Location of hinge line for tilted lens plane

[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 (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.