[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-seo-657882-105":3,"detail-sidebar-cat-0-en-105":80,"doc-detail-657882-en":130},{"code":4,"msg":5,"data":6},0,"ok",{"site_id":7,"language":8,"slug":9,"title":10,"keywords":11,"description":12,"schema_data":13,"social_meta":73,"head_meta":75,"extra_data":77,"updated_unix":79},105,"en","an-analytical-representation-of-the-2d-generalized-balanced-power-diagram","An Analytical Representation of the 2d Generalized Balanced Power Diagram","","Tessellations help model microstructure in cellular and polycrystal materials. Classical Voronoi and Laguerre (power) tessellations use convex cell boundaries, limiting accuracy for anisotropic grains with curved, possibly non-convex interfaces. Elliptic-distance generalizations yield more varied structures, yet existing GBPD computations rely largely on discretized label-image forms. This work derives an analytic 2D representation of GBPD vertices and edges and proposes a complete algorithm to compute the full diagram.",{"@graph":14,"@context":72},[15,34,55],{"@type":16,"itemListElement":17},"BreadcrumbList",[18,23,27,31],{"item":19,"name":20,"@type":21,"position":22},"https://docshare.wps.com","Home","ListItem",1,{"item":24,"name":25,"@type":21,"position":26},"https://docshare.wps.com/document/","Document",2,{"item":28,"name":29,"@type":21,"position":30},"https://docshare.wps.com/document/research-report/","Research & Report",3,{"item":32,"name":10,"@type":21,"position":33},"https://docshare.wps.com/document/an-analytical-representation-of-the-2d-generalized-balanced-power-diagram/657882/",4,{"url":32,"name":10,"@type":35,"image":36,"author":41,"headline":10,"publisher":44,"fileFormat":47,"inLanguage":8,"description":12,"dateModified":48,"datePublished":49,"encodingFormat":47,"isAccessibleForFree":50,"interactionStatistic":51},"DigitalDocument",{"url":37,"@type":38,"width":39,"height":40},"https://docshare.wps.com/thumbnails/an-analytical-representation-of-the-2d-generalized-balanced-power-diagram/657882.png","ImageObject",300,407,{"name":42,"@type":43},"วิน","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-10-30","2026-10-26",true,{"@type":52,"interactionType":53,"userInteractionCount":22},"InteractionCounter",{"@type":54},"ViewAction",{"@type":56,"mainEntity":57},"FAQPage",[58,64,68],{"name":59,"@type":60,"acceptedAnswer":61},"Why are classical Voronoi and Laguerre tessellations considered too restrictive for some microstructures?","Question",{"text":62,"@type":63},"Their cells are convex polytopes, which makes them restrictive for data with anisotropic grains and curved boundaries.","Answer",{"name":65,"@type":60,"acceptedAnswer":66},"What does the generalized balanced power diagram (GBPD) build upon?",{"text":67,"@type":63},"GBPD extends the anisotropic Voronoi diagram by weighting its elliptic generators, leading to more diverse cell shapes and sizes.",{"name":69,"@type":60,"acceptedAnswer":70},"What is the main contribution of this work regarding the GBPD?",{"text":71,"@type":63},"It derives an analytic representation of GBPD vertices and edges in 2D and uses it to propose a novel algorithm to compute the entire diagram.","https://schema.org",{"og:url":32,"og:type":74,"og:title":10,"og:site_name":45,"og:description":12},"article",{"robots":76,"canonical":32},"index,follow",{"doc_id":78,"site_id":7},657882,1792993339,{"code":4,"msg":81,"data":82},"success",[83,87,91,95,100,105,110,114,119,122,126],{"id":22,"doc_module":4,"doc_module_name":25,"category_name":84,"show_sort_weight":85,"slug":86},"Story & 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Representation of the 2d Generalized Balanced Power Diagram  \nChristian Jung 1 Claudia Redenbach 1  \n1 RPTU Kaiserslautern-Landau  \nGottlieb-Daimler-Stra􀀙e 48, 67663 Kaiserslautern, Germany  \nAbtract  \nTessellations are an important tool to model the microstructure of cellular and polycrystal materials. Classical tessellation models include the Voronoi diagram and Laguerre tessellation whose cells are polyhedra. Due to the convexity of their cells, those models may be too restrictive to describe data that includes possibly anisotropic grains with curved boundaries. Several generalizations exist. The cells of the generalized balanced power diagram are induced by elliptic distances leading to more diverse structures. So far, methods for computing the generalized balanced power diagram are restricted to discretized versions in the form of label images. In this work, we derive an analytic representation of the vertices and edges of the generalized balanced power diagram in 2d. Based on that, we propose a novel algorithm to compute the whole diagram.  \nKeywords  \nVoronoi diagram, Laguerre tessellation, anisotropy, non-convexity, curved boundary, conic sections  \n1 Introduction  \nTessellations play an important role in many 􀀌elds of research including computational geometry, computer graphics, materials science, and stochastic geometry. A tessellation can be interpreted as a subdivision of space into subsets, called cells. A well-studied class of tessellations are Voronoi-like tessellations which are generated by a 􀀌nite set of points in Rd. Every point in space is assigned to the generator it is closest to w.r.t. a given distance measure d. Depending on the choice of d, a wide range of tessellation models emerges. An overview over distance functions and tessellations that have been studied in this context is given in [11] .  \nIn case of the Euclidean distance, the Voronoi diagram [30] is obtained. Assigning a weight wi 2 R >0 to every generator, the power distance between two points x; pi 2 Rdis de􀀌ned by dist(x; pi; wi) = jj x 􀀀 pi jj2 􀀀 wi. The tessellation model induced by the  \npower distance is called power diagram or Laguerre tessellation. It can be seen as a 􀀌rst generalization of the Voronoi diagram with its weights allowing to control the size of the cells. The cells of both the Voronoi diagram and the Laguerre tessellation are convex polytopes with their boundaries being subsets of hyperplanes. Algorithms for computing these models are implemented in, for example, CGAL [9], voro++ [44] or Neper [29] . Voronoi and Laguerre tessellations have been used to reconstruct and stochastically model 2d and 3d cellular materials such as polycrystalline structures in aluminum alloys or foams [25, 34, 24, 35, 43, 40, 26] . Several algorithms exist to optimize the con􀀌guration of generators such that the resulting tessellation describes the given structure as good as possible with respect to some discrepancy measure. These optimization techniques include simulated annealing [23], cross-entropy methods [41, 32] and the MetropolisHastings algorithm [39] . Furthermore, Laguerre tessellations of pre-de􀀌ned cell areas can be generated [6] .  \nHowever, the cell convexity makes these models quite restrictive when it comes to approximating real-world microstructures which may exhibit curved cell boundaries and signi􀀌cant anisotropy. Non-convex cell shapes can be obtained with the Apollonius diagram (also additively-weighted Voronoi diagram or Johnson-Mehl tessellation) . It is induced by an additively-weighted Euclidean distance. Similarly, the multiplicativelyweighted Voronoi diagram is induced by Euclidean distances that are weighted by a scalar factor. Both models have curved cell boundaries. Analytic descriptions of their cells'boundaries are known [15, 48] and algorithms for their simulation exist [12, 19, 20, 16] . The set Voronoi diagram [37] generalizes the Apollonius diagram.","cbCaitQZJmQyRzoE","https://ap.wps.com/l/cbCaitQZJmQyRzoE","pdf",944704,16,"English","# Introduction\n## Distance-based tessellations and Voronoi/Laguerre models\n## Limitations of convex cell assumptions\n## Curved-boundary generalizations (Apollonius/JM, set Voronoi, voxelized variants)\n## Elliptic distances and the anisotropic Voronoi diagram\n## Generalized Balanced Power Diagram (GBPD) and computational gap","[{\"question\":\"Why are classical Voronoi and Laguerre tessellations considered too restrictive for some microstructures?\",\"answer\":\"Their cells are convex polytopes, which makes them restrictive for data with anisotropic grains and curved boundaries.\"},{\"question\":\"What does the generalized balanced power diagram (GBPD) build upon?\",\"answer\":\"GBPD extends the anisotropic Voronoi diagram by weighting its elliptic generators, leading to more diverse cell shapes and sizes.\"},{\"question\":\"What is the main contribution of this work regarding the GBPD?\",\"answer\":\"It derives an analytic representation of GBPD vertices and edges in 2D and uses it to propose a novel algorithm to compute the entire diagram.\"}]","An Analytical Representation of the 2d Generalized Balanced Power Diagram | PDF"]