Leonard Evens

30-May-2006, 09:25

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?

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?