Simuorb: a new method for generating and describing the intersection points of clique-arrangements

Simuorb: a new method for generating and describing the intersection points of clique-arrangements | 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 Simuorb: a new method for generating and describing the intersection points of clique-arrangements Philippe Ryckelynck, Laurent Smoch This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8176544/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 7 You are reading this latest preprint version Abstract This work, which may be seen as a companion paper to [1], handles the way the intersection points made by the diagonals of a regular polygon are distributed. It was stated recently by the authors that these points lie exclusively on circles centered on the origin and also the way their respective radii depend on the four indices of the vertices of the initial regular n-gon which characterize the two straight lines underlying the intersection points. Because these four vertices are located at preset positions on the the regular n-gon inscribed in the unit circle whose path-length perimeter is constant, it allows the orbits to be characterized by 3 parameters instead of 4, describing roughly the lengths of the paths between the first three vertices, whether the quadrilateral described by these four vertices is simple or complex. This approach enables us to deal with the orbits gener- ated by the clique-arrangement, and to handle their cardinalities as well as the multiplicities of the associated intersection points. A reliable counting-algorithm based on this triplet strategy is provided in order to enumerate the intersection points without generating the associated graph. The orbits being simulated, we call this method Simuorb. The procedure is robust, fast and allows a compre- hensive understanding of what is happening in a clique-arrangement, whether it contains a large number of points or not. MSC Classification: 51M04 , 05C38 , 05C12 , 05C07 , 52-08 Clique-arrangements Geometric graphs Cyclic quadrilaterals Computational methods Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Editorial decision: Revision requested 08 May, 2026 Reviews received at journal 08 May, 2026 Reviewers agreed at journal 04 Apr, 2026 Reviewers invited by journal 19 Dec, 2025 Editor assigned by journal 19 Dec, 2025 Submission checks completed at journal 19 Dec, 2025 First submitted to journal 21 Nov, 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. 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-8176544","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":562819255,"identity":"08bcba2b-c9d4-4b70-9171-25c7c2be6a5d","order_by":0,"name":"Philippe Ryckelynck","email":"","orcid":"","institution":"University of the Littoral Opal Coast","correspondingAuthor":false,"prefix":"","firstName":"Philippe","middleName":"","lastName":"Ryckelynck","suffix":""},{"id":562819256,"identity":"b4923f1b-a93b-4355-aca0-fd44842935a6","order_by":1,"name":"Laurent Smoch","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABHUlEQVRIie3PsUoDMRjA8S8ErkvaWwOV3CscBOqgD3PhoC6HIF0cHAJCnLoWj4o+hFAcHHIcOB34AF28pZPDSZcKLZh4J4i93uyQ/xK+D34kAXC5/mGD3wOGSwAPAOmonqVd/M2rj7AhRb3RUWQ3SNpFN0GqHjV0kV4xqj6ethDM8mx9cZ+fD2gEutxs2fFNJiF73ick4WlahBAux/EwXeQTzxLzMH5UCAl61fKwhOO+MmSYjHB/kQtFXitLxC0IiSu9T/x3jneGBHNL7izRUBO/NLe0EGpuQYbA0hJpSE82hIoDZDVBU8XJ91/Iy1lDxpxTWkrdQnw/foRPxVgwj7M1uToRD9eAys0pY9SPs7cW8hNpXx8GLpfL5erqCw5XaI1i1kJFAAAAAElFTkSuQmCC","orcid":"","institution":"University of the Littoral Opal Coast","correspondingAuthor":true,"prefix":"","firstName":"Laurent","middleName":"","lastName":"Smoch","suffix":""}],"badges":[],"createdAt":"2025-11-21 20:53:11","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-8176544/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8176544/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":98733159,"identity":"7d2c1957-f865-4f39-a477-0a017cb1ae48","added_by":"auto","created_at":"2025-12-22 05:54:36","extension":"json","order_by":0,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":3981,"visible":true,"origin":"","legend":"","description":"","filename":"b480d6d806a14d4a8dc9068e3bdee581.json","url":"https://assets-eu.researchsquare.com/files/rs-8176544/v1/af6272e4d6330a9a6e5a120c.json"},{"id":98778328,"identity":"7027d112-7c38-49d1-880c-2aa218dc5b98","added_by":"auto","created_at":"2025-12-22 12:29:10","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":2641460,"visible":true,"origin":"","legend":"","description":"","filename":"SimuorbRyckelynckSmoch21112025.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8176544/v1_covered_00b05c63-6baa-4175-86b0-1467f02025dc.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"\u003cp\u003eSimuorb: a new method for generating and describing the intersection points of clique-arrangements\u003c/p\u003e","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"numerical-algorithms","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"numa","sideBox":"Learn more about [Numerical Algorithms](http://link.springer.com/journal/11075)","snPcode":"11075","submissionUrl":"https://submission.nature.com/new-submission/11075/3","title":"Numerical Algorithms","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"Clique-arrangements, Geometric graphs, Cyclic quadrilaterals, Computational methods","lastPublishedDoi":"10.21203/rs.3.rs-8176544/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8176544/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThis work, which may be seen as a companion paper to [1], handles the way the intersection points made by the diagonals of a regular polygon are distributed. It was stated recently by the authors that these points lie exclusively on circles centered on the origin and also the way their respective radii depend on the four indices of the vertices of the initial regular n-gon which characterize the two straight lines underlying the intersection points. Because these four vertices are located at preset positions on the the regular n-gon inscribed in the unit circle whose path-length perimeter is constant, it allows the orbits to be characterized by 3 parameters instead of 4, describing roughly the lengths of the paths between the first three vertices, whether the quadrilateral described by these four vertices is simple or complex. This approach enables us to deal with the orbits gener- ated by the clique-arrangement, and to handle their cardinalities as well as the multiplicities of the associated intersection points. 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