Neural Adaptive Tension for Multi-Geometry Curve Subdivision: A Unified Approach

Neural Adaptive Tension for Multi-Geometry Curve Subdivision: A Unified Approach | 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 Neural Adaptive Tension for Multi-Geometry Curve Subdivision: A Unified Approach Hassan Ugail, Newton Howard This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9262382/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 Curve subdivision is pivotal in computer graphics for generating smooth geometric objects from control polygons. Traditional methods rely on a global tension parameter, limiting adaptability across diverse curvatures and geometries. This paper introduces a shared learned tension predictor, employing a 140K-parameter network to predict per-edge insertion angles, enabling adaptive curve subdivision across Euclidean, spherical, and hyperbolic geometries. The network incorporates local intrinsic features and a trainable geometry embedding, ensuring specificity without architectural modifications. Theoretical guarantees on structural safety and conditional convergence are provided. Empirical evaluations demonstrate superior performance in bending energy and smoothness compared to fixed-tension baselines, with notable generalisation on out-of-distribution examples. Curve subdivision interpolatory schemes neural operators non-Euclidean geometry spherical geometry hyperbolic geometry Poincaré disk geometric deep learning convergence analysis log-exp subdivision adaptive tension 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. 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-9262382","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":624415794,"identity":"3511ce73-19d2-4492-b619-0e96310c9ad6","order_by":0,"name":"Hassan Ugail","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA80lEQVRIiWNgGAWjYBACAwhpIQdiSFQwMDA2gEUOENQiYQzWcoZ4LQwSiRuI1mLOwGP84keBRPp26cMPbxyouCfbwH74ATPPGdxaLBt4zCx7DCRyd/alGVscOFNs3MCTZsDMcwOPww7wmBnwALVsOMNgJv2xLSGxgSGHgZnnA34thn8MJNINzrB/kzgI0sL/hqAW48dAWxIMzvCYQbRIgGzB57DDbGXMMgYShjt7eIqBfkkwbpN4ZnBwDh7vGxxv3vzxzR8beXMe9o3AEEuQ7edPfvjgzTHcWhiYGdgkUATYGPDGCkQTHq+OglEwCkbBKAACACLnUMKMO1MYAAAAAElFTkSuQmCC","orcid":"","institution":"University of Bradford","correspondingAuthor":true,"prefix":"","firstName":"Hassan","middleName":"","lastName":"Ugail","suffix":""},{"id":624415795,"identity":"f3c67c98-047c-4081-97b7-82522119f68f","order_by":1,"name":"Newton Howard","email":"","orcid":"","institution":"Rochester Institute of Technology","correspondingAuthor":false,"prefix":"","firstName":"Newton","middleName":"","lastName":"Howard","suffix":""}],"badges":[],"createdAt":"2026-03-30 04:38:42","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-9262382/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9262382/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":109429902,"identity":"1b6d339d-322f-4a8e-9778-33941b18669a","added_by":"auto","created_at":"2026-05-18 04:10:35","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1511336,"visible":true,"origin":"","legend":"","description":"","filename":"TheVisualComputer.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9262382/v1_covered_eba040e3-ef42-45d6-bdcf-bf8dda7e5539.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Neural Adaptive Tension for Multi-Geometry Curve Subdivision: A Unified Approach","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":"Curve subdivision, interpolatory schemes, neural operators, non-Euclidean geometry, spherical geometry, hyperbolic geometry, Poincaré disk, geometric deep learning, convergence analysis, log-exp subdivision, adaptive tension","lastPublishedDoi":"10.21203/rs.3.rs-9262382/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9262382/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"Curve subdivision is pivotal in computer graphics for generating smooth geometric objects from control polygons. 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