Ab initio Binocular Formulation of Listing's Law

Ab initio Binocular Formulation of Listing's Law | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Article Ab initio Binocular Formulation of Listing's Law Jacek Turski This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7661442/v2 This work is licensed under a CC BY 4.0 License Status: Posted Version 2 posted You are reading this latest preprint version Show more versions Abstract Human eyes do not have perfectly aligned optical components; the fovea is displaced from the posterior pole, and the crystalline lens is tilted away from the eye's optical axis. Although important in the study of vision quality, it is used here in binocular and oculomotor vision research. In the binocular system with misaligned optics, the decomposition into torsion-free (geodesic) and torsional changes in the eye’s posture is computed in the framework of Rodrigues’ vector. It leads to the first consistently formulated binocular version of Listing’s law, which is important for oculomotor control by constraining the eye’s redundant torsional degree of freedom. It is also supported by 3D simulations. The Listing plane is replaced with the binocular eyes’ posture corresponding to the eye muscles’ natural tonus resting position, which serves as a zero-reference level for convergence effort. Modern ophthalmology studies, along with the results presented here, support binocular modification of Helmholtz's 1867 formulation of Listing’s law. It includes revising the elusive neurophysiological significance of the Listing plane. Further, the binocular constraints couple 3D changes in the torsional positions of the eyes within the binocular extension of Listing’s law, which was previously formulated ad hoc. Lastly, a precise noncommutativity rule underlying Listing’s law and the half-angle rule is discussed by specifying the configuration space of sequences of binocularly constrained eyes’ fixations, which is visualized in GeoGebra simulations. The results obtained in this study should be a part of the answers to the questions posted in the literature on the relevance of Listing’s law to clinical practices. Biological sciences/Neuroscience Biological sciences/Psychology Eye’s misaligned optics the asymmetric eye ocular torsion Rodrigues’ vector Listing’s law half-angle rule configuration space GeoGebra dynamic geometry Full Text Additional Declarations The authors declare no competing interests. Cite Share Download PDF Status: Posted Version 2 posted You are reading this latest preprint version Show more versions Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-7661442","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":597141655,"identity":"03251b93-69fb-4041-ae94-f680938f2acd","order_by":0,"name":"Jacek Turski","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAvklEQVRIiWNgGAWjYJACxgYDNjkQ48ADYpTzQLUYg7UkEK+FgSGxAcQjSou9RO7jjzMK+NLnhx1+CLTFTk63gZAtEulmkhsM2HI33k4zAGpJNjY7QFBLGhvjA5CW2QkgLQcStxGhhfkjUEu64ez0D0RrYQA5LEFeOodYW848Y5OcYcBmuEE6p+BAggERfmFvBzqs588xefnZ6Zs/fKiwkyOoBQqOMRiAVRoQpxwEahjkG4hXPQpGwSgYBSMMAACsqkFRGSIb1QAAAABJRU5ErkJggg==","orcid":"https://orcid.org/0000-0001-7968-3980","institution":"University of Houston-Downtown","correspondingAuthor":true,"prefix":"","firstName":"Jacek","middleName":"","lastName":"Turski","suffix":""}],"badges":[],"createdAt":"2025-09-19 23:15:09","currentVersionCode":2,"declarations":{"humanSubjects":false,"vertebrateSubjects":false,"conflictsOfInterestStatement":false,"humanSubjectEthicalGuidelines":false,"humanSubjectConsent":false,"humanSubjectClinicalTrial":false,"humanSubjectCaseReport":false,"vertebrateSubjectEthicalGuidelines":false},"doi":"10.21203/rs.3.rs-7661442/v2","doiUrl":"https://doi.org/10.21203/rs.3.rs-7661442/v2","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":104782645,"identity":"1c3ee2db-bf20-4ae5-86e8-b1aacabc931e","added_by":"auto","created_at":"2026-03-17 07:57:38","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1994091,"visible":true,"origin":"","legend":"","description":"","filename":"RSTurskiV22026.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7661442/v2_covered_597eba67-65f3-49ae-a584-c028926ea3bb.pdf"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"Ab initio Binocular Formulation of Listing's Law","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":true,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Eye’s misaligned optics, the asymmetric eye, ocular torsion, Rodrigues’ vector, Listing’s law, half-angle rule, configuration space, GeoGebra dynamic geometry","lastPublishedDoi":"10.21203/rs.3.rs-7661442/v2","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7661442/v2","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eHuman eyes do not have perfectly aligned optical components; the fovea is displaced from the posterior pole, and the crystalline lens is tilted away from the eye's optical axis. 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