Center of perspective and nodal point

[Leonard Evens]

Before going on, let me say that this is of little importance from a practical point of view, so if it offends you, don't proceed further.

It seems well known to those who engage in panoramic photographic that, in order to avoid parallax errors where adjacent shots are joined, you should roate about the center of the entrance pupil. For most lenses, with both sides of the lens in air and unit pupil magnification (or close to that), the center of the entrance pupil coincides with the front nodal point (or is close to it). But for some lenses, e.g., those of telephoto design, the pupil magnification may not be close to one, and so the entrance pupil will be some distance from the front nodal point.

This seems to be a contradiction. The nodal points are defined by the condition that for every off axis point in the subject, the ray from that point to the front nodal point makes the same angle with the lens axis as the corresponding exit ray from the rear nodal point to the image point. That means that subject points along the entrance ray will yield image points along the exit ray. That seems by definition to say the nodal point is the center of perspective and the proper point to rotate about.

I've looked for an explanation of this seeming paradox without much success. The only explanation I found which began to make sense suggested that the entrance pupil can in effect be treated as a peephole for a pinhole camera, but I didn't find the rest of the explanation convincing. After some thought, I've come up with the following explanation, but I'm not sure it is right. Anyway, here it is.

Consider the pencil of rays starting at the subject point and passing through the entrance pupil. They form a solid cone. with base the entrance pupil. The key point is that the front nodal ray may not be the central ray of this cone. In that case, the rear nodal ray will not be the central ray of the corresponding solid cone from the exit pupil to the image point. Now take into account that the film plane is never going to be set precisely at the image point. It certainly won't be if the subject point is not in the plane of exact focus, and if we are interested in parallax errors, at least one of two points which should line up will be at some distance from the plane of exact focus, albeit in the DOF region. That means we have to consider not points in the negative plane but image discs. One can make the argument that for two image points to appear to be lined up, the central rays of the corresponding cones to the exit pupil should be lined up, which means that the two image discs in the film plane will be concentric. If you rotate about the nodal point, you will mess up that arrangement.

So the conclusion would seem to be that it makes more sense to rotate about the center of the entrance pupil, although in point of fact it is not the actual center of perspective in the strict geometric sense usually studied in art theory. I haven't worked it out quantitatively, but it may be that you mess up the concentric relation of the image discs less by so doing.

I'm not sure I find my own explanation convincing. Perhaps it is obvious why the nodal point despite the equal angle property is not the center of perspective and I'm just missing something. If so, I would like an explanation. Of course, in most large format photography, the nodal point, the principal point (intersection of the principal plane with the lens axis) and the center of the entrance pupil are the same or at least extremely close to one another, so it doesn't make any difference.

A somewhat different question is the following. It is often asserted that the entrance and exit pupils can depend on the distance to the plane of exact focus (and hence to the corresponding image plane). This is equivalent to saying that the pupil magnification can vary when focusing. The entrance pupil is defined to be the image of the physical aperture as seen through the front elements of the lens and the exit pupil similarly with respect to the rear elements. Unless the distance between lens elements or distances to the physical aperture change while focusing, I don't see how the positions of the entrance and exit pupil can change. That may happen with zoom lenses or possibly some other small format lenses, but it doesn't happen with large format lenses. Am I missing something?

Any enlightening comments?

[Michael S. Briggs]

There are at several cases of panaromaric cameras. I think Leonard is addressing the case where a conventional camera is rotated to take a series of photos. It seems that the answer for this case is the entrance pupil. For a swing-lens cameras with stationary film, the lens is swung about its rear nodal point. This case is discussed in Applied Photographic Optics by Sidney Ray. He also mentions rotary cameras in which the camera rotates and the film is moved past a slit -- in this case he asserts that the correct rotation point is the front nodal point. Most of this is a diversion -- just we should be clear about which case is being discussed -- no doubt the case of swinging an otherwise conventional camera.

[Leonard Evens]

Michale Briggs said:
There are at several cases of panaromaric cameras. I think Leonard is addressing the case where a conventional camera is rotated to take a series of photos.

Yes that is what I am talking about.

It seems that the answer for this case is the entrance pupil.

Do you have a reference for that which explains why it should be the case?

For a swing-lens cameras with stationary film, the lens is swung about its rear nodal point. This case is discussed in Applied Photographic Optics by Sidney Ray. He also mentions rotary cameras in which the camera rotates and the film is moved past a slit -- in this case he asserts that the correct rotation point is the front nodal point.

I don't see how the latter case, a rotary camera, is different from rotating a fixed camera. In that case, he seems to be agreeing with me that the center of perspective is the front nodal point. That is just the paradox that is concerning me. The front nodal point is not the center of the entrance pupil if the pupil magnification is not one.

Ray's book seems to be the authoritative source, but I don't know where to get a copy of it, short of buying it---at least $149 used. Northwestern doesn't have a copy.

[Helen Bach]

Rotary cameras are different from rotating a normal camera and taking successive frames because the displacement of the image on the film isn't an issue in the latter case.

If the nodal points and entrance pupil are displaced, rotating the lens about the rear nodal point (ie for rotary cameras) only works perfectly for distant objects.
From the second edition of Ray:

[Struan Gray]

Leonard, I went though a similar epiphany a few years ago, after prompting by Yves Coulombe, who sadly doesn't seem to be posting these days. The dawning of the light has been preserved for posterity here:

http://www.photo.net/bboard/q-and-a-...?msg_id=003bju

[Sadly, I think the discussion in French that Yves refers to did not survive the change of forum at galerie-photo. If it did, I failed to find it.]

The point - ahem - is that although the nodal points are essential in locating the in-Gaussian-focus image of any point in subject space, the selection of which rays from the subject actually combine to form the image is done by the aperture. The privileged ray passing through the nodal points need not contribute at all.

If the front nodal point were the center of perspective you would expect that bodies in subject space lying on lines radiating out from the nodal point would be superimposed on the negative. However, for small stops a napkin sketch will quickly convince you that if the center of the entrance pupil is displaced from the nodal point, those bodies will put light onto the negative in different positions. Their perfect, in-focus representations are collinear with the exit pupil, but that doesn't mean that the ray bundles permitted to pass through the lens will necessarily pass through the film plane at the same place.

This is easy to see with the simplest possible asymmetric lens: a small hole in front of a singlet. The difficulty comes in accepting the results as general for complex, thick lenses, for wide apertures, or for LF-irrelevant exotics such as underwater objectives or oil-immersion microscope lenses.

"The flow of light" does also affect DOF and exposure compensation. Searching here and photo.net using "pupillary magnification" will turn up a host of discussions of the topic. Jeff Conrad's detailed DOF notes here at LF.info are clearer than anything I have read on or offline on that topic, and Emmanuel Bigler has posted some detailed descriptions of exposure compensation with asymmetric lenses at photo.net.

This is not just a theoretical exercise (although mostly so :-). Some of the LF aerial telephotos can have significant pupillary magnifications, as well as the early 200-400 mm telephotos from Meyer and - I think - Dallmeyer. Tessars are the classic example of a 'normal' lens that is asymmetric, but I doubt that they are asymmetric enough to cause problems in anything but oddball use. How many photographers make stitched macro panoramics on sheet film?

[Leonard Evens]

Struan,

Thanks for the explanation. I think I was trying to say something along the same lines.

The point is that any object points which are collinear with the center of the entrance pupil will produce image points which are collinear with the center of the exit pupil. So rotating about the center of the entrance pupil will not disturb parallax relations if we look at the cone with apex the object (image) point and base the entrance (exit pupil). Rotating about the front nodal point instead will disturb the collinearity of the rotated points with the center of the entrance pupil (and hence of the image points with the center of the exit pupil). What wasn't clear to me was that this actually made any difference for real lenses. As long as the blur circles of the image points in the film plane are within the coc, we shouldn't be able to tell the difference.

I've done some calculations, which I don't really trust, but they suggest that with a 500 mm lens and two points at roughly 10 and 20 meters from the lens, the parallax error from a 45 degree rotation might be of order of magnitude 1 degree. At a distance of 15 meters that would suggest a spread of about 260 mm. If the camera was focused at that distance, the magnification would be 29, so the two exit pupil rays would be off by about 260/29 ~ 9 mm where they intersect the film plane. That is a signficant shift. But, as I said, I don't really trust the calculation.

[C. D. Keth]

What sort of panoramas are you making that demand this sort of accuracy? I know from firsthand experience that for motion control in motion picture effects work (where the perspective must match to a T) it is accurate enough 99% of the time to use the center of the lens barrel at the front of the iris ring as the point of rotation. It's not millimeter accurate but it's plenty accurate to fool audiences and satisfy multi-million-dollar investors. It could also be (this is likely, actually) that I'm just not familiar enough optically with telephoto design lenses for LF work.

[Helen Bach]

What sort of panoramas are you making that demand this sort of accuracy?...

Well, Leonard's first sentence reads; "Before going on, let me say that this is of little importance from a practical point of view, so if it offends you, don't proceed further."

For me, this discussion is about understanding the principles behind what we do. This may or may not have practical benefit, but it certainly doesn't detract from anything. However, knowing that the entrance pupil is the centre of perspective of a lens, and given that the entrance pupil is extremely easy to locate in practice* makes me wonder why you don't do motion control using the entrance pupil instead of the front of the iris ring. I mean, it's just such a trivial matter in comparison to everything else to do with motion control, or to almost any aspect of cinematography...

*You look into the front of the lens and put your finger on the barrel where the aperture appears to be. How hard is that?

*******************

I find it a little bit puzzling that Leonard has re-written what I wrote in another recent discussion without mentioning the discussion, or my explanation of why it is the entrance pupil and not the front nodal point. No matter.

Though this is a re-write of that which has already been written, here is my version with less DIY required than my previous version. No disrespect to Struan for his excellent explanation. It's good to get things from two or more independent sources.

Perspective is a three-dimensional issue: it is not bound by the rules of in-focus images.

A ray passing through the front nodal point is sufficient to define the location of an object point in the plane of focus, but not of an object point that is not in focus, ie that forms a disc in the film plane.

A ray passing through the centre of the aperture defines the apparent centre of the disc, on the film, of an object point that is not in perfect focus. A ray that passes through the centre of the aperture is a ray that, before entering the lens, is directed towards the centre of the entrance pupil.

Therefore the rays that are directed towards the centre of the entrance pupil define the way in which the three-dimensional object space is translated into a two-dimensional image.

*******************

Swing lens cameras: The two types of swing-lens camera (lens swings, film stays still; lens and camera body swing, film moves in the camera) are analogous. In both cases the rotation should be about the rear nodal point, because swinging about the rear nodal point is the only way in which a distant in-focus point will stay in the same place on the film as the rotation occurs. The error introduced for near points, and for the change in the location of the entrance pupil (assuming that the entrance pupil is not coincident with the rear nodal plane), is limited by the width of the slit. The slit minimises the effective amount of rotation for any point on the film.

Best,
Helen

[Struan Gray]

...Their perfect, in-focus representations are collinear with the exit pupil, ....

I meant, of course, collinear with the rear nodal point.

Helen, Leonard, please restate at will. I am a great fan of learning by discussion, and wish there was more time for symposium-style dialogue in today's formal education systems. It's one reason I like online forums so much, and I wish there had been discussions like this in the archives when I first started to confuse myself with this stuff. It is easy to find statements of fact like those in Ray or the Lens Tutorial; much harder to find someone who can put things into context or explain with words and concepts instead of squiggles and equations.

The classic application for this degree of nitpicking is photogrammetry of, architectural interiors or historical artefacts. If you wish to have an accurate photographic record of your collection of Etruscan terracottas or, say, the House of the Satyr at Pompeii, you need to care about where your camera is seeing from.

[Struan Gray]

...Their perfect, in-focus representations are collinear with the exit pupil, ....

I meant, of course, collinear with the rear nodal point.

Helen, Leonard, please restate at will. I am a great fan of learning by discussion, and wish there was more time for symposium-style dialogue in today's formal education systems. It's one reason I like online forums so much, and I wish there had been discussions like this in the archives when I first started to confuse myself with this stuff. It is easy to find statements of fact like those in Ray or the Lens Tutorial; much harder to find someone who can put things into context or explain with words and concepts instead of squiggles and equations.

The classic application for this degree of nitpicking is photogrammetry of, architectural interiors or historical artefacts. If you wish to have an accurate photographic record of your collection of Etruscan terracottas or, say, the House of the Satyr at Pompeii, you need to care about where your camera is seeing from.