Does the nodal point of the lens really have to be on the axis of rotation?

Anyone taking an interest in panorama photography will soon read that “The nodal point of the lens must be directly on the axis of rotation”. But just how important is this? Intuitively it must be less important if the scene contains no objects close to the camera, but at what distance does it become significant? This short note answers the question. The calculation is first applied to a 35mm film camera, then extended to cover digital formats.

35mm Cameras

Consider the following scenario: a 35mm camera on a tripod with a panning head, the axis of rotation exactly vertical but with the nodal point of the lens offset from the axis of rotation. The scene comprises a single pointed fence post, at a distance of a few metres, viewed against a background of distant hills. If we view the scene with the camera rotated so that the post is at the left hand edge of the frame then rotate so that the post is at the right hand edge of the frame it will appear to move relative to the background. By calculating the magnitude of the effect we can assess just how important it is to position the nodal point on the axis of rotation.

Just one well-known formula from optics is required. When a simple convex lens forms an image of an object the magnification is:

magnification = (size of image) ÷ (size of object) = (image distance) ÷ (object distance)

The significant thing that happens when the camera is rotated is that the nodal point of the lens moves laterally (i.e. at right angles to the line of sight). Suppose this movement is 10mm – by calculating the effect for such a large movement we can scale it down for smaller values later. Ignore the fact that the camera has rotated – suppose that instead it had just moved laterally by 10mm. The effect on the image in the camera would be just the same as if we had kept the camera fixed and instead moved the post by 10mm. This is the case because the distant hills are effectively “at infinity” whether measured from the post or from the camera! We now apply our formula to calculate the movement of the image of the fence post on the film.

(distance moved by image) ÷ (distance moved by object) = (size of image) ÷ (size of object) = (image distance) ÷ (object distance)

Suppose that the fence post is 5m from the camera and that we are using a 35mm focal length lens:

magnification = 35mm/5m = 0.007 so the image moves 10mm x 0.007 = 0.07mm

To understand what this means we will use as our criterion the “circle of confusion”

The recommended “circle of confusion” for a 36mm x 24mm negative is usually quoted as 0.025mm or 0.033mm. This is the maximum amount of “unsharpness” considered acceptable at the film plane and in turn dictates the resolving power of the lens. It is the size of a spot of light which is, to the naked eye viewing a 10” x 8” print (i.e. a x8 enlargement), indistinguishable from a point.

The movement of 10mm with the nearest object at 5m will probably therefore produce a visible effect, being over twice the diameter of the circle of confusion. To err on the side of caution we could accept a value half of this, i.e. a movement of 5mm in an object at 5m. This will probably produce a “just noticeable difference” since its image will move by an amount about equal to the diameter of the circle of confusion. Notice that the ratio of acceptable movement (5mm) to distance of nearest object (5m) is one to one thousand – an easy to remember rule of thumb. But remember that this applies to a 35mm lens on a 35mm camera, longer lenses will magnify positioning errors proportional to their focal length.

Digital Cameras

A typical digital camera might have a sensor which is just one fifth the size of the 35mm film camera just considered. But in making a print of the same size we are, in effect, using a five times greater enlargement. When we use a 7mm focal length it is usually referred to as “equivalent to a 35mm lens on a 35mm camera” because it gives about the same field of view. We find that the factor of five cancels out and we arrive at the same conclusion: that the ratio of acceptable movement (5mm) to distance of nearest object (5m) is one to one thousand.

Conclusion

The importance of having the nodal point of the lens exactly on the axis of rotation depends entirely on the subject. If there are no objects nearer than, say, 10m then a 10mm movement of the nodal point between shots can probably be tolerated. This is reassuring since my early panoramas were made using an old pan and tilt head with significant offset of the nodal point – fortunately they contained nothing nearer than about 100m (see illustration at top). On the other hand, my more recent work on interiors has used a special bracket which allows adjustment of the camera position to put the nodal point exactly on the axis. The method of making this adjustment is the subject of another technical note.

Technical Notes

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