tilt or not to tilt

[Leonard Evens]

Nigel,

This is how it works. Let's talk about the case where the back is vertical and the lens is tilted down. Then as you may know, the Scheimpflug principle says that the film plane, the lens plane and the plane of exact focus all intersect in a line. But there is an additional line called the hinge line that is not so well known, despite Merklinger's attempts to publicize it. In the above case, this line is vertically below the lens and lies in the subject plane. The region in focus is a wedge centered on this plane of exact focus and bounded above and below by two (half) planes which also emanate from the hinge line. The angular opening of this wedge depends on the f-stop.

Saying the wedge is centered on the plane of exact focus is not correct in the sense of angular centering, and it requires some further explanation. At any fixed horizontal distance from the lens, consider a vertical plane. That intersects the two bounding planes in two lines, one above and one below. These are equidistant from the plane of exact focus in the vertical direction. Moreover, at the hyperfocal distance, which depends on the f-stop, that equal distance is (very close to) the distance from the lens to the hinge line. Hence by proportionality, the distances above or below the plane of exact focus can be calculated at any other distance.

In any case, since the region of good focus is a wedge, objects near the lens need to be close to the plane of exact focus to be sharp. That means that if there is any significant vertical extent close to the lens that needs to be in focus, you are unlikely to be able to accomplish it with a tilt.

Saying this in words so someone will understand it is not easy. You would be well advised to draw some diagrams to help you get it right. It is possible to learn to use tilts by rote by just repeating what happened to work in some other situation, but it is much easier if you understand the underlying three dimensional geometry.

[Capocheny]

Hi Leonard,

Wow! I'd say you did a bang-up job on explaining this to Nigel (and to the rest of us, for that matter.) I know people say it's an easy concept to grasp in practice... but in theory, it's hard to imagine.

It's too bad I don't know how to send attachments on this site because I have a movie that explains it in diagramatic form. It demonstrates what you're saying in literary form...

If you'd like to email me off-line... I'll be glad to send it to you. I know how to do it in my email program. I believe it's a substantial file size so... I'll do my best to attach and send it to you, ok?

Cheers

[Brian Ellis]

I don't understand Leonard's explanation and I'd like to. I can envision the wedge that's often talked about and of course I understand the Scheimpflug principles. But I can't envision the "hinge line" that Leonard discusses. If I have the camera back straight up and the lens pointed down as in the situation under discussioon what does this line look like? By that I mean is it a horizontal line out somewhere in the subject area extending from left to right across the entire area that I can see in the ground glass, or is it a line at the lens position somewhere above or below the lens, just where is it and what does it look like? What causes it to move?

Sorry to ask such an obtuse question but I tried reading Merklinger and couldn't make heads or tails of what he said. In fairness to myself, I didn't think his system was practical for field photography as opposed to studio where everything can be precisely measured so I didn't try very hard. But as Leonard says it would be nice (though I don't think really necessary) to understand the geometry and I clearly don't.

I no longer have my Merklinger book so I can't go there but I do have many of the common books on large format photography. Is there a diagram in any of them showing the hinge line in different situations? I also have most back issues of "View Camera" and "Photo Techniques" if there are articles in any of them that discuss this.

[John Hennessy]

Merklinger has some movies as part of one of his on-line papers (maybe the ones referred to above) which are extremely helpful to internalizing such a non-intuitive thing as tilts and swings. Everyone seems to know (even my barber) that the plane of sharp focus, the film plane and the lens board plane intersect. But few seem to know that another intersection is needed to fully define the plane of sharp focus. That is the hinge line. Remember high school geometry: two points define a line and two lines define a plane. The hinge line is the line (or point as drawn from the side of the camera) defined by (a) a vertical line (or plane edge to be precise) through the center of the lens and (b) another line parallel to the lens board one film-to-lens length in front of it.

I agree with Leonard that drawing all this out a few times (I did it years ago on huge graph paper and to scale with a few typical subjects) will show one why a little tilt is usually--maybe always--more effective than too much and why hyperfocal distance is important. The answer is that with 1 to 3 degrees of tilt, the wedge of things in focus opens up as wide as Madame Butterfly's fan; the bad news is that PSF is closer to vertical. Conversely. too much tilt causes the cone of image seen by the lens and wedge of sharpness to coincide more closely; the bad news is the wedge gets so narrow that near tall objects and distant low objects (or worse yet--falling away valleys) are hopelessly out of focus. I am from Missouri (the land of sceptics) so just remembering to focus on the near or to starve a cold or whatever didn't work for me.

[Leonard Evens]

Brian,

About the hinge line. I have to admit that although it is now perfectly clear to me, I had a hard time understanding it from Merklinger's exposition. I finally got it straight after working my way through Bob Wheeler's Notes. He doesn't spend much time discussing it, but what he says was crystal clear, at least to me. Then going back to Merklinger, what he said made sense. I think I did a reasonable job of explaining it in www.math.northwestern.edu/~len/photos/pages/dof_essay.pdf. But of course something always looks easier to you if you already understand it. You might look at my essay, skipping through the formulas and going to the section where I discuss the hinge line.

Let me give it another try in words. One problem is that the diagrams that people use are seldom three dimensional, and what is a line in space appears as a point in a two dimensional diagram which represents just one cross section. Keep in mind that generally two planes in space, if they are not parallel, intersect in a unique line. If you throw in a third plane, except for some special circumstances, it will intersect that line in a point. However, in the view camera configurations, the third plane intersects the other two in that same line, rather than a point. Keep in mind that picture of three planes intersecting in a single line.

Suppose the camera is vertical, and the lens is tilted down for simplicity. You say you can visualize the Scheimpflug line which is the common intersection of the film plane, the lens plane, and the subject plane. The Scheimfplug line is some distance below the actual frame in the film plane. Next consider a (vertical) plane passing through the lens and parallel to the film plane. (In actuality it will have to be some specific reference point depending on the lens, but let's ignore that and treat the lens as a point.) That vertical plane will also intersect the subject plane in a single line. That line is the hinge line. It is parallel to the Scheimpflug line. If you drop a plumb line perpendicular vertically from the lens, it will meet the hinge line at right angles. So the hinge line extends horizontally in either direction, centered on the plumb line.

Now try to imagine what happens as you move the rear standard back and forth, while keeping the lens tilt fixed. To be specific, suppose you move it back. The film plane will move further from the lens, so the Scheimplug intersection line will move down in the film plane. That is because the lens plane will remain fixed and increasing the distance from the film to the lens will force the intersection of the two planes to drop downward in the film plane. Since the subject plane also has to pass through that Scheimpflug line, it follows that the subject plane must move. The crucial fact then is that while it moves, it still passes through the hinge line. Hence, the result of moving the standard is to swing the subject plane about the hinge line. Think of the plane of exact focus as a door lying on its side with its hinges attached to the hinge line. It swings up and down but remaining fixed on its hinges as you move the rear standard back and forth.

There is no way to see the hinge line on the ground glass, as there is no way to see the Scheimpflug line on the gg, at least with any plausible lens. The angular displacement of the hinge line from the lens axis is 90 degrees and that of the Scheimpflug line is over 90 degrees. But that doesn't make it useless in practical situations. Of course, what you see on the gg is extremely important, but we shouldn't forget that there is actually a three dimensional scene out there that we are trying to photograph. These reference lines tell us some very important things about that geometry.

What causes the hinge line to move? Well it won't move if you keep the lens tilt fixed. That is why it is important. But changing the tilt will move it. If you increase the tilt, it will move closer to the lens and if you decrease the tilt, it will move further from the lens. When the tilt is zero, it will be, as we mathematicians put it, at infinity.

The above discussion is for one special configuration, the back vertical and the lens tilted down. If the back is vertical and the lens is tilted up, the hinge line will be above the lens. More generally, you can always find it by visualizing the position of the subject plane and the Scheimpflug line, and then imagining a plane through the lens, which is parallel to the film plane. That plane will intersect the subject plane in the hinge line. It will have about the same relative position as the Scheimpflug line.

The hinge line, despite its crucial importance for view camera movements, does not seem to receive any attention in any of the standard references, except for Merklinger, who makes it too complicated. I'm not sure exactly why it has been ignored, since as Merklinger points out, it was apparently known from the very beginning of the subject. Perhaps you have to be trained professionally, as a mathematician or engineer would be, to think in three dimensional terms to understand it. I was always surprised when teaching mathematics that so many bright students had so much trouble with such visualization. I would be surprised if many large format photographers lacked the basic geometric visualization skills, so it is a mystery to me.

[Sam Crater]

Thanks very much for the link to your paper, Leonard. It certainly explains the hinge line. Can you use it to explain how as the tilt approaches zero and the plane of focus is near the hyperfocal distance, there is so much more of the in-focus wedge behind the plane of exact focus than in front of it?

[Robert J Cardon]

I'd beg to differ with the claims of tremendous DOF at F32 for an 80mm. DOF is DOF, which is a function of focal length and aperature. An 80mm on a 4x5 80mm is still 80mm and gives you the same real DOF as an 80 on Med Format or 35mm, IMHO. If your near subject is 3' from the camera, with the far point on infinity, F32 simply isn't going to give you tack sharp focus everywhere, at least when no tilts are used. Sure the so-called acceptable COF on a big neg is larger than for a small neg, but if you're making 16x20+ prints, something will give in terms of sharpness - there are always trade-offs!

[Leonard Evens]

Robert,

You are ignoring the fact that the amount of enlargement necessary to produce a certain size print, say 16 x 20, is dependent on the format. You would magnify a 35 mm negative about 17 times to produce such a print and a 4 x 5 negative about 4 times (actually slightly more). Hyperfocal distances, DOF and other such matters depend on the diameter of the maximal acceptable circle of confusion in the film plane. This number depends in turn on what is acceptable in the print under specified viewing conditions and the degree of enlargment. Thus assume people are going to get very close to your 16 x 20 print. Then a plausible coc for the print would have diameter 0.2 mm. The resulting coc in the 4 x 5 negative would be about one fourth of that or 0.05 mm. The acceptable coc in the 35 mm film would be about one fourth of that or 0.0125 mm. The formula for hyperfocal distance is f^2/Nc where f is the focal length, N is the f-number and c is the diameter of the coc in the film. If you decrease c by a factor of four, you increase the hyperfocal distance by the same factor. A larger hyperfocal distance means less depth of field. But of course, 80 mm is a wide angle lens for 4 x 5 and a moderately long lens for 35 mm, so there are other considerations at play here.

Of course, if you use a 4 x 5 camera with an 80 mm lens, and enlarge some 24 x 36 mm section of it to 16 x 20 size, you will get the same results (except possibly for shifts) as with a 35 mm camera and an 80 mm lens. Or if you enlarged both the 4 x 5 and the 35 mm negatives , both from 80 mm lenses at the same aperture, the same amount, thus producing radically different size prints, you would get the same depth of field

[Robert J Cardon]

LE - Yes you are correct, all I was pointing out was that even with an 80 or a 75, there's not so much DOF that you will never need tilts to get the sharpness you need; nor can you be casual about your technique. I quickly learned that these wide lenses are not always cure-alls for the DOF problems encountered with a 210 and longer.

RJ

[Leonard Evens]

Sam,

I'm not sure just what you are asking. Perhaps this might clarify it.

As the lens tilt approaches zero, both the Scheimpflug line and the hinge line move downward "to infinity". The film plane, the lens plane, and the subject plane approach being parallel. The vertical distance from the lens to the hinge line approaches infinity. In the limit, the nearer limiting plane approaches a plane parallel to the final subject plane and, if the subject plane is at the hyperfocal distance, the nearer plane is at half the hyperfocal distance from the lens. The farther limiting plane also approaches a plane parallel to the subject plane, but that plane lies "at infinity".

(The lines and planes at infinity are hard to visualize. Mathematicians understand it through a generalization of ordinary geometry called projective geometry. )

One thing that may be confusing is how the distances to the two limiting half planes above and below the subject plane change. Consider a line along the horizontal lens axis BEFORE tilting, and suppose you maintain a focus so that the subject plane intersects this line at the hyperfocal distance from the lens as you decrease the lens tilt. Consider also the vertical distance J from the lens to the hinge line. Then at the hyperfocal distance, the vertical distance from upper and lower limiting planes to the subject plane are slightly more than the distance J. As the tilt angle gets smaller, that distance J get larger, so the upper limiting plane and subject plane both approach the vertical. But at any tilt angle, the lower limiting plane is almost parallel to the untilted lens axis since it is essentially distance J below it. (Draw a picture.) That means it never intersects the untilted lens axis, and the back DOF in the horizontal direction is infinite. As the tilt angle gets smaller and smaller, the bottom limiting plane just drops lower and lower, and in the limit becomes the plane at infinity.

Perhaps it might clarify things to consider the situation when the lens is tilted down but the tilt angle is very small (as indeed may often be the case for real cameras). In that case, the hinge line is very far below the lens. For example, if the focal length is 150 mm, and the tilt angle is .01 degrees, the hinge line would be about 860 meters below the lens. Assume you have focused so the subject plane passes through the hyperfocal distance at lens level, and suppose the hyperfocal distance is about 7 meters. The subject plane would make an angle with the vertical of a little less than half a degree. The near limiting plane would make an angle of less than one quarter a degree and would pass through the lens level just about at 3 1/2 meters from the lens. It would continue to rise and as it passed through the plane at the hyperfocal distance of 7 meters, it would be about 860 meters above the lens level. The farther limiting plane would start well below lens level at the hinge line, about 860 meters down. At the hyperfocal distance, it would still be about 860 meters below the lens level and would extend outward from there. The back horizontal focus behind the hyperfocal distance would be infinite at any lens tilt.