Projective quadrics and orbital elements

Projective quadrics and orbital elements | 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 Projective quadrics and orbital elements Alberto José Marín Fdez. de Capel, Miguel Ángel Sánchez Granero This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-5907722/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 28 Mar, 2025 Read the published version in Astrophysics and Space Science → Version 1 posted 7 You are reading this latest preprint version Abstract Due to the fact that Keplerian orbits are conic sections, projective geometry gives a description of orbits based on projective hyperquadric properties. Using matrix algebra to describe hyperquadrics allows the construction of a new set of orbital elements using the eigenvalues and eigenvectors of the improper hyperquadric matrix and the construction of reference systems associated with the orbit. It ispossible to work directly with the orbit in Cartesian coordinates in the three-dimensional space as a 4x4 matrix, constructing the quadric associated with theorbital conic. Celestial Mechanics Orbital Elements Reference Systems Projective Quadric Eigenvalues Eigenvectors Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 28 Mar, 2025 Read the published version in Astrophysics and Space Science → Version 1 posted Editorial decision: Revision requested 24 Feb, 2025 Reviews received at journal 24 Feb, 2025 Reviewers agreed at journal 04 Feb, 2025 Reviewers invited by journal 04 Feb, 2025 Editor assigned by journal 01 Feb, 2025 Submission checks completed at journal 27 Jan, 2025 First submitted to journal 26 Jan, 2025 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. 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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-5907722","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":420141806,"identity":"f1f82dce-cd2b-4760-8cde-9da260fcb358","order_by":0,"name":"Alberto José Marín Fdez. de Capel","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA/ElEQVRIiWNgGAWjYFACxucfPlTY8DCwA9kJYBHmBgJamM0YZ5xJ42FghmthJKyFmbftMANYCwMxWvinHWZ7zNt2Xsacmcd0w8MdNgwGxw82MHz4g1uLxO1kdsM5527zWDbzmN1IPJPGYHAmsYFxZhsea27nH5B4U3abx+AwSAvQhQYHEhuYefG4Tf52MoMED9s5mJb/DAbnHzYw/8HjMIPbyWySPG0HYFoOMBjcANrCwIZbi+HtZGbDGWeSgVrYyoBaknkkbzxsONiLxy9yt5MZH3yosLM3ON687ebPNjs5vvPJBx/8wOMwDMADIg6QoGEUjIJRMApGARYAABgnVlhBc4Y0AAAAAElFTkSuQmCC","orcid":"","institution":"University of Almería","correspondingAuthor":true,"prefix":"","firstName":"Alberto","middleName":"José Marín Fdez.","lastName":"de Capel","suffix":""},{"id":420141807,"identity":"d0f3ffb7-bdda-42a9-b7a7-eb06f9d3d501","order_by":1,"name":"Miguel Ángel Sánchez Granero","email":"","orcid":"","institution":"University of Almería","correspondingAuthor":false,"prefix":"","firstName":"Miguel","middleName":"Ángel Sánchez","lastName":"Granero","suffix":""}],"badges":[],"createdAt":"2025-01-26 17:38:05","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-5907722/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-5907722/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1007/s10509-025-04420-z","type":"published","date":"2025-03-28T15:57:19+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":79605150,"identity":"7452fd9b-acb3-4ca3-b82c-d9d5b62dc4a7","added_by":"auto","created_at":"2025-03-31 16:10:50","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":492397,"visible":true,"origin":"","legend":"","description":"","filename":"amarinmsanchez.pdf","url":"https://assets-eu.researchsquare.com/files/rs-5907722/v1_covered_f6e9dde0-e276-4686-9b0a-407511a90ebd.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Projective quadrics and orbital elements","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"astrophysics-and-space-science","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"astr","sideBox":"Learn more about [Astrophysics and Space Science](https://www.springer.com/journal/10509)","snPcode":"10509","submissionUrl":"https://submission.nature.com/new-submission/10509/3","title":"Astrophysics and Space Science","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"Celestial Mechanics, Orbital Elements, Reference Systems, Projective Quadric, Eigenvalues, Eigenvectors","lastPublishedDoi":"10.21203/rs.3.rs-5907722/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-5907722/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"Due to the fact that Keplerian orbits are conic sections, projective geometry gives a description of orbits based on projective hyperquadric properties. 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