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S.J. Judge; Representing Rotations as Pairs of Reflections Elucidates the Geometry of 3-D Eye Rotations . Invest. Ophthalmol. Vis. Sci. 2003;44(13):2129.
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© ARVO (1962-2015); The Authors (2016-present)
Purpose: To develop more intuitive methods of understanding the geometry of thee-dimensional (3-D) rotation of the eyes. Methods: Study of the 3-D patterns of eye rotation goes back to Helmholtz, and is of considerable current interest. Eye rotations in 3-D are constrained to patterns such as Listing’s law, or its generalisation ‘L2’, according to the context. Our understanding of the patterns of 3-D eye rotations, and their effect on the retinal images, has been greatly advanced by the development of algebraic methods, notably by Tweed and colleagues, for calculating the effect of eye rotations. I show here how the geometry of 3-D rotations of the eye and their visual effects can be made easier to understand by use of the principle that a rotation through angle α can be achieved by a pair of reflections in planes with an angular separation α/2, and a common line that is the rotation axis. (Mathematically, the method is equivalent to decomposing the unit quaternions so successfully used to study three-dimensional eye rotations into pairs of pure quaternions - ones whose scalar part is zero - which represent the reflections.) My purpose here is to show how several of the well-known relationships describing patterns of 3-D eye rotation can be derived by purely geometric methods - essentially by drawing the appropriate diagram that makes the relationship ‘obvious’ - in something like the same way that classical Euclidean geometry often shows more directly than coordinate geometry why various relationship are true in 2 dimensions. Results: It is shown 1) that Listing’s law implies the so called ‘half-angle rule’ for composing rotations; 2) that the torsion, γ, associated with a given Helmholtz azimuth, α, and elevation, ß, is given by tan(γ/2) = -tan(α/2)tan(ß/2), and 3) that if the eyes move so as to maintain the projection of the visual plane on the horizontal retinal meridia of both eyes they will move in the pattern known as ‘L2’ in which in vergence,ϕ, the Listing planes of each eye rotate outward like barn doors so as to lie ϕ/2 apart. Conclusion: Decomposition of rotations into pairs of reflections is useful for clarifying why certain results are true in 3-D geometry, and may well be more generally useful for consideration of issues in the geometry of 3-D space.
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