In a previous report about distortion and globe effect in binoculars I had sketched the interesting history of pincushion distortion in binoculars. Since distortion is regarded an aberration, it would appear obvious that a successful optical design should lead to its complete elimination . The reality turned out to be more complex: A binocular is a visual instrument, and the visual impression of a "good" design is not solely the result of an absence of distortions within the instrument. Quite to the contrary, since our vision seems to induce a slight amount of barrel distortion, a fully corrected binocular should contribute a similarly strong amount of pincushion distortion to yield a perfect correction of the combined system, observer+binocular.
The need for a deliberate implementation of pincushion distortion had been realized at Carl Zeiss Jena by the mid 1940s, when the theoretical studies of Hermann Slevogt  and field studies of August Sonnefeld  eventually led to a paradigmatic shift in the construction of binoculars, away from distortion-free optics. The supplementary distortion facilitated a smooth panning of the binocular and avoided the emergence of the globe effect that infested earlier distortion-free designs. Most binocular manufacturers followed Zeiss' example, so that by the Mid 1950s a certain amount of distortion, according to the "angle-condition", had become the new industry standard.
The pincushion distortion that came along with the angle-condition did not remain without side effects: Toward the edges of field, straight lines are bending inwards, hampering the esthetic impression of the optic. About 2005, a few manufacturers began to re-introduce binoculars with very low distortion, the Japanese Kowa and Nikon among them. The globe effect, virtually forgotten over the decades, was thus raising its head once again, yielding an unpleasant panning behavior of these binoculars and promoting motion sickness with some of the rather sensitive observers. Swarovski began to experiment with low-distortion binoculars along with their introduction of the Swarovision line in 2008. User reports about the emergence of the globe effect in these binoculars were contradictory. In short, it was getting time to understand these phenomena better, and I intensified systematic studies and computer simulations of these effects, which I had begun in 2005. The results of these studies have subsequently been published in a scientific paper  and are also discussed in detail in my recent book .
These studies suggested the existence of an optimum amount of distortion, roughly half as strong as the angle condition previously chosen by Zeiss and many other manufacturers. The amount of distortion can be expressed through the distortion parameter k, and a value about k = 0.7 appears to represent the best compromise, allowing the elimination of the globe effect for at least 90% of the observers, and, at the same time, greatly reducing the side-effects of the residual pincushion distortion on straight lines.
The graphic displays the (relative-)distortion chosen by Swarovski and Zeiss for their recent models. The red curve toward the upper left corner of the figure corresponds to the rather strong pincushion distortion of the angle-condition. In green, the k = 0.7 curve indicates the optimum amount of distortion, being about half-way in between the distortion-free case (which would be the horizontal null-line) and the angle-condition. Zeiss' newly introduced Victory SF models are located very close to that optimum line. It therefore appears that Zeiss has left its old path of strong distortion, as defined with the angle condition, and has moved onto the "golden path of the middle", i.e. k = 0.7. In a recent interview on birdforum.net, product manager of Zeiss Sports Optics, Gerold Dobler, explains that the choice of distortion implemented to the SF is the result of an optimization procedure, based on field tests in which the perceived intensity of the globe effect has been investigated. This would imply that earlier predictions made in  and  are accurate and finally verified through independent field tests with volunteers.
Now that Nikon, Swarovski and apparently also Zeiss have dramatically deviated from the old distortion paradigm, with only Leica being left over as the last high-end manufacturer that still implements high amounts of pincushion distortion, it appears that we are currently experiencing another paradigmatic shift, away from the angle-condition of the 1940s, toward a reduced but nonzero residual pincushion distortion. The future will tell whether or not this shift will turn out to be as radical and complete as its predecessor, so that the era of the angle-condition can be declared over.
It is important to note that all distortion numbers presented here are evaluated from the data-sheets of the manufacturers and hence prone to errors. For example, the exact magnification of a binocular may deviate from the rounded number that is officially specified, and so does the apparent angle of view. In certain instances, the resulting errors may be substantial. I have tried to minimize the errors by using the specified field of view (in m/1000m) instead of the specified true angle of view, since the latter is usually less accurate. Another problem arises from the fact that the perception of the globe effect is not solely related to the k-factor, but also depending on the detailed shape of the distortion curve (from the center toward the edge), and these curves are not published by the manufacturers. In any case, all numbers presented here should be taken with a grain of salt, because the available data are simply not sufficiently accurate to allow for a precise determination of the k-values. Manufacturers are therefore encouraged to add their exact distortion values to their data sheets, just as it is common practice within the camera industry. Additionally, the specification sheets should contain sufficiently accurate numbers. If the magnification of a binocular amounts to 8.2, then that should be specified, instead of a rounded "8".
I shall repeat that the distortion parameter about k ~ 0.6 - 0.8 is a necessary, but not sufficient condition for a good panning behavior. For example, the Zeiss SF seems to do well with its 10x42 version, while the 8x42 is showing a rather pronounced globe effect. Even though its integrated distortion value is fine, its distortion curve is unfortunately bending down toward the edge of field and thus generating the impression of a 'rolling' image.
Last modified: January 2023