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It motivates the problems of partitioning space into regions associated with points and constructing triangulations of the convex hull with desirable properties. The chapter defines bisector lines and half-planes, extends the idea to bisector planes and half-spaces in higher dimensions, and explains key relationships underpinning Voronoi cells and Delaunay triangulations, including geometric projections tied to convex hulls.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":30,"@type":76,"position":81},"https://docshare.wps.com/document/technology/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/chapter-8-dirichletvoronoi-diagrams-8-fundamental-concepts/657522/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/chapter-8-dirichletvoronoi-diagrams-8-fundamental-concepts/657522.png","ImageObject",300,407,{"name":92,"@type":93},"Sophia Brooks","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-29","2026-10-26",true,{"@type":102,"interactionType":103,"userInteractionCount":19},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What is the main purpose of Voronoi diagrams and Delaunay triangulations in this chapter?","Question",{"text":112,"@type":113},"They are introduced as important tools in computational geometry, used to partition space into regions for sites and to construct triangulations of convex hulls with useful geometric properties.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"How are bisector lines defined for two distinct points in a plane?",{"text":117,"@type":113},"For points a and b, the bisector line is the set of points orthogonal to segment (a,b) that passes through its midpoint, consisting of points equidistant from a and b.",{"name":119,"@type":110,"acceptedAnswer":120},"What changes when moving from 2D to 3D in the bisector construction?",{"text":121,"@type":113},"The locus of points equidistant from two points becomes a bisector plane, obtained by an orthogonality condition similar in spirit to the 2D bisector line derivation.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},657522,1793097740,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":29,"category_name":30,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":19,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":139,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":144,"read_time":145},962084925636,"https://ap-avatar.wpscdn.com/davatar_994ba38a5ba835b3df7d355c54d3ed8d","Chapter 8  \nDirichlet–Voronoi Diagrams and Delaunay Triangulations  \n8.1 Dirichlet–Voronoi Diagrams  \nIn this chapter we present the concepts of a Voronoi diagram and of a Delaunay triangulation. These are important tools in computational geometry and Delaunay triangulations are important in problems where it is necessary to fit 3D data using surface splines. It is usually useful to compute a good mesh for the projection of this set of data points onto the xy-plane, and a Delaunay triangulation is a good candidate.  \nOur presentation of Voronoi diagrams and Delaunay triangulations is far from thorough. We are primarily interested in defining these concepts and stating their most important properties. For a comprehensive exposition of Voronoi diagrams, Delaunay triangulations, and more topics in computational geometry, our readers may consult O’Rourke [31], Preparata and Shamos [32], Boissonnat and Yvinec [8], de Berg, Van Kreveld, Overmars, and Schwarzkopf [5], or Risler [33] . The survey by Graham and Yao [23] contains a very gentle and lucid introduction to computational geometry.  \nIn Section 8.6 (which relies on Section 8.5), we show that the Delaunay triangulation of a set of points, P, is the stereographic projection of the convex hull of the set of points obtained by mapping the points in P onto the sphere using inverse stereogrgaphic projection. We also prove that the Voronoi diagram of P is obtained by taking the polar dual of the above convex hull and projecting it from the north pole (back onto the hyperplane containing P) . A rigorous proof of this second fact is not trivial because the central projection from the north pole is only a partial map. To give a rigorous proof, we have to use projective completions. But then, we need to define what is a convex polyhedron in projective space and for this, we use the results of Chapter 5 (especially, Section 5.2) .  \nSome practical applications of Voronoi diagrams and Delaunay triangulations are briefly discussed in Section 8 .7.  \nLet E be a Euclidean space of finite dimension, that is, an affine space E whose underlying  \n146 CHAPTER 8 . DIRICHLET–VORONOI DIAGRAMS  \nFigure 8 . 1: The bisector line L of a and b  \n−→  \nvector space E is equipped with an inner product (and has finite dimension) . For concreteness, one may safely assume that E = Em , although what follows applies to any Euclidean space of finite dimension. Given 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. It is also often useful to find triangulations of the convex hull of P having some nice properties. We 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 . Given 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. It is called the bisector line of a and b. The bisector line of two points is illustrated in Figure 8.1.  \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. In any orthogonal frame, letting m = (x, y ), a = (a1 , a2 ), b = (b1 , b2 ), the equation of this line is  \n(b1 − a1 )(x − (a1 + b1 )/2) + (b2 − a2 )(y − (a2 + b2 )/2) = 0 , which can also be written as  \n(b1 − a1 )x +(b2 − a2 )y = (b21 + b22)/2 − (a21 + a22)/2 .  \nThe closed half-plane H (a, b) containing a 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 + a22)/2 ,  \n8.1. DIRICHLET–VORONOI DIAGRAMS 147  \nand the closed half-plane H (b, a) containing b and with bou","cbCaibrTikcvo3iV","https://ap.wps.com/l/cbCaibrTikcvo3iV","pdf",604286,49,"English","# 8.1 Dirichlet–Voronoi Diagrams\n## Voronoi diagrams and Delaunay triangulations overview\n## Bisector lines and half-planes in 2D\n## Post office problem and Voronoi regions V(pi)\n## Bisector planes and half-spaces in 3D\n## Applications and further sections (8.5–8.7)","[{\"question\":\"What is the main purpose of Voronoi diagrams and Delaunay triangulations in this chapter?\",\"answer\":\"They are introduced as important tools in computational geometry, used to partition space into regions for sites and to construct triangulations of convex hulls with useful geometric properties.\"},{\"question\":\"How are bisector lines defined for two distinct points in a plane?\",\"answer\":\"For points a and b, the bisector line is the set of points orthogonal to segment (a,b) that passes through its midpoint, consisting of points equidistant from a and b.\"},{\"question\":\"What changes when moving from 2D to 3D in the bisector construction?\",\"answer\":\"The locus of points equidistant from two points becomes a bisector plane, obtained by an orthogonality condition similar in spirit to the 2D bisector line derivation.\"}]","Chapter 8 - Dirichlet–Voronoi Diagrams - 8 - Fundamental Concepts | PDF",1792981955,123]