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It introduces generators as a finite set of points in a metric space, defines Voronoi regions via distance comparisons, and states that planar regions are convex polytopes under Euclidean distance. The document also demonstrates practical Python workflows, including plotting generators, computing Voronoi diagrams, and using Delaunay triangulation for related computations.",{"@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/spherical-voronoi-diagrams-a-pydata-london-story-unfolded/665408/",{"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/spherical-voronoi-diagrams-a-pydata-london-story-unfolded/665408.png","ImageObject",300,407,{"name":92,"@type":93},"bocahgabuttt","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-30","2026-10-29",true,{"@type":102,"interactionType":103,"userInteractionCount":81},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What is a Voronoi diagram and how are Voronoi regions defined?","Question",{"text":112,"@type":113},"Given a metric space (X, d) and a finite set of generator points p, the k-th Voronoi region consists of all points closer to pk than to any other generator. The collection of these regions forms the Voronoi diagram.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"What geometric property holds for planar Voronoi regions in Euclidean space?",{"text":117,"@type":113},"If the metric space is R2 with the Euclidean distance, every Voronoi region in a Voronoi diagram is a convex polytope.",{"name":119,"@type":110,"acceptedAnswer":120},"How does the document compute and visualize Voronoi diagrams using Python?",{"text":121,"@type":113},"It plots generator points, then uses SciPy’s Voronoi computation to generate the Voronoi diagram and visualizes the result with plotting routines. It also demonstrates Delaunay triangulation as part of the computational workflow.","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},665408,1793302648,{"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":81,"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},962094248839,"https://ap-avatar.wpscdn.com/davatar_994ba38a5ba835b3df7d355c54d3ed8d","Spherical Voronoi Diagrams A PyData London story unfolded  \nNikolai Nowaczyk  \n08/05/2016  \n Nikolai Nowaczyk  Voronoi  08/05/2016 1 / 31   \n1 Planar Voronoi Diagrams  \n Example: Traveling in London  Formal description  \n Realization in Python  \n2 Spherical Voronoi Diagrams  \n Example: Traveling around the world  Problem description  \n PyData story  \n Realization in Python  \n Applications  \n3 Appendix  \nNikolai Nowaczyk  Voronoi  08/05/2016 2 / 31   \nOutline  \n1 Planar Voronoi Diagrams  \n Example: Traveling in London  Formal description  \n Realization in Python  \n2 Spherical Voronoi Diagrams  \n Example: Traveling around the world  Problem description  \n PyData story  \n Realization in Python  \n Applications  \n3 Appendix  \nNikolai Nowaczyk  Voronoi  08/05/2016 3 / 31   \nVoronoi diagrams in the plane  \nInput: Generator Points  \n Nikolai Nowaczyk  Voronoi  08/05/2016 4 / 31   \nVoronoi diagrams in the plane  \n[http://alexbeutel.com/webgl/voronoi.html](http://alexbeutel.com/webgl/voronoi.html)  \nOutput: Voronoi Regions (coloured)  \n Nikolai Nowaczyk  Voronoi  08/05/2016 4 / 31   \nMathematical definition  \nDefinition (Voronoi diagram)  \nLet (X ; d) be a metric space (for instance R2 ) and p = (p 1 ; : : : ; pK ) be a finite collection of K distinct points called generators.  \n1 For each 1 􀀔 k 􀀔 K , the set  \nRk := fx 2 X j 8j  k : d (x ; pk) 􀀔 d (x ; pj)g  \nis the k-th Voronoi region.  \n2 The collection (Rk)1􀀔k􀀔K is the Voronoi diagram defined by p.  \nNikolai Nowaczyk  Voronoi  08/05/2016 5 / 31   \nMathematical definition  \nDefinition (Voronoi diagram)  \nLet (X ; d) be a metric space (for instance R2 ) and p = (p 1 ; : : : ; pK ) be a finite collection of K distinct points called generators.  \n1 For each 1 􀀔 k 􀀔 K , the set  \nRk := fx 2 X j 8j  k : d (x ; pk) 􀀔 d (x ; pj)g  \nis the k-th Voronoi region.  \n2 The collection (Rk)1􀀔k􀀔K is the Voronoi diagram defined by p.  \nTheorem  \nIf (X ; d) is R2 with the Euclidean distance function, then every Voronoi region of every Voronoi diagram is a convex polytope.  \nNikolai Nowaczyk  Voronoi  08/05/2016 5 / 31   \nPython: Plotting the generator points  \np o i n t s = numpy . a r r a y ([[ 1 . 5 , 1 . ] , . . . ] ) imp ort m a t p l o t l i b . py p l o t a s p l t  \np l t . p l o t ( p o i n t s [ : , 0 ] , p o i n t s [ : , 1 ] , ' o ' , c o l o r= ' b l a ck ' ) [p l t . show](p l t . show) ( )  \nNikolai Nowaczyk  Voronoi  08/05/2016 6 / 31   \nPython: Calculating Voronoi diagram  \nfrom s c i py . s p a t i a l import Vo ro n o i , vo ro n o i _ p lot _ 2 dv o r = V o r o n o i ( p o i n t s )  \nvo ro n o i _ p lot _ 2 d ( vo r , ax ) [p l t . show](p l t . show) ( )  \nNikolai Nowaczyk  Voronoi  08/05/2016 7 / 31   \nPython: Delaunay triangulation  \nfrom s c i py . s p a t i a l import De l a u n ay t r i = De l a u n ay ( p o i n t s )  \nax . t r i p l o t ( p o i n t s [ : , 0 ] , p o i n t s [ : , 1 ] , t r i . s i m p l i c e s . co py ( ) , c o l o r = ' b lu e ' ) [p l t . show](p l t . show) ( )  \nNikolai Nowaczyk  Voronoi  08/05/2016 8 / 31","cbCaichR7AxSZ1Z7","https://ap.wps.com/l/cbCaichR7AxSZ1Z7","pdf",1909981,35,"English","# Planar Voronoi Diagrams\n## Example: Traveling in London\n## Formal description\n## Realization in Python\n# Spherical Voronoi Diagrams\n## Example: Traveling around the world\n## Problem description\n## PyData story\n## Realization in Python\n## Applications\n# Appendix\n## Outline","[{\"question\":\"What is a Voronoi diagram and how are Voronoi regions defined?\",\"answer\":\"Given a metric space (X, d) and a finite set of generator points p, the k-th Voronoi region consists of all points closer to pk than to any other generator. The collection of these regions forms the Voronoi diagram.\"},{\"question\":\"What geometric property holds for planar Voronoi regions in Euclidean space?\",\"answer\":\"If the metric space is R2 with the Euclidean distance, every Voronoi region in a Voronoi diagram is a convex polytope.\"},{\"question\":\"How does the document compute and visualize Voronoi diagrams using Python?\",\"answer\":\"It plots generator points, then uses SciPy’s Voronoi computation to generate the Voronoi diagram and visualizes the result with plotting routines. It also demonstrates Delaunay triangulation as part of the computational workflow.\"}]","Spherical Voronoi Diagrams - A PyData London story unfolded | PDF",88]