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{{short description|Geometric description of eye positioning and movement}}
'''Listing's law''', named after German [[mathematician]] [[Johann Benedict Listing]] (1808–1882), describes the [[Three-dimensional space|three-dimensional]] orientation of the [[human eye|eye]] and its [[rotation|axes]] of rotation. Listing's law has been shown to hold when the head is stationary and upright and gaze is directed toward far targets, i.e., when the eyes are either fixating, making [[saccades]], or pursuing moving visual targets.▼
▲'''Listing's law''', named after German
Listing's law (often abbreviated L1) has been generalized to yield the ''binocular extension of Listing's law'' (often abbreviated L2) which also covers [[vergence]].▼
▲Listing's law (often abbreviated L1) has been generalized to yield the ''[[Binocular vision|binocular]] extension of Listing's law'' (often abbreviated L2) which also covers [[vergence]].
It was proposed by Listing based on its geometric beauty, and never published it. It was first published by [[Christian Georg Theodor Ruete|Ruete]] in a 1855 textbook. [[Hermann von Helmholtz|Helmholtz]] first found empirical justification based on measurements of [[Afterimage|afterimages]].<ref name="wong-2004" />
== Definition ==
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Listing's law can be deduced without starting with the orthogonality assumption. If one assumes that all achieved eye orientations can be reached from some chosen eye orientation and then rotating about an axis that lies within some specific plane, then the existence of a unique primary orientation with an orthogonal Listing's plane is assured.<ref name="tweed-vilis-1990"/>
The expression of Listing's law can be simplified by creating a coordinate system where the origin is primary position, the vertical and horizontal axes of rotation are aligned in Listing's plane, and the third (torsional) axis is orthogonal to Listing's plane. In this coordinate system, Listing's law simply states that the torsional component of eye orientation is held at zero. (Note that this is not the same description of ocular torsion as rotation around the line of sight: whereas movements that start or end at the primary position can indeed be performed without any rotation about the line of sight, this is not the case for arbitrary movements.) Listing's law can also be formulated in a [[coordinate-free]] form using [[geometric algebra]].<ref>
Listing's law is the specific realization of the more general
=== Donders' law ===
{{Anchor|Donders' law}}
For any one gaze direction the eye's 3D spatial orientation is unique and independent of how the eye reached that gaze direction (previous gaze directions, eye orientations, temporal movements). It is implied by Listing's law. Note that it is theoretically possible for Listing's law to be false, but Donders' law to be true.
=== Listing's plane ===
The Listing's plane of a subject can be measured by recording the vector of rotation that would cause the eye to rotate from its primary position to a rotated position. It is orthogonal to the line of sight at the primary position. The line of sight is typically horizontal, but does not necessarily point straight ahead (perpendicular to the [[coronal plane]]). Instead, it points towards the nose or the temples by as much as 15 degrees, across subjects. Also, within each subject, the primary position tilts towards the temples when viewing distant objects due to [[vergence]]. The tilt angle is 0.72° per degree of vergence<ref name=":0">{{Cite journal |last=Kapoula |first=Z. |last2=Bernotas |first2=Marijus |last3=Haslwanter |first3=Thomas |date=1999-05-01 |title=Listing’s plane rotation with convergence: role of disparity, accommodation, and depth perception |url=https://doi.org/10.1007/s002210050727 |journal=Experimental Brain Research |language=en |volume=126 |issue=2 |pages=175–186 |doi=10.1007/s002210050727 |issn=1432-1106}}</ref>
The plane has thickness (standard deviation) of about 1 degree.<ref name=":0" />
=== Listing's half-angle rule ===
Let <math>\hat n</math> be the gaze direction when the eye is in the primary position.
Consider the scenario: The eye is looking at a certain direction <math>\hat v</math>, then it turns towards a different direction <math>\hat v'</math>. If the eye follows Listing's law, then orientation of the eye is uniquely determined in both gaze directions, and so there exists a unique rotation that turns the eye from the first orientation to the second.
It is a theorem of geometry that, for any <math>\hat v, \hat v'</math>, the rotation axis is in the plane perpendicular to <math>\frac{\hat n + \hat v}{2}</math>. This is Listing's half-angle rule. This can be proved by noting that a rotation by <math>\theta</math> is composed of two reflections across two planes <math>\theta/2</math> apart. The plane is called the velocity plane (or displacement plane). Listing's plane is the velocity plane of the primary position <ref name="wong-2004" /><ref>{{Cite journal |last=Tweed |first=D. |last2=Cadera |first2=W. |last3=Vilis |first3=T. |date=1990 |title=Computing three-dimensional eye position quaternions and eye velocity from search coil signals |url=https://pubmed.ncbi.nlm.nih.gov/2321369/ |journal=Vision Research |volume=30 |issue=1 |pages=97–110 |doi=10.1016/0042-6989(90)90130-d |issn=0042-6989 |pmid=2321369}}</ref>
== Purpose ==
{{unsourced section|date=November 2023}}
There has been considerable debate for over a century whether the purpose of Listing's law is primarily motor or perceptual. Some modern neuroscientists{{snd}}
== Common misconceptions ==
# It is often assumed that the primary position is at the mechanical center of the eye's range of movement. Primary position can only be determined by measuring Listing's plane. Direct measurements show that the location of primary position (and thus the orientation of Listing's plane) varies between subjects. Primary position is generally close to center, but it may be rotated slightly up or down, left or right.
# It is often misunderstood that Listing's law says that the eye only rotates about axes in Listing's plane. This is incorrect. Listing's plane only provides the orientations of the eye relative to primary position, expressed as an angle of rotation about some axis in Listing's plane (normally using the right-hand rule, where one curls the fingers of the right hand in the direction of rotation and the thumb then points in the direction of the rotation vector). This is not the same as the axes that the eye actually rotates about; in fact, Listing's law requires that the rotation axis of most saccades lies ''outside'' of Listing's plane,<ref>
The axes of rotation associated with Listing's law are only in Listing's plane for movements that head toward or away from primary position. For all other eye movements towards or away from some non-primary position, the eye must rotate about an axis of rotation that tilts out of Listing's plane. Such axes lie in a specific plane associated with this non-primary position. This plane's normal lies halfway between the primary gaze direction and the gaze direction of this non-primary position. This is called
== Modifications and violations ==
[[File:Flight dynamics with text ortho.svg|thumb|Yaw, pitch, roll.]]
Listing's law is not obeyed when the eyes counter-rotate during head rotation to maintain gaze stability, either due to the [[Vestibulo-ocular reflex]] (VOR) or the [[optokinetic reflex]]. Here the eye simply rotates about approximately the same axis as the head (which could even be a pure torsional rotation). This generally results in slow movements that drive the eye torsionally out of Listing's plane. However, when the head translates without rotating, gaze direction remains stable but Listing's law is still maintained.▼
=== Half-Listing's law strategy ===
Listing's law persists but takes on a torsional bias when the head is held at a tilted posture and the eyes counter-roll, and when the head is held steady upward or downward Listing's plane tilts slightly in the opposite direction.{{clarify|date=July 2013}}▼
▲Listing's law is
▲Listing's law persists
Perfect VOR would stabilize retinal image but cause violation to Listing's law, As a compromise, eye motion follows the half-Listing's law strategy, where instead of following the [[Listing's law#Listing's half-angle rule|Listing's half-angle rule]] (a geometric consequence of Listing's rule), eyes react to head motion in VOR by rotating around a modified velocity plane. The modified velocity plane makes an angle with Listing's plane that is 1/4, instead of 1/2, of the angle between the gaze direction and the primary direction.<ref name="wong-2004" />
Listing's law does not hold during sleep.<ref name="wong-2004">{{cite journal|author=A.M. Wong|title=Listing's law: clinical significance and implications for neural control|journal=Survey of Ophthalmology|year=November–December 2004|volume=49|number=6|pages=563–575|doi=10.1016/s0039-6257(04)00134-1}}</ref>▼
=== Other violations ===
Listing's law holds during fixation, saccades, and smooth pursuit. Furthermore, Listing's law has been generalized to the ''binocular extension of Listing's law'' which holds also during vergence.<ref name="wong-2004"/>▼
When larger "gaze saccades" are accompanied by a head movement, Listing's law cannot be maintained constantly because the eyes move much faster than the head. The eye typically reaches the destination in 80 ms, but the head needs about 300 ms. In this case, the eyes start at the position following Listing's law, then arrive at the destination violating it, then as the head continues to move into position, the eyes retain their orientation, until the head reaches the destination, and the eyes end up following Listing's law again in the end. The temporary violation can reach up to 15 degrees of torsion relative to Listing's law. The data can be explained by assuming that the eyes take the fastest possible path to their final orientation, with no constraints on torsion, except that it stays less than 15 degrees.<ref>{{Cite journal |last=Tweed |first=Douglas |last2=Haslwanter |first2=Thomas |last3=Fetter |first3=Michael |date=1998-08-28 |title=Optimizing Gaze Control in Three Dimensions |url=https://www.science.org/doi/10.1126/science.281.5381.1363 |journal=Science |language=en |volume=281 |issue=5381 |pages=1363–1365 |doi=10.1126/science.281.5381.1363 |issn=0036-8075}}</ref>
▲Listing's law does not hold during sleep.<ref name="wong-2004">{{cite journal |
▲Listing's law holds during fixation, saccades, and smooth pursuit. Furthermore, Listing's law has been generalized to the ''binocular extension of Listing's law'', which holds also during vergence.<ref name="wong-2004"/>
=== Adaptation ===
Listing's law can be violated in neurological conditions, such as acute unilateral fourth nerve palsy. However, there is an adaptive mechanism that ensures Listing's law, so that chronic patients of unilateral fourth nerve palsy satisfy Listing's law again. The adaptation fails under central fascicular palsy, as even chronic patients suffer from deviation from Listing's law.<ref name="wong-2004" />
=== Binocular extension ===
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=== Shape and thickness ===
Certain slight physiological deviations from Listing's rule are commonly described in terms of the "shape" and "thickness" of Listing's plane:<ref>{{cite journal |author1=Joseph F. X. Desouza |author2=David A. Nicolle |author3=Tutis Vilis |url=
* the "shape" specifies in how far it is indeed a (flat) plane or more generally a somewhat curved surface,
* the "thickness" specifies in how far eye movements indeed lie precisely within the plane (or surface) or may lie just slightly next to it.
== Visual consequences ==
Since Listing's law and its variants determine the orientation of the eye(s) for any particular gaze direction, it therefore determines the spatial pattern of visual stimulation on the retina(s). For example, since Listing's law defines torsion as zero about a head-fixed axis, this results in
== Physiology ==
{{unsourced section|date=November 2023}}
In the 1990s there was considerable debate about whether Listing's law is a neural or mechanical phenomenon. However, the accumulated evidence suggests that both factors play a role in the implementation of different aspects of Listing's law.
The horizontal recti muscles of the eyes only contribute to horizontal eye rotation and position, but the vertical recti and oblique muscles each have approximately equal vertical and torsional actions (in Listing's plane coordinates). Thus, to hold eye position in Listing's plane, there needs to be a balance of activation between these muscles so that torsion cancels to zero.
The [[Extraocular muscles|eye muscles]] may also contribute to Listing's law by having position-dependent pulling directions during motion, i.e., this might be the mechanism that implements the
Higher gaze control centers in the frontal cortex and superior colliculus are only concerned with pointing gaze in the right direction and do not appear to be involved in
== Pathology ==
Damage to any of the physiology described above can disrupt Listing's law
The influence of [[strabismus surgery]] on the Listing's planes of the two eyes is not fully understood. In one study, patients' eyes showed greater adherence to Listing's rule after the operation, however, the relative orientation of the Listing's planes of the two eyes had changed.<ref>{{cite journal |
== Measurement ==
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== Discovery and history ==
Listing's law was named after German [[mathematician]] [[Johann Benedict Listing]] (1808–1882).
Listing's law was first confirmed experimentally by the 19th == References ==
{{Reflist}}
*
== Further reading ==
* Agnes M. F. Wong
== External links ==
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