Listing's law

Listing's law, named after German mathematician Johann Benedict Listing (1808–1882), describes the three-dimensional orientation of the eye and its 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 (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 states that the eye does not achieve all possible 3D orientations and that, instead, all achieved eye orientations can be reached by starting from one specific "primary" reference orientation and then rotating about an axis that lies within the plane orthogonal to the primary orientation's gaze direction (line of sight / visual axis). This plane is called Listing's plane.

It can be shown that Listing's law implies that, if we start from any chosen eye orientation, all achieved eye orientations can be reached by starting from this orientation and then rotating about an axis that lies within a specific plane that is associated with this chosen orientation. (Only for the primary reference orientation is the gaze direction orthogonal to its associated plane.)

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.

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.

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