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/profond...heimpflug.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 !