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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>{{cite journal |doi=10.1016/0893-6080(94)90056-6 |title=Invariant body kinematics: I. Saccadic and compensatory eye movements |year=1994 |last1=Hestenes |first1=David |author-link1=David Hestenes |journal=Neural Networks |volume=7 |issue=1 |pages=65–77 |url=
Listing's law is the specific realization of the more general "Donders' law"
=== 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}} who have tended to emphasize optimization of multiple variables{{snd}} consider Listing's law to be the best compromise between motor factors (e.g., taking the shortest possible rotation path) and visual factors (see below for details).
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== 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" />
=== Other violations ===
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 |last=Wong |first=Agnes M. F. |title=Listing's law: clinical significance and implications for neural control |journal=Survey of Ophthalmology |date=November–December 2004 |volume=49 |number=6 |pages=563–575 |doi=10.1016/s0039-6257(04)00134-1 |pmid=15530944}}</ref>
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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== 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.
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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 ==
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== Further reading ==
* Agnes M. F. Wong
== External links ==
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