Location of hinge line for tilted lens plane

[Leonard Evens]

Warning. This is highly technical, and it won't be of any practical interest, except possibly for people doing tilts with lenses of highly asymetric design, such as telephoto lenses.

If you fix the tilt angle, then it is known that, as you focus, the plane of exact focus rotates about a line, called the hinge line. The hinge line is also where the planes bounding the DOF region meet. If the back is vertical, the hinge line is located a certain distance, determined by the focal length and tilt angle, below the lens. Every analysis of this that I've seen, including my own, treats the lens as a point. The one exception is a diagram in Scheimpflug's original patent, as shown by Merklinger, which seems to suggest that the hinge line is in a plane through the tilt axis---see below---and parallel to the film plane. My intuition suggests that this should be right, at least if the pupil magnification is close to 1. (Otherwise, for DOF calculation, one would presumbly have to tilt about a line through the entrance pupil and use the plane, parallel to the film plane, containing the entrance pupil, but that is just a guess.)

A related question is where the tilt axis should be. Merklinger's discussion seems to suggest it should be the front nodal point, and my guess is that is right. But Merklinger's further comments suggest that one might use some other axis, even one in the film plane, and that would change the location of the hinge line.

My problem is that I've just read what Sidney F. Ray has to say about the matter briefly in Applied Photographic Optics, 3rd ed. In the text, he suggests that one use the rear nodal point to tilt about, and in a diagram, he shows the wedge shaped DOF region as being centered on a line in the film plane. I am pretty sure his diagram is wrong, unless one tilts about an axis in the film plane, since it conflicts with what Wheeler, Merklinger, and everyone else I know has looked at it, including myself, thinks. But I've been wrong before.

Any ideas? If someone has already done the analysis, I would like to know before I try to plunge in.

P.S. Just reading through various section of Ray, I've already found two other errors in formulas, which look almost like typos. This would not be unusual in a book of such scope, and I suspect there are a lot more. (The copy editor who did my mathematics book told me people still find errors in recent editions of Dickens novels.) I've written to the publisher to see if there is a list of errata, but if anyone already has one, please let me know.

[Andy Eads]

Leonard,
This might be a moment to advance the field of knowledge...and you are just the guy to do it. I have a Nikkor process lens at home that came with the official Nikon measurements for nodal points, true focal length, etc. If you can find such a lens or have one already, you could set up an experiment. It would seem that a person with your analytical skills could make the necessary measurements and work out the mathematical descriptions for what is really happening. I bet there would be no typos in your formulas too.
Andy

[Jeff Conrad]

Every analysis of this that I've seen, including my own, treats the lens as a point. The one exception is a diagram in Scheimpflug's original patent, as shown by Merklinger, which seems to suggest that the hinge line is in a plane through the tilt axis---see below---and parallel to the film plane. My intuition suggests that this should be right, at least if the pupil magnification is close to 1.
I assume that you're referring to Fig. 21 in Focusing the View Camera; I've always assumed that this figure is correct, but I've considered the internodal distance insignificant in the scheme of anything I've ever photographed, so, like Merklinger, I've ignored it in any analysis that I've done. I've never quite decided where I should use the nodal planes and where I should use the pupils.

I've never really worried about where the tilt axis should be (in most cases, there isn't a choice); when you tilt the lens, the image goes out of focus regardless of the location of the tilt axis, so refocusing always is needed anyway. It seems to me that the final geometry is the same regardless of the means used to achieve it. I set tilt visually using Bond's method, so a rough idea of the location of the PoF pivot axis is fine for me.


In the text, he suggests that one use the rear nodal point to tilt about, and in a diagram, he shows the wedge shaped DOF region as being centered on a line in the film plane. I am pretty sure his diagram is wrong, unless one tilts about an axis in the film plane,
To which text and diagram are you referring?

I've found a fair number of typos in this book (though I've not attempted to compile a list), in Chapter 58 and elsewhere; for example, I think Eq. 58.1 should be

theta = arcsin(f/u tan psi)

where psi is the angle between the image plane and the PoF.

[Emmanuel BIGLER]

Leonard. You should have a look at figures 6, 7 and 8 in this article.
http://www.galerie-photo.com/profondeur-de-champ-et-scheimpflug.html

No need to read French. the diagrams suffice (IMHO ;-)

Figure 7 is nothing but yours.
Figure 8 is the direct extension to a thick asymmetric compound lens.

This article was due to be translated into English... ahem last year. Still on my 'TODO' list.

Basically the derivation is as follows, see figure 8.
- planes defining the limits of acceptable sharpness in image space are parallel. Nothing changes w/respect to your derivation. This is actually the most difficult part of the derivation.
- yes, the position of the exit pupil defines the actual circle of confusion...
- ... however in object space the corresponding slanted planes are strictly conjugate w/respect to the three parallel planes near the image plane
- hence, to find them a simple symbolic ray-tracing yields the result on fig.8 as re-posted here. Parallel rays on output have to cross the hinge in the focal plane on input, whatever the thicknkess and asymmetry of the lens could be.

[Leonard Evens]

Jeff,

I agree that for most large format lenses, and for most practical situations, none of this makes any significant difference. But it is still possible that for pupil magnification different from 1 and/or for extreme close-up photography, there could be some observable effects. In any case, I would like to understand it all.

I've been thinking some more about the matter, and I've come to some tentative conclusions.

First, with respect to Scheimpflug's Rule, at least with pupil magnification 1, it has to be based on the front nodal point since that is the geometric center of perspective, and the principal planes are used to determine image positions. I still haven't figured out just what effect the distance between the principal planes does to the Rule, but it seems to be some sort of shift.

Second, I agree that tilting about some other axis, parallel to the film plane, should not yield a signficant difference in the important features. After all, many view cameras, including mine, use base tilts rather than axial tilts and we know from practical experience, this doesn't make any real difference in the final result. Some thought shows that what happens when you shift the tilt axis is that you shift the position of the front nodal point as well as tilting. However, you can compensate for that by movements horizontally and vertically, so you can bring the front nodal point back to where it was originally. The movements will be small in anything but extreme close-ups compared to subject distances, The horizontal movements will not make any significant difference, but the vertical movements will change what is included in the frame, just as normal rise and fall do. If you did not compensate for such a rise/fall, you would change the position of the exact subject plane, but not by enough to matter except in extreme close-ups.

Ray's statement and the diagram are in Section 22.6 and Figure 22.10. In view of the above remarks, I think his contention that you should tilt about an axis through the rear nodal point is innocuous as far as the main points are concerned. It might be justified if you are trying to maintain the position of the film plane, but since the front nodal point would move, I think you would have to change the position of the film plane anyway, to preserve focus. I hadn't looked yet at Chapter 58, and I see from what he says there why he is thinking of using the rear principal plane and nodal point. I have to think more about the effect of the separation of the principal planes. I think he is all wet about Equatiion 58.1. He might be referring to the fact that the sin of the tilt angle is the focal length divided by the distance to the hinge line in the principal plane.

[Emmanuel BIGLER]

Another diagram to try and explain why the limit planes of acceptable sharpness are parallel in image space.

One of the tricks is to check for image sharpness not on the actual slanted image plane but on a kind of venitian blind with the small wood plates parallel to the exit pupil. With this trick we avoid entering into difficult considerations about elliptical Cocs ;-) and we solve the problem like on Leonard's diagram without any equation !!

[Leonard Evens]

Some more thoughts:

I think one way to look at how the separation of the principal planes affects things is as follows. Suppose pupil magnification one to keep things simple. The object space and the image space should be considered two seprate spaces. You go from points in one space to points in the other space by adding (or subtracting) the vector connecting the centers of the two principal planes. The object side of the object space is real, but what is on the film side is virtual. Similarly, the iamge side of the image space is real, but what appears on the other side is virtual. There is a virtual film plane on the film side of the object space, and there is a virtual subject plane on the subject side of the image space. These are shifted versions of the real film plane and the real subject plane. Then the real subject plane, the front principal plane, and the virtual film plane all intersect in a line. As you move the virtual film plane, the subject plane rotates about the hinge line with is in the usual position with respect to the front nodal point. Similarly, the virtual subject plane, the rear principal plane, and real film plane intersect in a line. If you move the real film plane, the virtual subject plane rotates about a hinge line in the usual position relative to the rear nodal point.

[Leonard Evens]

Emmanuel.

You diagrams are helpful, but there is something I don't understand. Planes 0, 1, 2, and 3 are transformed into parallel planes 0', 1', 2', and 3' by taking image points. So doesn't it make sense to choose the film plane somewhere in the middle corresponding to the plane of exact focus in the DOF region and have the endpoints of the cones on the other planes corresponding to the boundaries of the DOF region? Then, the cones intersect the film plane in ellipses, and you are within the DOF region if the image ellipses are sufficiettly small.

[Jeff Conrad]

Ray's Figure 22.10 definitely looks wrong to me--the apex of the DoF wedge should be directly below the object nodal point.

I don't think it really matters where the lens is tilted, but it would seem to me that unless you tilt about the center of the entrance pupil, you get a slight vertical displacement on the object side, just as with a panorama. However, it also would seem that, unless you tilt about the center of the exit pupil, you get a slight vertical displacement on the image side.


The object space and the image space should be considered two seprate spaces.
This seems reasonable. When the lens is tilted, the spaces are displaced along the lens axis, so there is a slight vertical displacement (the product of the internodal distance and the sine of the tilt angle).

The immediately preceding text in Section 58.3.3 leads me to believe that Ray's Eq. 58.1 was intended to apply to the object distance; if that indeed was the case, the equation is correct as I stated it (the derivation is quite simple).

[Doug Dolde]

I'd like to see somebody doing all this calculating in the field when trying to get it all in focus. Just use your lupe and focus the dang thing !

[Jerry Fusselman]

Doug, you forgot the message in the first paragraph of the thread:


Warning. This is highly technical, and it won't be of any practical interest, except possibly for people doing tilts with lenses of highly asymetric design, such as telephoto lenses.

That's a perfectly clear. Personally, I am very interested in this topic.

[Jeff Conrad]

Doug,

I also don't think any of us implied that we would make such calculations in the field; I, for certain, could not cope with it. If you read Leonard Evens's paper (http://www.math.northwestern.edu/~len/photos/pages/dof_essay.pdf) (PDF), you'll see that his field procedure is simple and practical. The purpose of the math beforehand usually is to determine what things one must worry about and what things can safely be ignored. Fortunately, it appears that quite a bit can be ignored.

In an earlier post, I mentioned that I don't need to know the exact position of the plane-of-focus pivot axis (yet another name for it ...). Nonetheless, it's still helpful to have a rough idea of what's going on so that one does not go nuts fiddling with the tilt and focus. To me, it's important to know that the pivot axis is below the lens rather below the image plane. If nothing else, the DoF at the near part of the image is less, so that getting correct focus for that area is critical.

[Leonard Evens]

Doug,

You didn't read the disclaimer at the beginning.

In fact, in the field, I seldom even use a loupe. I have a pair of +5 diopter reading glasses I got my ophthamologist to prescribe and those usually allow get me to get close enough to the gg for all practical purposes. That is particularly true since I use the focus spread method described elsewhere in the LF Photography webpage. I'm surprised you are wasting your time fiddling with a loupe. :) Also, if you read the questions from beginners who can't seem to find the right tilt angle no matter what they do, "focus the dang thing" is not much help as advice.

Note that it is also possible that the theoretical disccussion here may have some consequences for special situations such as when using telephoto lenses with pupil magnification different from 1. Or, it may not. We won't know until we finish the analysis. But as I noted originally, that is not the main point of the discussion.

All large format photographers, even the most practical of us, use a combination of theory, intutition, and practical experience. There is no way to learn the theory and then get it right without some intutition and a lot of trial and error. On the other hand, some understanding of the theory is essential even to begin, and often more understanding can save some time doing it by trial and error. It is just not true that you just look at the gg and all becomes obvious. You have to know what to look for.

[Leonard Evens]

Getting back to the main issue: the location of the hinge line.

As Emmanuel's diagrams make clear, the hinge line in object space is just where we thought it should be, in a plane parallel to the film plane passing through the center of the front principal point, which in air is the front nodal point. Its distance from that point is f/sin(phi), where f is the focal length and phi is the tilt angle. It is parallel to the line of intersection of the subject plane, the front principal plane, and the (virtual) film plane, identified by Scheimfplug. But it is not identical with it as Ray's diagram seems to suggest.

Also, as Emmanuel's diagrams make clear, each plane parallel to the film plane is the optical image, using the principal planes, of a plane through the hinge line. Choosing such a plane parallel to the film plane, but some distance from it, an image point in it is the apex of a cone with its base the exit pupil, which need not be in the rear principal plane. One then does the analysis of its elliptical trace on the film plane---call it the defocus ellipse---in the usual way, except the height of the cone will not be the same as the distance of the image point to the rear principal plane. This could change the maximal distance you can be from the film plane and still have an acceptable defocus ellipse, but otherwise the analysis should proceed in the same manner. To see exactly how this will work quantitatively will require going back and looking at that analysis again, which in time I will do. The problem is that, unlike the untitlted case, the sizes of the defocus ellipses depend on their location. So if you set an upper bound on the size of the defocus ellipse, the corresponding image points (the apexes of the cones) don't lie in a plane. When all this is translated back to object space, the surfaces bounding the region of adequate focus won't be planes. I found that except in some very special circumstances, the departure of these surfaces from planes was very small and could be ignored. Displacement of the exit pupil from the rear principal plane may change that somewhat, but I doubt if the conclusion would be very different. Instead of asking for the largest possible DOF region, one could choose to be satisfied with a slightly smaller acceptable DOF region bounded by planes. I suspect that there will also have to be some modification of estimates which will involve the pupil magnification as in the untilted case.

If pupil magnification is not 1, we've seen in other threads that perspective relations of what appears to be in line with what will be affected. Points in line with the front nodal point won't generally produce concentric defocus discs in the film plane, and their centers could in fact be separated by signficant amounts. The same thing should be true in the tilted case except that we have defocus ellipses rather than circles.

I don't believe any of my lenses have a pupil magnfication not close to one, so I can't do any experiments, and that is frustrating. I wonder if anyone who has been reading this thread has any observational evidence which might bear on the matter.

Also, I wonder if anyone has ever tried to stitch together panoramics made with a view camera with a tilted lens. Just how the wedge shaped DOF regions fit together, particularly close to the lens might produce some interesting effects, particularly if the pupil magnification differs signgicantly from one, and you are rotating the camera about the entrance pupil which won't be centered over the hinge line in that case.

[Emmanuel BIGLER]

Jeff Conrad has pointed to me that my diagram refers to the Depth of Focus problem (Dofocus) and not the Depth of Field (Dofield)

So I have redrawn the diagrams for both cases in the image space. The locus of acceptable sharpness for Dofield and Dofocus are actually planes, and they are parallel to the ideal plane. The formal difference for Dofield is that the mid-plane is located at the harmonic mean between both limit planes. For Dofocus the mid-plane is .. in the middle.

This does not change in its principle the ray tracing that solves the position of the Dofield hinge in object space.

The difference is small in practice between both drawings, it has been exaggerated for clarity, and can be neglected. Assume that your Dofield or Dofocus limits are 148 and 152 mm from the exit nodal point, this corresponds to the case of a far distant object with a 150mm lens, with a CoC of 90 microns and a f-number = 22. The best mid-plane for the Dofield problem is located at 149.97 mm instead of 150. 30 microns, sligtly more than one mil, whereas the ANSI tolerance for film holder is 7 mils, 180 microns.

So if we want to summarise : we try and find the minimum sensible hypothesis to solve the problem without algebra, only simple ray tracing.

1/ we neglect diffraction and we assume that symbolic ray tracing of gaussian optics are valid even at high angles,

2/ we neglect elliptical defocusing spots i.e. within the validity of hypothesis #1, we judge sharpness not on film but on a small platelet centered on film but lifted/rotated to be parallel to the exit pupil ; this allows us to keep the same circle of blur in the whole film plane. Yes this is strange but if we accept this, all the rest is simplified.

With those two hypothesis, the 3-ray diagram as in Leonard's original article can be easily extended to the general case of a thick coupound asymmetrical lens.

Now is there something useful in practice to extract from this theory ? Certainly yes, in the sense 'everybody knew it but we explain why' ;-);-)

Imagine that for some reason you need to tilt and you need to focus on a subject in low light level. Assume that the subject is close to a plane. For example a painting in a church for which you definitely need to tilt ;-) strange, but aren't we here strange guys using strange cameras ?

This is a typical "slanted" Dofocus problem.

Since it is difficult to see where the the best focusing position is, we assume that we can guess a reasonable value for the tilt angle. The diagram shows that if the tilt angle is correct, looking for the proper plane does not need any correction of the tilt angle, a simple translation applied to the rear standard suffices since those planes are parallel. So you defocus on both sides until it is equally blurred and you take the mid-point.

I do not have however a good procedure to find the proper tilt angle, but if the blur is uniform, we know that the tilt angle is good, and that we only need an adjustement of the longitudinal translation, taking mid-point should deliver the best sharpness even if it is very difficult to see.

It is well-known that you should focus as quickly as possible. If you spend too much time focusing, your eyes get strained and it becomes worse. So the mid-point in between equal blur is probably an efficient method to properly focus in low ligh levels.