in Table IIb. I have used the symbol, θ, to represent this angle. I don’t want to dwell on this matter, Tables IIa and IIb are included just to help out a bit in those frequent circumstances where the film plane is vertical. Discussion The view camera provides us with two focus controls: the lens tilt, and the back extension—that is, adjustment of the film-to-lens distance. Most users would consider the back extension to be the main focusing tool. As we have seen, however, the main thing that adjusting the back does is to change the orientation of the plane of sharp focus. Adjusting the back focus causes the 38 Merklinger: FOCUSING THE VIEW CAMERA plane of sharp focus to rotate about the hinge line. Lens tilt, on the other hand, changes the distance (J) between the lens and the plane of sharp focus. Thus lens tilt is, in a sense, the true distance-regulating tool. The distance that lens tilt regulates is a bit strange in that it is measured in a direction parallel to the film. We usually think of focusing distance as measured perpendicular to the film. I have not found it difficult to adapt to this different frame of reference. In fact, the advantages far outweigh the disadvantages. Instead of fiddling with seemingly endless cycles of adjust the tilt, focus, adjust the tilt again, refocus etc., I find I can now usually set the tilt—even before the camera is on the tripod—then set the back, and that’s it, done, so far as focus is concerned! No more iterations. It would be handy to do away with Tables III and IV by having a distance scale (instead of an angle scale) on the tilt adjustment mechanism. A true distance scale would require a different scale for each focal length. A J/f scale would be slightly less convenient, but one scale would serve for all lenses. An illustration of a tilt scale for a 180 mm lens may be found on page 56, at the end of Chapter 6. These are the basic principles. The topics of perspective and distortion will be covered in the next chapter. And after that we’ll use the principles described here to analyze the depth of field situation. Chapter 5: PERSPECTIVE and DISTORTION 39 Chapter 5 PERSPECTIVE and DISTORTION In Chapter 5 we will be addressing some aspects of perspective. Let me say at the outset that I am not an expert on perspective. Furthermore the terminology (language) I use here is not universal. The rules of perspective are, in my humble view, somewhat arbitrary; they are man-made standards rather than natural physical laws. The accepted conventions for perspective predate photography and even the camera obscura. Furthermore the rules have changed with time. My purpose in discussing perspective here is not primarily to establish what is correct perspectiveÑalthough I will express opinions. Rather I wish to explain how the relative positions of lens and film influence geometrical distortion and hence apparent perspective. For rectilinear lenses, tilting the camera lens has no inherent impact upon the apparent perspective of the image. Swinging or tilting a lens is a focusing operation. For the purpose of analyzing perspective, we can indeed best think of the lens simply as a pin-hole. A corollary to this is that when adjusting a view camera, we can consider the action of focusing and the action of controlling perspective quite independently. These two actions do not affect one another. These actions only become coupled if some other constraintÑlike the covering power of the lensÑlimits the degree of adjustment one should properly make. That being said, the next logical question is: Well, what does control perspective? And, what is it anyway. Perspective in its most general terms means giving visual clues to the viewer of a two-dimensional image, so that he or she can interpret the image in three dimensions. These clues include such nuances as relative size, sharpness, and contrast. Contrasty objects are generally perceived to be closer than objects of low contrast, for example. Fuzzy objects are assumed to be closer than or more distant than the main, clearly delineated subject. The type of perspective I will address here, however, is restricted to geometrical shapes and sizes. In the context of photography with rectilinear lenses, there are two effects of interest: image magnification and image distortion. One might even argue that there is only oneÑmagnificationÑbut the degree of magnification may be different in different directions. I find it easier to think in terms of relative magnificationÑhow does image size change if I put the object here rather than thereÑand distortionÑdo the proportions (relative height and width) of the object change if it is moved from here to there. Both effects depend mostly upon one factor: how far off the principal axis is the image? Although the term Òprincipal axisÓ was defined in Chapter 3, we havenÕt needed it up until now. To recap, the principal axis is a line through the lens 40 Merklinger: FOCUSING THE VIEW CAMERA PRINCIPAL AXIS PIN-HOLE LENS FILM PLANE FIGURE 33: The image of a spherical object (the moon, say) will be imaged near the principal axis as a circle. At a large angles off the principal axis (more than 30°) the same object at the same distance will be imaged as a somewhat larger ellipse. The image is magnified because the effective focal length of the lens is greater at this angle, and it is elongated because the imaging rays strike the film obliquely. FIGURE 34: HereÕs a photograph of six table-tennis balls and five round white dots taken with a 15 mm lens on a 35 mm camera. The balls in the corners show significant stretch while the ball in the very center is quite round. The dots, on the other hand are round no matter where they areÑso long as the film plane is parallel to the object planeÑas was the case here. Chapter 5: PERSPECTIVE and DISTORTION 41 and perpendicular to the film plane. Only near the principal axis are objects imaged in the proportions that reflect the true state of affairs. When a small light ray bundle of circular cross-section strikes the film at an angle, the image on the film will not be circular, but rather elliptical: a circle that has been stretched in one direction. In one direction, perpendicular to a line from the image to the intersection of the principal axis and the lens plane, the image has the ÔrightÕ dimension. In the other direction, parallel to a line from image to the intersection of the principal axis and the lens plane, the image is too long. This is what I mean by distortion. How much too long depends upon how obliquely the ray bundle strikes the film. On the film plane near the principal axis, objects are imaged with negligible stretch. Far from the principal axis, the stretch can be significant. Figure 33 illustrates. What really matters here is the angle measured at the lens between the principal axis and the rays forming the particular image of interest. Within 25° to 30° of the principal axis, the stretch is almost unnoticeable. Beyond that it becomes increasingly noticeable; the stretch will reach a factor of two for rays 60° off the principal axis. The implication of this phenomenon for perspective is that the flattening of a circle tends to make us think we are looking at the circle from an oblique angle. If we are looking at a sphere which gets flattened, however, we are confused: the image does not look real. In real life a sphere is always round, no matter how we look at it. Figure 34 shows some table-tennis balls and some flat white dots mounted on a sheet of black cardboard. They are photographed from a distance of one foot with the widest-angle rectilinear lens I have: a Pentax 15 mm lens for 35 mm cameras. In this photograph the dots all look fine. The balls look fine near the center of the image, but rather strange near the corners. Your eyes may notice a certain discrepancy between Figures 33 and 34. In Figure 33, the ellipseÕs smallest dimension is larger than the circle. In Figure 34, the ÔwidthÕ of the ellipsoidal table-tennis balls is just about the same as diameter of the round table-tennis balls. Why the difference? The answer is that I drew Figure 33 to represent the images of two spherical objects each subtending the same angle at the lens. If the objects are of equal size, they must be at the same radial distance from the lens for this to be the case. Off the principal axis the image magnification is greater (as is the distortion) and so the image is larger. For Figure 34, the balls at the edges and in the corners are farther from the lens than is the ball at the center of the picture. One might expect therefore that their images would be smaller, but the magnification effect exactly cancels the size reduction due to subject distance. If we concern ourselves only with the extent of an object measured in a direction parallel to the film plane, image magnification depends simply upon lens-to-image distance, A, divided by the lens-to-subject distance, Z. For images of three-dimensional objects, we need to consider the lens-to-image Merklinger: FOCUSING THE VIEW CAMERA 42 FILM PLANE Pin-Hole LENS h PLANE OF SHARP FOCUS (Arbitrary) Ah Z HINGE LINE PLANE AT DISTANCE, Z, PARALLEL TOFILM PTF PLANE A Z FIGURE 35: Image magnification depends upon the lens-to-image distance divided by the lens-to-subject distance. There are subtle differences in how this works out for two-dimensional and three-dimensional subjects. PIN-HOLE LENS PRINCIPAL AXIS FILM PLANE FIGURE 36: If the image of a spherical object lies off the principal axis and also extends over a large range of off-principal axis angles, the image will be egg-shaped. The portion of the image farthest from the lens is magnified more than are other parts of the image nearer the lens. Chapter 5: PERSPECTIVE and DISTORTION 43 slant range divided by the lens-to-subject slant range. The shortest possible lens-to-image distanceÑand hence minimum magnificationÑwill occur along the principal axis. 60° off the principal axis, image magnification will be twice what it is at the principal axis, assuming objects are uniformly distant from the lens. Figure 35 illustrates some of these considerations. These elongation and magnification effects compound one another: the farther off the principal axis (in angular terms) an object lies, the greater is its magnification, and, the more stretch it undergoes. Even worse, for large objects subtending a significant angle at the lens, the magnification and stretch can be significantly greater at one side of the object than at the other. Figure 36 shows how a large image of a spherical object can become egg-shaped. What has happened here is that a) since the image is off the principal axis, the sphere is imaged as an ellipsoid and b) since the ellipsoid covers a wide range of magnifications, the end of it farthest from the principal axis is magnified significantly more than the end near the principal axis is. Users of ordinary cameras equipped with wide angle lenses will be familiar with these Òwide angle distortionÓ effects. A beach ball looks normal in the center of a picture, but more like a football near the corners of the image. And if the beach ball is close enough to the camera, so that itÕs image is relatively large, the changing magnification from one end of the ÔfootballÕ to the other will make it look more egg-shaped than football-shaped. The Ôdistortion controlÕ offered by view cameras is really nothing more than manipulation of the Ôwide-angle-distortionÕ described above. What makes the view-camera different from the normal SLR, say, is that in the case of the view camera we see only a portion of the total picture. If the view camera photographer selects that portion near the principal axis, we see a relatively normal image. But he can equally well select the very outside of the lens coverage circle, where stretch is at a maximum, and where magnification is varying most rapidly. And so he can turn beach balls in the very center of the final image into eggsÑor vice versa. Although lenses made for view cameras usually have lots of covering power, even lenses of modest covering power can be made to yield excessive distortion. All the view camera photographer needs to do is to swing the camera back. Figure 37 shows how. In this case there is no image at all where the principal axis meets the film plane. But the film is illuminated at a steep angle, and hence the distortion effects are significant. In this case the photographer might have a focusing problem, but then again, for the right subject, this geometry might be exactly what he needs. I note in passing that there is an exposure factor to consider also when the imaging rays fall obliquely on the film. The image intensity is reduced because the same bundle of light is spread over a larger area of film. If the light rays fall at a 45° angle to the film, one must open the lens by one stop (or double the exposure time) to compensate. At 60° off the principal axis, the compensation required is two stops. 44 Merklinger: FOCUSING THE VIEW CAMERA PRINCIPAL AXIS LENS LENS AXIS FILM PLANE FILM FIGURE 37: If the image of a spherical object lies off the principal axis and also extends over a large range of off-principal axis angles, the image will be egg-shaped. The portion of the image farthest from the lens is magnified significantly more than are other parts of the image nearer the lens. FIGURE 38: 38 a) show a relatively normal photograph of a globe. 38 b) shows the same object from essentially the same camera position, but with the camera back swung by 40°. The world no longer appears spherical, and the books seem to be lying on a slope. Chapter 5: PERSPECTIVE and DISTORTION 45 Figure 38 shows exactly the situation just described. Figure 38a shows a quite normal photo of a globe. For this photograph I used only back rise and front fall to obtain a downward viewing angle. The globe is very slightly taller than it is wide as a result, but one can hardly notice it. Figure 38b shows what happens when I swing the back by 40°. It was also necessary in this case to swing the front by 20° to maintain reasonable focus. I would have used greater back swing to exaggerate the effect even more, but the camera bellows would not allow that without cut-off of the image. Figure 38b was also taken from slightly farther away than Figure 37a, just to let the globe fit comfortably in the frame. When reproducing images of flat subjects such as drawings or paintings, it is important that the film plane and the object planes are parallel. This will ensure that any distortion of the image by the camera will exactly compensate the natural perspective distortion that exists when we look at a drawing at an angle. If we were to look objectively at the object from the position of the lens, we would see that the extreme corners looked small and foreshortenedÑthat is, squashed in one direction. But the intentional distortion introduced by a rectilinear lens exactly compensates and ensures that the final image looks ÔrightÕ. This is why the flat white dots in Fig 34 still look circular. Perhaps the most common examples of a distorted but ÔcorrectÕ images are pictures of buildings. By convention, tall buildings are represented in art and photography with vertical lines that do not converge, and in some cases even with horizontal lines that do not converge. Preventing converging vertical lines is simple: keep the principal axis horizontal. This means keeping the film plane vertical. Keeping horizontal lines from converging is also simple: keep the principal axis perpendicular to the horizontal lines in the image. If we are photographing a building in such a way that we can see only one face of the building, we usually want to keep both horizontal and vertical lines from converging. This is done by holding the principal axis perpendicular to the visible face of the building. In other words, we keep the film plane parallel to the face of the building. We can position the lens anywhere, but keep the film parallel to the building. If, in another example, one can see two faces of a building (the front and one side, for example), the convention states that we maintain only the vertical lines parallel; we let the horizontals converge. We can still play with the orientation of the principal axis in the horizontal plane. The most natural image usually results when the principal axis is pointing at about the center of the building. There is some freedom, however, to exaggerate the apparent perspective. In general, the building will tend to look as though it is being viewed in a direction parallel to the principal axisÑeven when that is not the case. Figure 39 illustrates two images of a building from the same camera position. Figure 39 a) shows a building taken using front rise only. (Well, actually, I also tilted the lens down a bit to sharpen the foreground.) Vertical Merklinger: FOCUSING THE VIEW CAMERA 46 FIGURE 39: a) shows a ÔcorrectÕ image of a building. Front rise only was used for this photo. In b) the back was swung by almost 38° to give a rather odd and ÔincorrectÕ image. This second photo looks in some ways as though it was taken from a camera position to the photographerÕs right. But other clues indicate this is not the case: we can see the left side of the building, for example, and we know that most buildings have 90° corners. Swinging the back also stretches the image horizontally. M = 0.125 y x M = 0.17 y x M = 0.25 M = 0.5 LENS (Pin Hole) FILM PLANE a) b) FIGURE 40: Planes of constant magnification lie parallel to the film plane. In a) above, the film plane lies at right angles to the main line of sight of the camera, resulting in normal perspective. In b), the camera back has been swung by about 38°, yielding magnifications characteristic of the Ôbox cameraÕ at right. Note. how the magnification at point x is greater than that at y in the case of a), but the reverse is true in b). Chapter 5: PERSPECTIVE and DISTORTION 47 lines are vertical and horizontal lines converge: for the left face they converge to the left and for the ÔfrontÕ face they converge to the right. For Figure 39 b) I moved the back by about 38° (counterclockwise if one is looking down on the camera). The lens also needed to be swung to re-establish acceptable focus. This somewhat extreme back swing causes the horizontal lines for the ÔfrontÕ of the building to converge to the left instead of to the right. The result tends to make the photograph look as though it was taken from a position further to the photographerÕs right. This photograph is not ÔcorrectÕ; the viewer is confused by the mixed visual clues. The significance of planes in front of the camera but parallel to the film plane is that small spherical objects (or subject elements) on any one of these plane will be reproduced in the image at a consistent magnification. That is, an object on any one such plane will be the same size in the image, no matter where on the plane it lies. We can thus ÔmapÕ image magnification as a system of parallel lines. If we swing the back, we change the orientation of these constant-magnification planes in the object space. The new Ômagnification mapÕ is skewed. But it is precisely consistent with a different camera position. Thus swinging the camera back results in an image which has the magnification of one (phantom) camera position, but an arrangement of image elements characteristic of another (the true) camera position. See Figure 40. I have seen examples in books where it is stated that the proper view camera solution to photographing a wall obliquely will involve both lens swingÑto maintain focusÑand back swingÑto ÒrestoreÓ the proper perspective. I do not believe this is correct. I suggest that what has happened is that when the photographer adjusted the lens for proper focus, he discovered that the horizontal lines no longer converged to the degree he expected. The true cause was an effective change in focal length, but the photographer interpreted the result as a change in apparent perspective. To achieve the expected convergence of the horizontal lines in the image, he adjusted the back. This is not a serious error; for the most part, it is all a matter of art and taste. So, the lesson for this chapter is that, for the purposes of perspective, lenses act like a simple pin hole. All that matters is where the image is in relation to the principal axisÑa line perpendicular to the film plane and passing through the pin hole. Near the principal axis, images are reproduced more or less naturally. For images formed at large angles from the principal axis, the image is stretched in one direction and magnified. The view camera is able to reproduce this Ôwide angle distortionÕ even with lenses which are not wide-angle. As a vague guide, images often tend to look as though the cameraÕs line-of-sight were along the principal axis of the cameraÑeven when such is not actually the case. This has not been a complete guide to perspective; the major points are that lens orientation (swing and/or tilt) does not affect perspective, rather, it is the angle of incidence of the image rays on the film that determines the effect. 48 Merklinger: FOCUSING THE VIEW CAMERA The Ottawa River, looking west Here, I used about 9° of forward lens tilt in order to have the plane of sharp focus pass just under the feet of the tripod. The plane of sharp focus falls a bit as it proceeds away from the camera, passing through the tree just right of center. Chapter 6: DEPTH of FIELD 49 Chapter 6 DEPTH OF FIELD There are at least two ways to think about depth of field. In the traditional treatment of the depth of field, one uses the following logic. First establish an image resolution criterion—expressed by the maximum permissible size of the circle of confusion. Then determine the depth of focus needed to achieve that circle of confusion for the lens aperture being used. Finally, calculate where objects may lie in order that their images are at the extreme limits of the permissible depth of focus. Another possible philosophy is that described in The INs and OUTs of FOCUS. In this “object field method”, one first asks what resolution is required at the object. Then one calculates what is necessary to achieve it in terms of the physical lens aperture diameter and the distance to the plane of sharp focus. The object field method has the advantages that we can adjust the required resolution to suit the requirements of the individual picture, and that precisely the same calculations work for all formats and all focal lengths. Both of these methods can be applied to view cameras. And they are, of course, completely consistent with one another. First, we’ll look at the traditional method. This method is probably more familiar to you. This scheme is, however, the more complex of the two, especially where view cameras are concerned. As we’ll later see in the Chapter 7, the simplest route to the image-based solution is via the object-based method—but we’ll look at the traditional methods first. Image-Based Depth of Field Following the steps outlined for the traditional method above, we set the maximum diameter of the circle of confusion as a. The permissible focus error is then (approximately) g = Na, either side of the film plane, where N is the f-number. Imagining that the film plane were moved to one extreme limit of the depth of focus and then the other, the plane of sharp focus moves through a wedge-shaped region of space in front of the camera. This wedge describes the region within which objects will be imaged at the film with a circle of confusion smaller than diameter a. We need, then, to decide what diameter to use for the maximum diameter of the circle of confusion, and then how to describe that wedge. First we’ll consider the size of the circle of confusion. One of the often-used Merklinger: FOCUSING THE VIEW CAMERA 50 LENS PLANE FRONT FOCAL PLANE f LENS AXIS A g Φ1 Φ2 Φ ZONE OF ACCEPTABLE DEFINITION α FILM PLANE PLANE OF SHARP FOCUS HINGE LINE PARALLEL-TO-FILM (PTF) PLANE SCHEIMPFLUG LINE FIGURE 41: For an allowable depth of focus ±g, the depth of field extends from Φ1 to Φ2: the shaded area above. numbers is one fifteen-hundredth of the focal length of the lens. This corresponds to the 1/30 mm number often used for 35 mm cameras, for example. It is indeed convenient to use some fraction of the lens focal length, because it will turn out that we can make one set of tables serve for all lenses that way. I offer the caution, however, that it may actually make more sense to use some fraction of the format diagonal rather than focal length. It will be easy to change criteria anyway, because the determining factor is the depth of focus which is simply the product of the f-number and the diameter of the circle of confusion. Thus one table can serve for many combinations of a and N, so long as the product of a and N remains unchanged. I propose two ways to describe the depth-of-field wedge. The straight forward way is to state the angles between the film plane and the depth of field limits. The angle for the ‘near’ limit of depth of field—on the lens side of the plane of sharp focus—will be called Φ1. The angle to the limit of depth of field on the far side of the plane of sharp focus will be called Φ2. See Figure 41. Angles are not always easy to estimate in the photographic environment, and so I propose a second alternative. Suppose we wish to determine the depth of field, about the plane of sharp focus, some specified distance in front of the camera. We can express the depth of field measured in a direction perpendicular to the plane of sharp focus as a fraction of the lens-to-plane of sharp focus distance. To keep things consistent, we’ll measure the lens-to-plane of sharp focus distance in a direction perpendicular to the film plane. The way of measuring things just described will not always seem Chapter 6: DEPTH of FIELD 51 LENS PLANE FRONT FOCAL PLANE PTF PLANE f LENS AXIS A g L1 L2 α FILM PLANE PLANE OF SHARP FOCUS Z L1 = mZ L2 = lZ FIGURE 42: For an allowable depth of focus ±g, the depth of field can also be described in a direction perpendicular to the Plane of Sharp Focus, shown here by the symbol L, and expressed as a fraction of the distance, Z. natural, but we have to be consistent. graphically in Figure 42. This scheme of things is shown Expressing depth of field as a fraction of the lens-to-plane of sharp focus distance has its limitations. It is possible for the depth of field to become too large to be described accurately this way. If the angle between Φ1 or Φ2 and Φ becomes equal to or greater than 90°, the depth of field, expressed as a fraction of the lens-to-plane of sharp focus distance, becomes infinite. In practice, this usually does not present the photographer with a problem. When the depth of field is that great, it is usually more than sufficient for the purpose. And if that is not the case, we can probably estimate angles well enough that the angular description of depth of field can be used instead. The depth of field tables at the back of this book, Tables VII to XVII, are organized in pairs for each f-stop from f/2.8 to f/90. The left-hand, or upper-most table shows the limits of depth of field expressed by angle. The right-hand page, or lower table, shows the depth of field as a fraction of the distance Z. The tables are calculated for a maximum circle of confusion of one-fifteen-hundredth (1/1500) of the lens focal length. But each table is actually valid for a number of combinations of f-number and circle of confusion size. A few of the applicable combinations are listed on each table. Note that each table applies for lenses of all focal lengths. Since the distance J is given as J/f, however, the appropriate value of J/f must be re-calculated if the lens focal length is changed. Merklinger: FOCUSING THE VIEW CAMERA 52 When changing lenses, care will be needed in choosing the right table to use. Let’s suppose we are using a 4⫻5 in. camera and presume the 150 mm lens is the ‘normal’ focal length. One-fifteen-hundredth of 150 mm is 0.1 millimeters. We thus might choose a = 0.1 mm. For the 150 mm lens we can then use the depth of field table for the main f-stop as marked on the table. If we switch to a 90 mm lens for that same camera, we will probably still wish to use a = 0.1 mm. But 0.1 mm is no longer one-fifteen-hundredth of the focal length. With the 90 mm lens, the ratio a/f is 0.1/90 or one-nine-hundredth of the focal length. The closest a/f value shown on the tables is 1/1000. If we look at the combinations of f-number and a/f values shown on the tables, we will see that the f-stop opposite a/f = 1/1000 is one stop larger than the f-stop for a/f = 1/1500. What this means in practice is that the same table that applies to the 150 mm lens at f/22 also applies to the 90 mm lens, but at f/16. The tables have a blank space in the bottom left corner. The purpose of that space is to provide a place for your notes on what combination of f-stop and lens focal length is appropriate in your case. Similarly, if you were using a 210 mm lens on that 4⫻5 camera, the f/22 table would apply, but at f/32 for the 210 mm lens. Thus in this case you might write in that space for the “f/22" table: “90 mm: f/16; 150 mm: f/22; 210 mm: f/32.” Although the depth of field fractions are given in the many tables, it can be instructive to show you the algebraic expressions for them. There are many ways to manipulate the formulae, but perhaps the most useful are the following: m= L1 = Z gD /A2 (6a) gD cos (Φ−α) 1+ fA or m= L1 = Z gJ sin (Φ−α) /fA 1+ and l= L1 = Z gJ sin (Φ−α)cos (Φ−α) (6b) f2 gD /A2 gD cos (Φ−α) 1fA (7a) Chapter 6: DEPTH of FIELD 53 or l= L1 = Z gJ sin (Φ−α) /fA 1- gJ sin (Φ−α)cos (Φ−α) . (7b) f2 These expressions tell us several things. First, there are many circumstances under which the depth of field is proportional to J or D. Increasing J or D increases depth of field, even when the distance Z remains unchanged. Thus using less lens tilt will usually increase depth of field. Another curious item is that when cos(Φ−α) is equal to zero, depth of field is precisely symmetrical about the plane of sharp focus. This situation will normally occurs when Φ is a little more than 90°. The depth of field is also symmetrical whenever J and D are quite small. The expressions given in Equations (6) and (7) lead exactly to the standard depth-of-field relations for ‘normal’ cameras when appropriate substitutions are made for zero lens tilt. Object-Based Depth of Field The object field method for dealing with depth of field does not require tables per se. There is a simple relationship between lens diameter, distance to the subject, distance to the plane of sharp focus and resolution spot size. That relation is: K=R S d (8a) L=D S d (8b) or where S is the resolution spot size, D is the distance from the lens to the plane of sharp focus, R is the distance along a ray from the lens to the plane of sharp focus and d is the physical diameter of the lens (at its working aperture). (In The INs and OUTs of FOCUS, S was called the “disk-of-confusion”.) The depth of field is either K or L as described following. If we work with distances measured along a ray from the lens, passing through some point on the plane of sharp focus, we will call the depth of field K. K is the depth of field on either side of the plane of sharp focus, measured along the