Singular points of offsets of plane curves: man-and-machine collaboration with CAS, DGS and AI

Singular points of offsets of plane curves: man-and-machine collaboration with CAS, DGS and AI | 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 Singular points of offsets of plane curves: man-and-machine collaboration with CAS, DGS and AI Thierry Dana Picard, Arie Haenel This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8661335/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 We explore envelopes of families of circles of constant radius centered on a plane curve, and offsets of plane curves, using automated commands for envelopes and geometric loci. The topology of envelopes and offsets is generally more complicated than that of the progenitor, so it is important to study the singularities.These can be cusps and crunodes (self-intersections). The work involves both analytic and algebraic methods (such as elimination), networking between different kinds of software, and automated commands to determine envelopes and geometric loci. Accurate plotting is a central issue, because in a neighborhood of a singular point the software may have difficulties and leave a blank gap. To address this issue, we demonstrate how the new Plot2D command provided byGeoGebra Discovery is effective. Finally, we analyze the output of generative AI for the above questions and discuss briefly how the pitfalls can and should be used to develop critical thinking and other skills. Generative AI can nevertheless be useful for rapid paraphrasing, suggesting candidate parametrizations, and proposing computational steps, provided that all results are verified by theorem based software and mathematical reasoning. The activities presented here have been proposed to undergraduates, both in regular classes and as individual students acting as undergraduate researchers, and to in-service teachers working toward an advanced degree. Dynamic Geometry Implicitization Envelopes Offsets Geometric Loci Singular points Automated methods Generative AI Full Text Additional Declarations No competing interests reported. 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. 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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-8661335","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":581777092,"identity":"d7e44161-53de-4ff0-8218-c95d04d6cdf3","order_by":0,"name":"Thierry Dana Picard","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAo0lEQVRIiWNgGAWjYFCCBMYHElDmgQdEamE2kGAwgGhJIFILG9ASAyibGA3m7LnHKixq/jDwtx9gJM4Wy553aTckjhkwSJxJINJhBjdyzG5INgAddoNYv4C0FIC0yJOkhQGkxYB4LWfeJUtIHDPmMTyT2ECkluO5Bz9L1MjJyR0/fPjDB2K0MDDwMDBLgEgGxgbiNIAUMxJp+CgYBaNgFIxUAADTFDGyLq0TRgAAAABJRU5ErkJggg==","orcid":"","institution":"Jerusalem College of Technology","correspondingAuthor":true,"prefix":"","firstName":"Thierry","middleName":"Dana","lastName":"Picard","suffix":""},{"id":581777093,"identity":"780a94aa-aacd-4ee7-bbf1-9c43bfb14e30","order_by":1,"name":"Arie Haenel","email":"","orcid":"","institution":"Jerusalem College of Technology","correspondingAuthor":false,"prefix":"","firstName":"Arie","middleName":"","lastName":"Haenel","suffix":""}],"badges":[],"createdAt":"2026-01-21 15:08:11","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-8661335/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8661335/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":103295799,"identity":"4324138d-6471-4a6b-ad23-939a0bd1c751","added_by":"auto","created_at":"2026-02-24 07:12:29","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":2074572,"visible":true,"origin":"","legend":"","description":"","filename":"AMAIADGsubmissionDPH.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8661335/v1_covered_9dffa261-6b63-47ea-9236-10ef8861e59a.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Singular points of offsets of plane curves: man-and-machine collaboration with CAS, DGS and AI","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"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":"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":"Dynamic Geometry, Implicitization, Envelopes, Offsets, Geometric Loci, Singular points, Automated methods, Generative AI","lastPublishedDoi":"10.21203/rs.3.rs-8661335/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8661335/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"We explore envelopes of families of circles of constant radius centered on a plane curve, and offsets of plane curves, using automated commands for envelopes and geometric loci. 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