A Unified Tangent-Based Parametrization of Conic Sections Generated from a Reference Circle

A Unified Tangent-Based Parametrization of Conic Sections Generated from a Reference Circle | 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 Research Article A Unified Tangent-Based Parametrization of Conic Sections Generated from a Reference Circle Asaad Ahmed Mohammed This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8497307/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract This study presents a unified geometric approach for determining the parameters of conic sections based on tangent geometry. Four generalized equations are derived that de scribe circles, ellipses, parabolas, and hyperbolas through the angular relationships between tangent lines, their points of contact, and the associated polar angle with respect to a central reference circle. The proposed framework establishes a direct geometric link between the tangent an gles and the emergence of different conic types, extending the classical polar-angle-based description of conic sections. The derived equations admit canonical conic forms, allowing the geometric parameters and classification of each conic to be identified within a single parametric structure. Moreover, the formulation reveals that all four conic sections intersect at the external point from which the tangents to the reference circle are drawn, providing a coherent ge ometric interpretation of their common origin. This tangent-based parametrization offers a mathematically consistent framework for analytical geometry and presents potential rele vance for applications in computational geometry and structural modeling. Pure Mathematics Conic sections tangent geometry reference circle canonical forms computa tional geometry Full Text Additional Declarations The authors declare no competing interests. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version 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-8497307","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":568148283,"identity":"aa914519-33f4-4c7b-9a25-21086c5089b9","order_by":0,"name":"Asaad Ahmed 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