[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-665660-105":59,"doc-detail-665660-en":130},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":123,"head_meta":125,"extra_data":127,"updated_unix":129},105,"en","voronoi-diagrams-and-fortunes-algorithm-lecture-11","Voronoi Diagrams and Fortune's Algorithm - Lecture 11","","CMSC 754 Lecture 11 introduces Voronoi diagrams and their role in computational geometry, defining Voronoi cells as regions closer to a given site than to any other. It explains why cells form (possibly unbounded) convex polyhedra via halfspaces and perpendicular bisectors, and highlights key applications including nearest-neighbor queries, computational morphology and medial axes, and center-based clustering. The lecture further covers natural neighbor interpolation and core planar properties: empty-circle behavior, Voronoi vertices, degree under general position, and the link between unbounded cells and the convex hull.",{"@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":20,"@type":76,"position":81},"https://docshare.wps.com/document/exam/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/voronoi-diagrams-and-fortunes-algorithm-lecture-11/665660/",{"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/voronoi-diagrams-and-fortunes-algorithm-lecture-11/665660.png","ImageObject",300,407,{"name":92,"@type":93},"Pentious","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-30","2026-10-29",true,{"@type":102,"interactionType":103,"userInteractionCount":8},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What is a Voronoi cell and how is it defined?","Question",{"text":112,"@type":113},"A Voronoi cell VP(pi) is the set of points closer to site pi than to any other site in P. It is defined using Euclidean distance comparisons against every other site.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"Why are Voronoi cells convex polyhedra?",{"text":117,"@type":113},"Points strictly closer to pi than to pj form an open halfspace bounded by the perpendicular bisector of pi and pj. A Voronoi cell equals the intersection of such halfspaces over all j≠i, and intersections of convex objects remain convex.",{"name":119,"@type":110,"acceptedAnswer":120},"How are nearest-neighbor queries answered using a Voronoi diagram?",{"text":121,"@type":113},"Compute the Voronoi diagram for P, then locate the Voronoi cell that contains the query point q. The generating site of that cell is the closest point in P to q.","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},665660,1793305517,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":8,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":44,"language":139,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":140,"faqs":141,"seo_title":142,"seo_description":67,"update_tm":129,"read_time":143},1374404730887,"https://ap-avatar.wpscdn.com/davatar_6f874abed73319feea01a86fa6f0fab8","CMSC 754: Lecture 11  \nVoronoi Diagrams and Fortune's Algorithm  \nReading: Chapter 7 in the 4M's. A nice applet illustrating the execution of this algorithm can be  \nfound at [http://www.raymondhill.net/voronoi/rhill-voronoi.html](http://www.raymondhill.net/voronoi/rhill-voronoi.html).  \nVoronoi Diagrams: Voronoi diagrams are among the most important structures in computational geometry. Throughout, let  \nkp 􀀀 qk = Xi1 (pi 􀀀 qi)2 ! 1=2  \ndenote the standard Euclidean distance between two points p; q 2 Rd. Let P = fp1 ; p2 ; : : : ; png be a set of points in Rd , which we call sites. De􀀌ne VP(pi), called the Voronoi cell, for pi, to  \nbe the set of points q in space that are closer to pi than to any other site, that is, VP(pi) = fq 2 Rd : kpi 􀀀 q k \u003C kpj 􀀀 q k; 8j  ig;  \nWhen P is clear from context, we will omit it and refer to this simply as V (pi) . Clearly, the Voronoi cells of two distinct points of P are disjoint. The union of the closure of the Voronoi cells de􀀌nes a cell complex, which is called the Voronoi diagram of P , and is denoted Vor(P )  \n(see Fig. 1(a)) .  \n(a) (b) Fig. 1: Voronoi diagram Vor(P ) of a set of points.  \nThe cells of the Voronoi diagram are (possibly unbounded) convex polyhedra. To see this, observe that the set of points that are strictly closer to one site pi than to another site pj is equal to the open halfspace whose bounding hyperplane is the perpendicular bisector between pi and pj . Denote this halfspace h (pi; pj) . It is easy to see that a point q lies in V (pi) if and only if q lies within the intersection of h(pi; pj) for all j  i. In other words,  \nV (pi) =\\ h(pi; pj)  \nj i  \nLecture 11 1 Spring 2020  \nCMSC 754 Dave Mount  \n(see Fig. 1(b)) . Since the intersection of convex objects is convex, V (pi) is a (possibly unbounded) convex polyhedron.  \nVoronoi diagrams have a huge number of important applications in science and engineering. These include answering nearest neighbor queries, computational morphology and shape analysis, clustering and data mining, facility location, multi-dimensional interpolation.  \nNearest neighbor queries: Given a point set P , we wish to preprocess P so that, given a query point q, it is possible to quickly determine the closest point of P to q. This can be answered by 􀀌rst computing a Voronoi diagram and then locating the cell of the diagram that contains q. (In the plane, this can be done by building the trapezoidal map of the edges of the Voronoi diagram. Each trapezoid lies within a single Voronoi cell, and can be labeled with the generating point.)  \nComputational morphology and shape analysis: A useful structure in shape analysis is called the medial axis. The medial axis of a shape (e.g., a simple polygon) is de􀀌ned tobe the union of the center points of all locally maximal disks that are contained within the shape (see Fig. 2) . If we generalize the notion of Voronoi diagram to allow sites that are both points and line segments, then the medial axis of a simple polygon can be extracted easily from the Voronoi diagram of these generalized sites.  \n(a) (b) (c)  \nFig. 2: (a) A simple polygon, (b) its medial axis and a sample maximal disk, and (c) center-based clustering (with cluster centers shown as black points) .  \nCenter-based Clustering: Given a set P of points, it is often desirable to represent the union of a signi􀀌cantly smaller set of clusters. In center-based clustering, the clusters are de􀀌ned by a set C of cluster centers (which may or may not be required to be chosen from P ) . The cluster associated with a given center point q 2 C is just the subset of points of P that are closer to q than any other center, that is, the subset of P that lies within q's Voronoi cell (see Fig. 2(c)) . (How the center points are selected is another question.)  \nNeighbors and Interpolation: Given a set of measured height values over some geometric terrain. Each point has (x; y) coordinates and a height value. We would like to interpolate the height value of so","cbCaiejNMNreezwT","https://ap.wps.com/l/cbCaiejNMNreezwT","pdf",388167,"English","# Voronoi Diagrams and Fortune's Algorithm - Lecture 11\n## Voronoi diagram definitions and cell complex\n## Applications: nearest neighbors, morphology, clustering, interpolation\n## Properties in the plane: empty circles, vertices, degree, convex hull","[{\"question\":\"What is a Voronoi cell and how is it defined?\",\"answer\":\"A Voronoi cell VP(pi) is the set of points closer to site pi than to any other site in P. It is defined using Euclidean distance comparisons against every other site.\"},{\"question\":\"Why are Voronoi cells convex polyhedra?\",\"answer\":\"Points strictly closer to pi than to pj form an open halfspace bounded by the perpendicular bisector of pi and pj. A Voronoi cell equals the intersection of such halfspaces over all j≠i, and intersections of convex objects remain convex.\"},{\"question\":\"How are nearest-neighbor queries answered using a Voronoi diagram?\",\"answer\":\"Compute the Voronoi diagram for P, then locate the Voronoi cell that contains the query point q. The generating site of that cell is the closest point in P to q.\"}]","Voronoi Diagrams and Fortune's Algorithm - Lecture 11 | PDF",23]