A class of geometric curve subdivision schemes on surfaces of constant curvature

A class of geometric curve subdivision schemes on surfaces of constant curvature | 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 class of geometric curve subdivision schemes on surfaces of constant curvature Taoufik Ahanchaou, Mohamed Bellaihou, Aziz Ikemakhen This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7021424/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 12 Jan, 2026 Read the published version in The Visual Computer → Version 1 posted 9 You are reading this latest preprint version Abstract G 1 curves are visually smooth and aesthetic curves. They play an important role in the domain of computer-aided design (CAD) and 3D modeling. Interpolatory geometric subdivision schemes is a methodological approach for attaining these curves while mitigating undesirable artifacts. In this paper, we introduce a novel geometric interpolatory subdivision scheme, named the angle-based 6-point geometric scheme, designed for curves on surfaces of constant curvature (Euclidean, spherical, and hyperbolic). This proposed scheme incorporates a tension parameter. We show that the scheme converges if the tension parameter belongs to a specific interval, and yields G 1 -continuous limit curves. This allows us to manipulate a whole family of curve subdivision schemes on constant curvature surfaces according to our needs. Numerical tests indicate the possibility of selecting the parameter within a well-defined range to achieve G 2 -continuity across all three surface models. Experimental examples are presented to illustrate the convergence and G 1 property of this scheme, accompanied by substantial applications showcasing its effectiveness. MSC Classification: 65D17 , 65D18 Geometric Subdivision scheme Spherical curve Hyperbolic curve Discrete geodesic curvature G1-continuity Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 12 Jan, 2026 Read the published version in The Visual Computer → Version 1 posted Editorial decision: Revision requested 09 Oct, 2025 Reviews received at journal 02 Oct, 2025 Reviewers agreed at journal 28 Sep, 2025 Reviews received at journal 14 Sep, 2025 Reviewers agreed at journal 11 Aug, 2025 Reviewers invited by journal 11 Aug, 2025 Editor assigned by journal 03 Jul, 2025 Submission checks completed at journal 02 Jul, 2025 First submitted to journal 01 Jul, 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-7021424","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":500532550,"identity":"bcaf74d0-9f5b-45dd-a4e8-cd4f6d2c6fc9","order_by":0,"name":"Taoufik Ahanchaou","email":"","orcid":"","institution":"FSTG","correspondingAuthor":false,"prefix":"","firstName":"Taoufik","middleName":"","lastName":"Ahanchaou","suffix":""},{"id":500532551,"identity":"a126b99c-6925-4c44-8c9c-be767622df53","order_by":1,"name":"Mohamed Bellaihou","email":"data:image/png;base64,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","orcid":"","institution":"FSTG","correspondingAuthor":true,"prefix":"","firstName":"Mohamed","middleName":"","lastName":"Bellaihou","suffix":""},{"id":500532552,"identity":"e13105ec-bec6-49cc-87da-19be69bdefc1","order_by":2,"name":"Aziz Ikemakhen","email":"","orcid":"","institution":"FSTG","correspondingAuthor":false,"prefix":"","firstName":"Aziz","middleName":"","lastName":"Ikemakhen","suffix":""}],"badges":[],"createdAt":"2025-07-01 14:38:18","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-7021424/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-7021424/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1007/s00371-025-04324-7","type":"published","date":"2026-01-12T16:29:31+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":100614737,"identity":"4979f9ab-b0aa-4a7b-8986-75ef3506d27f","added_by":"auto","created_at":"2026-01-19 17:23:39","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":5156044,"visible":true,"origin":"","legend":"","description":"","filename":"ForSubmissionAhanchaou.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7021424/v1_covered_05282151-c235-44cd-a463-ddac8293cbc2.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"A class of geometric curve subdivision schemes on surfaces of constant curvature","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":"the-visual-computer","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"tvcj","sideBox":"Learn more about [The Visual Computer](http://link.springer.com/journal/371)","snPcode":"371","submissionUrl":"https://submission.nature.com/new-submission/371/3","title":"The Visual Computer","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"Geometric Subdivision scheme, Spherical curve, Hyperbolic curve, Discrete geodesic curvature, G1-continuity","lastPublishedDoi":"10.21203/rs.3.rs-7021424/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7021424/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eG\u003csup\u003e1\u003c/sup\u003e curves are visually smooth and aesthetic curves. 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