[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81919-en":3,"doc-seo-81919-105":31,"detail-sidebar-cat-0-en-105":93},{"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":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":28,"seo_description":14,"update_tm":29,"read_time":30},81919,8796095461564,"Liam","https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d",8,"Research & Report","Intrinsic Meshing of Closed Surfaces Using Geodesic Distances","A method constructs intrinsic triangulations of closed discrete surfaces where edges follow shortest geodesic paths and faces form geometric primitives inherited from the underlying mesh. Given a watertight input triangulation, intrinsic meshes are built via local optimization on-surface—edge swaps, edge splits, edge collapses, and triangle splits—without changing the original geometry. Element size is governed by a characteristic length field and quality by intrinsic, angle-based criteria. Geodesic distances are computed exactly with continuous Dijkstra plus A* acceleration, reducing cost to about 3% of standard propagation. Refinement and coarsening are supported, and the approach enables direct high-order mesh generation, validated on Thingi10K.","Intrinsic Meshing of Closed Surfaces Using Geodesic Distances  \nTim Gabriel 1,2 , Jean-Fran¸cois Remacle2 , and Christophe Geuzaine 1  \n1 Universit´e de Li`ege, Li`ege, Belgium , {tim. gabriel, [cgeuzaine](cgeuzaine}@uliege. be)[}](cgeuzaine}@uliege. be)[@uliege. be](cgeuzaine}@uliege. be)  \n2 Universit´e Catholique de Louvain, Louvain-la-Neuve, Belgium ,  \n[jean-francois. remacle@uclouvain. be](jean-francois. remacle@uclouvain. be)  \narXiv :2607 .04989v 1 [ cs .CG] 6 Jul 2026  \nAbstract  \nWe present a method for constructing intrinsic triangulations of closed discrete surfaces, in which edges correspond to shortest geodesic paths and faces decompose into geometric primitives inherited from the underlying mesh. Starting from a watertight input triangulation, the method progressively builds an intrinsic mesh through local optimization operations—edge swaps, edge splits, edge collapses, and triangle splits—performed directly on the surface without modifying the original geometry. Element size is controlled via a characteristic length field, and quality is enforced through angle-based criteria derived from intrinsic distances. Geodesic distances are computed exactly using a continuous Dijkstra approach, accelerated by an A* search strategy that reduces computation to roughly 3% of the cost of standard propagation. The framework supports both refinement and coarsening, overcoming a key limitation of prior intrinsic methods based on developable triangles. As a by-product, the intrinsic triangulation provides a natural foundation for direct high-order mesh generation, bypassing the classical pipeline of first constructing a linear mesh and subsequently curving it. The method is validatedon the Thingi10K dataset across nearly 5,000 geometrically complex models.1  \nKeywords: intrinsic mesh, discrete isogeometric triangulation, geodesic, shortest path, A* search, coarsening, direct high-order meshing  \n1 Introduction  \nThe numerical representation of surfaces plays a crucial role across a wide range of domains, including geometric modeling and design, rendering, surface analysis, geometric processing, and computational physics. Broadly speaking, two main categories of surface representations can be distin-  \n1 The implementation code is openly available within the official Gmsh repository at [https://gitlab.onelab.info/](https://gitlab.onelab.info/)[ ](https://gitlab.onelab.info/)[gmsh/gmsh](gmsh/gmsh.)[.](gmsh/gmsh.)  \nguished: continuous representations and discrete representations.  \nThe first category consists of continuous surfaces, typically originating from design blueprints or analytical descriptions. In computer-aided design (CAD), surfaces are most commonly defined by explicit parametric mappings  \n(x, y, z) = x(u, v),  \nwhere (u, v) belong to a parametric domain, usually rectangular. In practice, these parametric patches are often trimmed, meaning that only a subset of the parameter domain is retained. The trimming curves are themselves typically defined as parametric curves in the (u, v) space. As a result, CAD models are generally composed of collections of trimmed parametric patches that must be assembled to form a complete surface.  \nOther types of continuous representations also exist. For instance, implicit surfaces are defined as the zero level set of a scalar function  \nf (x, y, z) = 0,  \nas commonly encountered in level-set methods, signed distance functions, or algebraic surface descriptions.  \nThe second major category corresponds to discrete surface representations. These arise naturally from numerical approximation, geometric processing, or measurement procedures such as scanning and reconstruction. Typical examples include polygonal meshes (most commonly triangle meshes), triangle soups, and point clouds. Such representations are sometimes referred to as organic surfaces, emphasizing that they do not rely on an underlying analytical description but rather on sampled geometric data.  \nFrom the perspective of com","cbCaioh6I90IkdLZ","https://ap.wps.com/l/cbCaioh6I90IkdLZ","pdf",6714018,5,1,24,"English","en",105,"# Abstract\n# Introduction\n## Surface representations and simulation requirements\n## Continuous vs. discrete geometry\n## Isogeometric triangulations and practical meshing challenges","[{\"question\":\"How does the method define edges and faces in the intrinsic triangulation?\",\"answer\":\"Edges correspond to shortest geodesic paths on the surface, and faces decompose into geometric primitives inherited from the underlying mesh.\"},{\"question\":\"What operations are used to build the intrinsic mesh without altering geometry?\",\"answer\":\"Starting from a watertight triangulation, the method progressively applies local on-surface optimization operations such as edge swaps, edge splits, edge collapses, and triangle splits.\"},{\"question\":\"How are geodesic distances computed efficiently and accurately?\",\"answer\":\"Geodesic distances are computed exactly using a continuous Dijkstra approach, accelerated with an A* search strategy that reduces computation to roughly 3% of standard propagation.\"}]","Intrinsic Meshing of Closed Surfaces Using Geodesic Distances | 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does the method define edges and faces in the intrinsic triangulation?","Question",{"text":77,"@type":78},"Edges correspond to shortest geodesic paths on the surface, and faces decompose into geometric primitives inherited from the underlying mesh.","Answer",{"name":80,"@type":75,"acceptedAnswer":81},"What operations are used to build the intrinsic mesh without altering geometry?",{"text":82,"@type":78},"Starting from a watertight triangulation, the method progressively applies local on-surface optimization operations such as edge swaps, edge splits, edge collapses, and triangle splits.",{"name":84,"@type":75,"acceptedAnswer":85},"How are geodesic distances computed efficiently and accurately?",{"text":86,"@type":78},"Geodesic distances are computed exactly using a continuous Dijkstra approach, accelerated with an A* search strategy that reduces computation to roughly 3% of standard 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