[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-665659-105":59,"doc-detail-665659-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-delaunay-triangulations-orourke-chapter-5-outline","Voronoi Diagrams - Delaunay Triangulations - O’Rourke, Chapter 5 - Outline","","Chapter 5 outline on Voronoi Diagrams and Delaunay Triangulations, beginning with planar-graph preliminaries. It proves Euler’s formula for connected planar graphs using the cases of trees and induction, then discusses triangulating planar graphs by adding a virtual point outside to convert faces into triangles. The chapter connects structural planar constraints to bounds on edges and faces and introduces nearest-neighbor graphs for a set of points in Euclidean space, including a basic computation-time estimate.",{"@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":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/voronoi-diagrams-delaunay-triangulations-orourke-chapter-5-outline/665659/",{"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-delaunay-triangulations-orourke-chapter-5-outline/665659.png","ImageObject",300,407,{"name":92,"@type":93},"Cipher","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-30","2026-10-29",true,{"@type":102,"interactionType":103,"userInteractionCount":14},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What does the chapter outline cover before Voronoi and Delaunay concepts?","Question",{"text":112,"@type":113},"It starts with planar-graph preliminaries, including a claim about Euler’s formula and its proofs, then explains triangulation of planar graphs and resulting edge/face bounds.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"How is Euler’s formula justified in the preliminaries section?",{"text":117,"@type":113},"The outline shows it holds for trees, then uses induction: if the graph has a cycle, remove an edge on the cycle to reduce vertices/edges/faces while preserving the relation through the inductive step.",{"name":119,"@type":110,"acceptedAnswer":120},"How does the outline transform a planar graph into one with triangular faces?",{"text":121,"@type":113},"It triangulates interior polygons, adds a virtual point outside, and triangulates the exterior polygon so the new planar graph has controlled changes in vertices, edges, and faces.","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},665659,1793305513,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":14,"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":129,"read_time":144},687208528416,"https://ap-avatar.wpscdn.com/davatar_9964176cb1d06d4a9deccf72a44ae3dc","# Voronoi Diagrams\n\nand  \nDelaunay Triangulations  \nO’Rourke, Chapter 5  \nOutline  \n• Preliminaries  \n• Voronoi Diagrams / Delaunay Triangulations  \n• Lloyd’s Algorithm  \n| Preliminaries   \u003Cbr>Claim:  \u003Cbr>Given a connected planar graph with 􀜸 vertices, 􀜧edges, and 􀜨 faces * , the graph satisfies:  \u003Cbr>􀜸 − 􀜧 + 􀜨 = 2   |\n| --- |\n| \u003Cbr>|\n| *The “external” face also counts. (Can think of this as a graph on the sphere.)   |\n\n\n| Preliminaries  \u003Cbr>Proof:  \u003Cbr>1. Show that this is true for trees.  \u003Cbr>2. Show that this is true by induction.   | |\n| --- | --- |\n| \u003Cbr>|  |\n\n\n| Preliminaries  \u003Cbr>Proof (for Trees) :  \u003Cbr>If a graph is a connected tree, it satisfies:  \u003Cbr>􀜸 = 􀜧 + 1 .   | |\n| --- | --- |\n| Since there is only one (external) face:  \u003Cbr>􀜸 − 􀜧 + 􀜨 = (􀜧 + 1) − 􀜧 + 1  = 2    \u003Cbr>     |  |\n\n## Preliminaries\n\nProof (by Induction) :  \nSuppose that we are given a graph 􀜩 .  \n􀀤 If it’s a tree, we are done.  \n􀀤 Otherwise, it has a cycle.  \nRemoving an edge on the cyclegives a graph 􀜩′ with:  \n􀀤 The same vertex set (􀜸′ = 􀜸)  \n􀀤 One less edge (􀜧′ = 􀜧 − 1)  \n􀀤 One less face (􀜨′ = 􀜨 − 1)  \nBy induction:  \n2 = 􀜸′ − 􀜧′ + 􀜨′ = 􀜸 − 􀜧 + 􀜨  \n## Preliminaries \n\nNote:  \nGiven a planar graph 􀜩 , we can get a planargraph 􀜩 ′ with triangle faces:  \n􀀤 Triangulate the interior polygons  \n􀀤 Add a“virtual point” outside  \nand triangulate the  \nexterior polygon.  \n## Preliminaries\n\nNote:  \nThe new graph has:  \n􀀤 􀜸′ = 􀜸 + 1 , 􀜧′ ≥ 􀜧 , 􀜨′ ≥ 􀜨  \n􀀤 􀜸′ − 􀜧′ + 􀜨′ = 2  \n􀀤 3􀜧′ = 2􀜨′  \n| Preliminaries  \u003Cbr>Note:  \u003Cbr>The new graph has:  \u003Cbr>􀀤 􀜸′ = 􀜸 + 1 , 􀜧′ ≥ 􀜧 , 􀜨′ ≥ 􀜨  \u003Cbr>􀀤 􀜸′ − 􀜧′ + 􀜨′ = 2  \u003Cbr>􀀤 3􀜧′ = 2􀜨′  \u003Cbr>This gives:  \u003Cbr>􀜧′ = 3􀜸′ − 6⇓  \u003Cbr>􀜧 ≤ 3􀜸 − 3   |  | 􀜨′ = 2􀜸′ − 4  \u003Cbr>⇓  \u003Cbr>􀜨 ≤ 2􀜸 − 2   | |\n| --- | --- | --- | --- |\n| The number of edges/facesofa planar graph islinear in the number of vertices.   |  |  |  |\n\n## Preliminaries\n\nDefinition:  \nGiven a set of points {􀝌1, … , 􀝌 􀯡} ⊂ ℝ􀯗 , the nearest-neighbor graph is the directed graph with an edgefrom 􀝌 􀯜 to 􀝌 􀯝 , whenever:  \n􀝌 􀯞 − 􀝌 􀯜 ≥ 􀝌 􀯝 − 􀝌 􀯜 ∀1 ≤ 􀝇 ≤ 􀝊 .  \n   \nNaively, the nearest-neighborcan be computed in 􀜱(􀝊2 ) timeby testing all possible neighbors.","cbCaiogeIoFvN3BA","https://ap.wps.com/l/cbCaiogeIoFvN3BA","pdf",2071907,52,"English","# Preliminaries\n## Claim and proof of Euler’s formula\n## Proof for trees\n## Proof by induction (cycle edge removal)\n## Triangulating a planar graph\n## Nearest-neighbor graph definition and computation cost","[{\"question\":\"What does the chapter outline cover before Voronoi and Delaunay concepts?\",\"answer\":\"It starts with planar-graph preliminaries, including a claim about Euler’s formula and its proofs, then explains triangulation of planar graphs and resulting edge/face bounds.\"},{\"question\":\"How is Euler’s formula justified in the preliminaries section?\",\"answer\":\"The outline shows it holds for trees, then uses induction: if the graph has a cycle, remove an edge on the cycle to reduce vertices/edges/faces while preserving the relation through the inductive step.\"},{\"question\":\"How does the outline transform a planar graph into one with triangular faces?\",\"answer\":\"It triangulates interior polygons, adds a virtual point outside, and triangulates the exterior polygon so the new planar graph has controlled changes in vertices, edges, and faces.\"}]","Voronoi Diagrams - Delaunay Triangulations - O’Rourke, Chapter 5 - Outline | PDF",131]