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It also notes historical origins and applications such as facility location and path planning.",{"@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/technology/","Technology",3,{"item":32,"name":10,"@type":21,"position":33},"https://docshare.wps.com/document/dirichletvoronoi-diagrams-and-delaunay-triangulations-chapter-9-overview/665658/",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/dirichletvoronoi-diagrams-and-delaunay-triangulations-chapter-9-overview/665658.png","ImageObject",300,407,{"name":42,"@type":43},"Adam","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-10-30","2026-10-29",true,{"@type":52,"interactionType":53,"userInteractionCount":26},"InteractionCounter",{"@type":54},"ViewAction",{"@type":56,"mainEntity":57},"FAQPage",[58,64,68],{"name":59,"@type":60,"acceptedAnswer":61},"What are Voronoi diagrams and Delaunay triangulations used for?","Question",{"text":62,"@type":63},"They are important tools in computational geometry. Delaunay triangulations are especially useful when fitting 3D data with surface splines, after projecting points onto the xy-plane.","Answer",{"name":65,"@type":60,"acceptedAnswer":66},"How is the bisector line of two points defined in 2D?",{"text":67,"@type":63},"For two distinct points a and b, the bisector line is the locus of points equidistant from a and b. It is the line orthogonal to the segment (a, b) and passing through its midpoint.",{"name":69,"@type":60,"acceptedAnswer":70},"What is the region V(pi) in the Dirichlet–Voronoi setting?",{"text":71,"@type":63},"V(pi) is the set of points x in the space such that d(x,pi) ≤ d(x,pj) for all other points pj. Equivalently, V(pi) is the intersection of half-planes or half-spaces H(pi,pj) over all j ≠ i.","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},665658,1793305512,{"code":4,"msg":81,"data":82},"success",[83,87,91,95,100,104,109,114,119,122,126],{"id":22,"doc_module":4,"doc_module_name":25,"category_name":84,"show_sort_weight":85,"slug":86},"Story & Novel",90,"story-novel",{"id":26,"doc_module":4,"doc_module_name":25,"category_name":88,"show_sort_weight":89,"slug":90},"Literature",80,"literature",{"id":33,"doc_module":4,"doc_module_name":25,"category_name":92,"show_sort_weight":93,"slug":94},"Exam",70,"exam",{"id":96,"doc_module":4,"doc_module_name":25,"category_name":97,"show_sort_weight":98,"slug":99},5,"Comic",60,"comic",{"id":101,"doc_module":4,"doc_module_name":25,"category_name":29,"show_sort_weight":102,"slug":103},6,50,"technology",{"id":105,"doc_module":4,"doc_module_name":25,"category_name":106,"show_sort_weight":107,"slug":108},7,"Healthcare",40,"healthcare",{"id":110,"doc_module":4,"doc_module_name":25,"category_name":111,"show_sort_weight":112,"slug":113},8,"Research & Report",30,"research-report",{"id":115,"doc_module":4,"doc_module_name":25,"category_name":116,"show_sort_weight":117,"slug":118},9,"Religion & Spirituality",20,"religion-spirituality",{"id":117,"doc_module":4,"doc_module_name":25,"category_name":120,"show_sort_weight":117,"slug":121},"World Cup","world-cup",{"id":123,"doc_module":4,"doc_module_name":25,"category_name":124,"show_sort_weight":123,"slug":125},10,"Lifestyle","lifestyle",{"id":127,"doc_module":4,"doc_module_name":25,"category_name":128,"show_sort_weight":96,"slug":129},19,"General","general",{"code":4,"msg":81,"data":131},{"doc_id":78,"user_id":132,"nickname":42,"user_avatar":133,"doc_module":4,"category_id":101,"category_name":29,"doc_title":10,"doc_description":12,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":26,"is_deleted":4,"is_public":22,"is_downloadable":22,"audit_status":22,"page_count":139,"language":140,"language_code":8,"site_id":7,"html_lang":8,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":12,"update_tm":79,"read_time":144},1374404737137,"https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d","Chapter 9  \nDirichlet–Voronoi Diagrams and Delaunay Triangulations  \n9.1 Dirichlet–Voronoi Diagrams  \nIn this chapter we present very briefly the concepts of a Voronoi diagram and of a Delaunay triangulation.  \nThese are important tools in computational geometry, and Delaunay triangulations are important in problems where it is necessary to fit 3D data using surface splines.  \nIt is usually useful to compute a good mesh for the projection of this set of data points onto the xy-plane, anda Delaunay triangulation is a good candidate.  \nOur presentation will be rather sketchy. We are primarily interested in defining these concepts and stating their most important properties without proofs.  \n390 CHAPTER 9 . DIRICHLET–VORONOI DIAGRAMS  \nFor a comprehensive exposition of Voronoi diagrams, Delaunay triangulations, and more topics in computational geometry, consult O’Rourke [?], Preparata and Shamos [?], Boissonnat and Yvinec [?], de Berg, Van Kreveld, Overmars, and Schwarzkopf [?], or Risler [?] .  \nThe survey by Graham and Yao [?] contains a very gentle and lucid introduction to computational geometry.  \nFor concreteness, one may safely assume that we work in the affine space E = Em , although what follows applies to any Euclidean space of finite dimension.  \nGiven a set P = {p1 ,..., pn } of n points in E, it is often useful to find a partition of the space E into regions each containing a single point of P and having some nice properties.  \n9.1. DIRICHLET–VORONOI DIAGRAMS 391 It is also often useful to find triangulations of the convex hull of P having some nice properties.  \nWe shall see that this can be done and that the two problems are closely related. In order to solve the first problem, we need to introduce bisector lines and bisector planes.  \nFor simplicity, let us first assume that E is a plane i.e. , has dimension 2 .  \nGiven any two distinct points a, b ∈ E, the line orthogonal to the line segment (a, b) and passing through the midpoint of this segment is the locus of all points having equal distance to a and b.  \nIt is called the bisector line of a and b. The bisector line of two points is illustrated in Figure 9 .1.  \n392 CHAPTER 9 . DIRICHLET–VORONOI DIAGRAMS  \nFigure 9 . 1: The bisector line L of a and b  \nIf h = ~~1~~2 a + ~~1~~2 b is the midpoint of the line segment (a, b), letting m be an arbitrary point on the bisector line, the equation of this line can be found by writing that hm is orthogonal to ab.  \nIn any orthogonal frame, letting m = (x, y ), a = (a1 , a 2 ), b = (b1 , b2 ), the equation of this line can be written as (b1 − a1 )x +(b2 − a2 )y = (b21 + b22) /2 − (a21 + a 22) /2 .  \n9.1. DIRICHLET–VORONOI DIAGRAMS 393 The closed half-plane H(a, b)containing a and with boundary the bisector line is the locus of all points such that (b1 − a1 )x +(b2 − a2 )y ≤ (b21 + b22) /2 − (a21 + a 22) /2 , and the closed half-plane H (b, a) containing b and with boundary the bisector line is the locus of all points such that  \n(b1 − a1 )x +(b2 − a2 )y ≥ (b21 + b22) /2 − (a21 + a 22) /2 .  \nThe closed half-plane H(a, b)is the setof all points whose distance to a is less that or equal to the distance to b, and vice versa for H (b, a) . Thus, points in the closed half-plane H(a, b) are closer to a than they are to b.  \n394 CHAPTER 9 . DIRICHLET–VORONOI DIAGRAMS We now consider a problem called the post office problem by Graham and Yao [?] .  \nGiven any set P = {p1 ,..., pn } of n points in the plane (considered as post offices or sites), for any arbitrary point x, find out which post office is closest to x.  \nSince x can be arbitrary, it seems desirable to precompute the sets V(pi ) consisting of all points that are closer to pi than to any other point pj  pi.  \nIndeed, if the sets V (pi ) are known, the answer is any post office pi such that x ∈ V (pi) .  \nThus, it remains to compute the sets V (pi) . For this, if x is closer to pi than to any other point pj  pi , then x is on the same side as pi with respect to the bisector line of pi and p","cbCaidzI4KRKS7Ja","https://ap.wps.com/l/cbCaidzI4KRKS7Ja","pdf",275125,52,"English","# 9.1 Dirichlet–Voronoi Diagrams\n## Bisector lines and half-planes\n## Bisector planes and half-spaces\n## Voronoi regions V(pi) and closest-site partition\n## Historical context and applications","[{\"question\":\"What are Voronoi diagrams and Delaunay triangulations used for?\",\"answer\":\"They are important tools in computational geometry. Delaunay triangulations are especially useful when fitting 3D data with surface splines, after projecting points onto the xy-plane.\"},{\"question\":\"How is the bisector line of two points defined in 2D?\",\"answer\":\"For two distinct points a and b, the bisector line is the locus of points equidistant from a and b. It is the line orthogonal to the segment (a, b) and passing through its midpoint.\"},{\"question\":\"What is the region V(pi) in the Dirichlet–Voronoi setting?\",\"answer\":\"V(pi) is the set of points x in the space such that d(x,pi) ≤ d(x,pj) for all other points pj. Equivalently, V(pi) is the intersection of half-planes or half-spaces H(pi,pj) over all j ≠ i.\"}]","Dirichlet–Voronoi Diagrams and Delaunay Triangulations - Chapter 9 - Overview | PDF",131]