Listing’s law: eye movement and near vision

In the first article we looked at the underlying basis of Listing’s and related laws and the relevance of eye position. In this article we will differentiate the eye movement from eye position and look at the extension of Listing’s Law to near vision.

Eye movements

If an eye moves from tertiary position A to another tertiary position B, then altogether three rotation vectors are required to define this process: rotation vector rA, which is necessary to describe the initial position in position A, rB to describe the final position in position B, as well as a third rotation vector rA?B to describe the movement that takes place in between.

Therefore, we differentiate between eye positions described by the static rotation vectors rA and rB and eye movements described by the dynamic rotation vector rA?B.

There is often a misunderstanding that, according to Listing’s law, it is not only the static rotation vectors rA and rB of the eye positions but also all dynamic rotation vectors rA?B that generally must lie in Listing’s plane for eye movements.

This, however, is not the case. Listing’s plane only qualifies those movements rA and rB that describe the eye position (with the primary gaze direction as a reference). The dynamic rotations, which superimpose two tertiary (or secondary) eye positions, cannot in fact all lie in Listing’s plane.

In order to understand this seemingly paradoxical fact, it is necessary to understand an essential difference between spatial translations and rotations. Translations, corresponding, for example, to the vector ?t = (?x, ?y, ?z) = (2,1,4) can uniquely be broken down into components that, if they are executed one after the other, lead to the same result as the direct displacement.

In addition, the information ?x = 2, ?y = 1, ?z = 4 is unique for any single definition of the final state without requiring information about the sequential order:

An object always comes to the same final state independently of whether the displacement ?x = 2 along the x-axis is executed first, or whether the displacement ?y = 1 along the y-axis is carried out first.

It is this fact which is quite different for rotations. Contrary to intuition, the following statements apply to rotations:

1. If two rotations are executed around different axes, then the result depends on the order (this non-interchangeability was explained in the first article).

2. A rotation about a given axis can indeed be made up of individual rotations about the x, y and z axes, but the values for the three angles are not unique, but rather they depend on the order of sequence.

3. If two rotations about two different axes (which both lie in Listing’s plane) are executed one after the other, the rotation axis of the combined rotation lies outside Listing’s plane.

4. If one wants to move the eye directly from one tertiary eye position to another tertiary eye position with a single rotation, you need an axis that lies outside Listing’s plane.

Points 1 and 2 were discussed in the last article with reference to Figure 1. Point 3 can also be explained using the example in Figure 1. Both individual rotation axes that are used, ie both the horizontal as well as the vertical axis, when taken individually, lie in Listing’s plane. However, the result of executing them one after the other is a rotation that is described by an axis that lies outside Listing’s plane:

The mathematical rules for the composition of rotation vectors¹ show that sequence A corresponds to rotation vector rA = (1,1,1) and sequence B corresponds to rB = (1,1,-1).

This means that vector rA contains +1 as the z-component in accordance with torsion in a clockwise direction. Analogously, vector rB contains -1 as the z-component corresponding to torsion in a counter-clockwise direction. Hence, neither of them lies in Listing’s plane (in particular, rB does not equal rA as further confirmation of point 1).

Point 4 is the inverse of point 3. To get from one tertiary position to another, one can, for example, rotate first about a Listing’s axis back to the primary position, and then about a second Listing’s axis to the new tertiary position. Hence, two Listing’s rotations are executed one after the other about different axes, and according to point 3, the resulting rotation axis lies outside of Listing’s plane.

Listing’s law expressible through eye movements

Some authors1,2 describe Listing’s law with the help of a Helmholtz sequence or a Fick sequence of rotations that, together with an additional torsional movement, lead to the same eye position as a result (see Figure 2). This is an alternative to descriptions relating to rotation vectors in Listing’s plane.

It is true that a Helmholtz sequence without additional torsion cannot lead to an eye position defined in accordance with Listing, but it can represent the gaze direction itself. Therefore, an entire rotation about the Listing rotation vector r0 = (0.166, 0.276, 0.0) can be, at least with regard to the representation of the gaze direction, be broken down into an initial horizontal eye movement to a secondary position and a subsequent (hypothetically torsion-free) vertical eye movement to the final tertiary position. The rotation vector of the horizontal eye movement that lies in Listing’s plane is given by rl = (0.0, 0.268, 0.0), corresponding to a rotation of ?y = 30.0° about the y-axis.

To reach the desired gaze direction from this secondary position, one needs a rotation about the x-axis by an angle of ?x = 20.27°. One can then calculate that the eye would arrive rotated by -?z = +5.48° compared to the actual position that is required according to Listing.

The entire sequence of individual rotations that are necessary to get the eye from the primary position to the correct tertiary position can be given by three angles (?z = -5.48°, ?y = 30.0°, ?x = 20.27°), whose corresponding rotations must be executed in the given order. However, there is no direct connection between these angles and the eye movements that are actually executed.

The torsional movement is not executed jerkily at the beginning or end of the eye movements, but rather is undertaken uniformly during the second eye movement (see Figure 3). The first eye movement is indeed torsion-free, because it starts in the primary position and takes place about an axis in Listing’s plane (ie the y-axis). If the eye is then to move via a vertical eye movement from this secondary position to the final tertiary position and reach the position defined according to Listing there, then the related rotation axis cannot lie in Listing’s plane (see point 4 above).

In fact, one can calculate that the required second rotation axis is given by r2 = (0.154, 0.007, -0.041),  and the last component, -0.041, says that r2 in Figure 3 protrudes out of the front of Listing’s plane. Hence, the eye actually executes a torsional movement that, from the perspective of an observer, is counter-clockwise (blue in Figure 3), and hence compensates for the rotation in the clockwise direction, which would take place without torsional movement according to the Helmholtz sequence.

The torsion angle ?z = -5.48° is necessary for the specifically selected sequence of eye movements to reach the eye position required according to Listing. Hence, the torsion angle is not an absolute characteristic of the eye position alone, but instead depends on the selected description. We must remind ourselves that, for the description of the identical position (including the torsional position) through a direct eye movement about a Listing’s axis, the involved torsion angle is equal to zero.

It is interesting to compare a Fick sequence of eye movements, ie starting with the vertical eye movement. Completely analogous observations (as with the Helmholtz sequence) result in the three angles (?z = +4.98°, ?x = 17.46°, ?y = 31.61°) in this order also leading to the correct position in the tertiary position. The required torsional angle is therefore not just different than for the Helmholtz sequence, it even has a different sign (another effect of the non-interchangeability of rotations – even the two other angles ?x and ?y are not exactly the same as for the Helmholtz sequence). The torsion angle differs depending upon the eye movement used to reach the eye position as defined according to Listing.

Extension of Listing’s law for near vision

While eye positions after version movements for far vision have been known for a long time and are described by Listing’s law, the eye positions after vergence movements for near vision have only been studied systematically in the past twenty years or so.2,3,4,5 It has been found that deviations from Listing’s law occur, and these become greater as the convergence angle increases. One common explanation for this is that fusion stimuli could be the cause for the corrected eye position for convergence.

The gaze directions (and hence the lines of sight) are no longer parallel. It can be shown mathematically that a horizontal line in space can no longer be reproduced on corresponding retinal positions if each of the eyes were to take its position individually according to Listing’s law L1 in accordance with its own gaze direction.

Experiments have shown that, instead, the eye positions can be described through rotations about axes filling out a plane tilted to the outside compared to Listing’s plane (see Figure 4b). The eye positions achieved in this way actually make a single binocular image of the spatial horizontals possible.

This law is often referred to as ‘Listing’s law for near vision’ or ‘L2’ and comprises Listing’s law L1 as a special case for vanishing vergence. Besides tilting the Listing’s planes, one can illustrate the effect of rule L2 directly based on the rotation of axes crosses. The differences between the cylinder axis positions for L2 (Figure 5b) and the conventional Listing’s law L1 (Figure 5a) become greater as the gaze direction is raised or lowered and as the convergence increases. For a vanishing lowering or raising of the gaze direction or for simple horizontal eye movements, there is no difference between the position according to L2 and according to L1. As the convergence disappears, the L2 law changes over to the limiting case L1 again for all gaze directions.

The differences between the cylinder axis positions for L2 and L1 are therefore dependent on gaze angle and convergence and may be up to 4.5°. Therefore, when L2 is taken into consideration, considerable improvements to the lens can be achieved in connection with astigmatic correction for near vision, compared with L1. This concerns both the visual acuity in the near reference point as well as the fields of vision in near vision.6,7

Summary

The position of the eyeball may be accurately defined by the extended Listing’s law, both for far vision and near vision. However, the task of Rodenstock as a lens manufacturer has been to apply this in the actual design of lenses. The effective cylinder axis in the lens may be specifically adapted in this way.

Rodenstock has implemented this procedure for a long time in manufacturing high-quality lenses, taking Listing’s law for far vision into account.

Listing’s law for near vision has been used since 2011 using the EyeLT concept. If Listing’s law is not applied, then a systematic error occurs that increases as the gaze and convergence angle increase.7

Read more

Optimisation of progressive lenses: Listing’s law

Dr Wolfgang Becken is senior principal research physicist  (optics), Andrea Welk is research engineer (optics), Johannes Schubart is development engineer (optics applications), llka Schwarz is senior manager development (optics applications) and Dr Gregor Esser is director research and development (optics) for Rodenstock.

References

1 Haslwanter T, Mathematics of Three-dimensional Eye Rotations, Vis. Res. 35, 12, 1727-1739 (1995)

2 Hooge I T,  Van den Berg A V, Visually evoked cyclovergence and extended Listing’s law, J Neurophysiol, Vol. 83, 2757–2775 (2000)

3 Mok D, Ro A, Cadera W,  Crawford J D, Vilis T, Rotation of Listing’s Plane During Vergence, Vis. Res. 32, 11, 2055–2064 (1992)

4 van Rijn L J, Van den Berg A V, Binocular Eye Orientation During Fixations: Listing’s Law Extended to Include Eye Vergence, Vis. Res. 33, 5/6, 691–708 (1993)

5 Tweed D, Cadera W, Vilis T, Computing three-dimensional eye position quaternions and eye velocity from search coil signals, Vis. Res. 30, 1, 97–110 (1990)

6 Uttenweiler D, Butz C, Near refraction and aberometry, Optician, 8 2013, 20 – 23 (2013)

7 Nicke K, Welk A, Schwarz I und Esser G,Brillengläser der Zukunft - Schritt 1, Der Augenoptiker, 6 2011, 53-56 (2011)