[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84834-en":3,"doc-seo-84834-105":29,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},84834,8796095462418,"Noah","https://ap-avatar.wpscdn.com/avatar/80000253c1241d02b47?x-image-process=image/resize,m_fixed,w_180,h_180&k=1778826106357471780",8,"Research & Report","A Heisenberg Subdivision Scheme with Central Smoothness Loss","An interpolatory subdivision scheme is introduced for control polygons valued in the three-dimensional Heisenberg group, the simplest noncommutative model geometry. Existing points are preserved at each refinement, while new points are inserted via a coordinate rule whose central correction arises from the group law. The horizontal coordinates follow the classical Dyn–Gregory–Levin four-point scheme, whereas the central coordinate gains a closed-form adjustment from signed area data. The limit curve’s horizontal smoothness matches the classical case, while the central part converges to a Zygmund-class function with a logarithmic modulus of continuity, and under a sharp verifiable forcing condition the limit is not C1; numerics confirm the effect.","A HEISENBERG SUBDIVISION SCHEME WITH CENTRAL  \nSMOOTHNESS LOSS  \nHassan Ugail  \nCentre for Visual Computing and Intelligent Systems University of Bradford Bradford, United Kingdom  \nAlfonso Carriazo  \nDepartment of Geometry and Topology University of Seville Seville, Spain  \narXiv :2607 .05446v1 [math .NA] 4 Jul 2026  \nJuly 8, 2026  \nABSTRACT  \nWe introduce an interpolatory subdivision scheme for control polygons that take values in the threedimensional Heisenberg group, the simplest noncommutative model geometry. The scheme keeps existing points at every refinement step and inserts new ones by a coordinate rule whose central correction comes from the group law. The two horizontal coordinates are refined by the classical four-point scheme of Dyn, Gregory and Levin, while the central coordinate acquires a closed-form correction built from a signed area of neighbouring horizontal data. Our main finding concerns the regularity of the limit curve. The horizontal part is exactly the classical four-point limit and inherits its smoothness. The central part behaves very differently. We prove that it converges toa continuous limit that belongs to the Zygmund class, with a logarithmic modulus of continuity.  \nUnder an explicit and verifiable condition on the central forcing, this logarithmic bound is sharp, because the scaled first differences then grow linearly with the refinement level, and the limit fails to be continuously differentiable. The effect is confirmed numerically. The correction is harmless at any single refinement step, but its repeated injection at every scale is what impacts smoothness.  \nThe example serves as a caution for nonlinear and group-valued subdivision, where a geometrically natural correction can impact regularity.  \nKeywords noncommutative subdivision · Heisenberg group · four-point scheme · interpolatory refinement · Zygmund regularity · loss of C 1 smoothness  \n1 Introduction  \nSubdivision schemes are a central tool in approximation theory, geometric modelling, and computational mathematics. Starting from a discrete control polygon, they generate refined data by repeated local insertion rules and, under suitable conditions, converge to smooth limit curves. Among the most studied examples are interpolatory schemes, which preserve all existing data and insert new points by stencil averages. In the Euclidean setting, the four-point scheme of Dyn, Gregory and Levin [1] is the canonical example. It is explicit, local, and interpolatory, and its C 1 regularity for parameter ω = 1/16 is well understood [2, 3, 4] .  \nSubdivision is one of several constructive paradigms in computer-aided geometric design for generating curves and surfaces from sparse control data. Related traditions include partial differential equation methods for surface design [5, 6], interactive and boundary-driven surface construction [7, 8, 9], harmonic and biharmonic Bézier patches [10, 11], and parameterised geometric modelling across visual computing and engineering [12, 13, 14, 15] . These methods share with subdivision the aim of producing controlled limiting geometry from compact descriptions. The present work stays within the subdivision tradition and asks how a local geometric rule behaves once the underlying data space is no longer Euclidean.  \nMany applications involve manifold-valued or group-valued data, for which component-wise linear refinement is geometrically inappropriate. Nonlinear subdivision schemes address this by adapting a linear scheme to curved data  \nA PREPRINT-JULY 8, 2026  \nwhile attempting to preserve convergence and smoothness. The proximity framework of Wallner and Dyn [16], with later developments by Wallner and Grohs [17, 18, 19], transfers regularity from a linear reference to its nonlinear counterpart under a quantitative closeness assumption. Other intrinsic constructions refine through geodesic averaging or through global refinement rules on Riemannian manifolds [20, 21, 22, 23] . These are, for the most ","cbCaiotW69yLsrI9","https://ap.wps.com/l/cbCaiotW69yLsrI9","pdf",708107,1,18,"English","en",105,"# Abstract\n# 1 Introduction","[{\"question\":\"What subdivision strategy is proposed for data in the Heisenberg group?\",\"answer\":\"The scheme is interpolatory: it keeps all existing control points at every refinement and inserts new points using a rule derived from the Heisenberg group law. The central coordinate gets an additional closed-form correction, while the horizontal coordinates are refined by the classical four-point scheme.\"},{\"question\":\"How does the horizontal part of the limit curve behave?\",\"answer\":\"The horizontal coordinates coincide with the classical four-point limit and inherit its smoothness properties. This means the horizontal smoothness is preserved exactly as in the Euclidean four-point scheme.\"},{\"question\":\"Why does the central coordinate lose smoothness in the limit?\",\"answer\":\"The central correction is harmless at a single refinement step, but repeated injection across all scales accumulates and changes regularity. Under an explicit verifiable condition on the central forcing, the scaled first differences grow linearly with the refinement level, leading to a limit that is not continuously differentiable; the phenomenon is also confirmed numerically.\"}]",1784198611,45,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"a-heisenberg-subdivision-scheme-with-central-smoothness-loss","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/a-heisenberg-subdivision-scheme-with-central-smoothness-loss/84834/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What subdivision strategy is proposed for data in the Heisenberg group?","Question",{"text":75,"@type":76},"The scheme is interpolatory: it keeps all existing control points at every refinement and inserts new points using a rule derived from the Heisenberg group law. The central coordinate gets an additional closed-form correction, while the horizontal coordinates are refined by the classical four-point scheme.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the horizontal part of the limit curve behave?",{"text":80,"@type":76},"The horizontal coordinates coincide with the classical four-point limit and inherit its smoothness properties. This means the horizontal smoothness is preserved exactly as in the Euclidean four-point scheme.",{"name":82,"@type":73,"acceptedAnswer":83},"Why does the central coordinate lose smoothness in the limit?",{"text":84,"@type":76},"The central correction is harmless at a single refinement step, but repeated injection across all scales accumulates and changes regularity. Under an explicit verifiable condition on the central forcing, the scaled first differences grow linearly with the refinement level, leading to a limit that is not continuously differentiable; the phenomenon is also confirmed numerically.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":45,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":45,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":45,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":45,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":45,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":45,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":45,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]