Voronoi in Toronto and Beyond - Mapping the Spatial Web

William B. Davis | @willy_maps

William is an information designer based in Toronto and co-founder of mapto.ca. He has a background in GIS and journalism and is currently working at Sun Life Financial.





This template is for data journalists and digital storytellers of any kind. No coding experience is required. If you are planning to include some custom map layers, you will need some familiarity with Mapbox Studio.

To configure and publish a story, you will need:

• A Mapbox access token. Sign up for a free account at mapbox.com to get one.

• A text editor. Atom, Sublime Text, and Visual Studio Code are all fine choices.


Jonathan Critchley | @Afrikanadese‍

Jonathan is a software developer based in Halifax, Nova Scotia, with a background in Urban Planning and GIS. He currently works for RATIO.CITY, a Toronto based technology company that helps city builders make data-driven decisions for land acquisition using maps to source, analyze and validate sites.



This interactive web map was created for Lightsavers using NASA World at Night map tiles and the Carto service.

The map displays projects across Canada where LED streetlight conversions and streetlight control systems have been completed, are being installed or have been approved for installation.

Users can click on a city to learn more about how many streetlights are LED, the energy and maintenance savings and potential greenhouse gas reductions, and if adaptive controls have been integrated.

The project required the design and creation of custom popups including icons.






Steven Fortune

Bell Laboratories

I am Technical Manager of the Algorithms Research group within the Computing Systems Principles department of Bell Laboratories, Alcatel-Lucent. Our group performs fundamental research in theoretical computer science, particularly algorithms. We also engage in practical algorithmic work relevant to Alcatel-Lucent’s technical problems.

Much of my past research has been in computational geometry, particularly robustness of geometric algorithms. For several years I worked on the WISE project, which uses geometric algorithms to predict radio propagation in indoor and outdoor environments. Other work includes a particularly efficient algorithm for approximating the roots of ill-conditioned, high-degree polynomials. More recent work includes the algorithmic innards of design tools for state-of-the-art optical networks.

Bibliography in bibtex format or html format. Recent Papers, most with links to gzipped postscript files.

I am associate editor of SIAM Journal on Computing and International Journal of Computational Geometry with Applications (and previously ACM Transactions on Graphics).


Patricio Gonzalez Vivo (1982, Buenos Aires, Argentina) is a New York based artist and developer. He explores interstitial spaces between organic and synthetic, analog and digital, individual and collective. In his work he uses code as an expressive language with the intention of developing a better together.

Patricio studied and practiced psychotherapy and expressive art therapy. He holds an MFA in Design & Technology from Parsons The New School, where he now teaches. Currently he works as a Graphic Engineer at Mapzen making openSource mapping tools.

WebSite - Twitter - GitHub - Vimeo - Flickr

Jen Lowe is an independent data scientist and data communicator at Datatelling where she brings together people + numbers + words. She teaches in SVA’s Design for Social Innovation program, cofounded the School for Poetic Computation, taught Math for Artists at NYU ITP, researched at the Spatial Information Design Lab at Columbia University, and contributed ideas at the White House Office of Science and Technology Policy. She’s spoken at SXSW and Eyeo. Her work has been covered by The New York Times and Fast Company. Her research, writing, and speaking explore the promises and implications of data and technology in society. She has a B.S. in Applied Math and a Master’s in Information Science. Often oppositional, she’s always on the side of love.




VORONOI DIAGRAMS: Which district school is closest to your home?

Marylu Tyndell

General Information

The Problem
Recently, school transportation in NJ had become an issue. The voters in each district were being given the opportunity to decide whether hazardous busing should be included in the school budget. “Hazardous busing” is considered to be the busing of students who live within a 1.5 mile radius of their elementary school and whose walk to school would be along a dangerous route, i.e., major highways, roads without sidewalks, etc. If the majority of the voters chose not to include this hazardous busing, the students would be forced to walk to school or to provide their own transportation.

The problem, therefore, would be to maximize the number of students who would have to walk to the schools and to minimize the walks for the students in the hazardous busing category. In other words, separate the sending districts into school zones according to a Voronoi diagram, where all homes in each school zone are closer to the school in that zone than any other school in the district.

Suggested Materials
Map of the school district

color pencils

color markers

rulers *

protractors *

  • (can also be done by compass and straightedge construction)

Knowledge of perpendicular bisectors and equidistance.

Activity Description

Guided Exploration
A. Use the map of your town to mark the current location of the elementary schools. Mark each school with a different color marker.

B. Draw segments between relevant pairs of schools. Construct the perpendicular bisectors for the segments. (Leave the perpendicular bisectors, but erase the segments.)

C. Identify the parts of the perpendicular bisectors that form the regions that each contain one school. The parts of the perpendicular bisector which you keep are the points closer to A or B than to any other point.

D. Have each student in your group mark the location of his/her home with color pencil.


  1. What do these regions represent in relation to the schools?

  2. Which school would you go to from your home? Why? Does the Voronoi diagram support your answer?

  3. Why might developers and planners be interested in this Voronoi diagram?

  4. Compare these regions with those used for the schools in real-life. What would prevent the town from using your Voronoi diagram for the school district?

Sample Results from Exploration
During the course of this lesson, I decided to get a large-scale view of the results for the district by compiling student information on one map. With three geometry classes participating, we located approximately 75 student homes on the map. We denoted each student’s home with a colored pencil ( the color matching the elementary school to which they would be assigned). The completed map showed each elementary school and all the students’ homes in color. Using this map, we could easily locate those students whose homes did not “fit” properly into the Voronoi diagram. This map was the reference for most of the follow-up questions.

A Mathematical Approach
The perpendicular bisector of a segment is the line that bisects and is perpendicular to the segment. The points on the perpendicular bisector (p.b.) are equidistant from the endpoints of segment. So, any points which are not on the p.b. are closer to one endpoint than the other. All the points that are closer to that site than any other would be in that site’s region in a Voronoi diagram.

In a Voronoi diagram, there may be many sites. Constructing the perpendicular bisector between each pair of sites will separate an area into regions. Each point in each region will be closer to the site in that region than to any other site in the diagram.

Further Problems for Review or Assessment
For different problem contexts, have the students consider the service areas for one of the following situations: firehouses, first aid stations, election polling places, video stores, pizza, parlors, hardware stores, etc.

Writing Assignment
Have the students write instructions on how to create a Voronoi diagram given 3 points as sites.

Followup Problems or Activities

  1. If the district were to build a new elementary school, where would you suggest that the best site would be? Show your suggested site on the map and explain how this decision was made.

  2. Create a Voronoi diagram for the fire stations in town. Look over the map and see if you can find any area of town where using the diagram would be a disadvantage. Show that area of the map and explain your answer in detail.

Teaching Notes


  1. Drysdale, Scot, Workshop on Voronoi Diagrams, Princeton University, July 1996.

  2. Rhoad, Milauskas,and Whipple,Geometry for Enjoyment and Challenge. McDougal, Littell & Co.,Chicago ,IL ,1991.

  3. Dickerson, Matthew, and Drysdale, Scot; Voronoi Diagrams and Proximity Problems; COMAP; Lexington, MA;1996

Solve gerrymandering once and for all by redistricting the United States using Voronoi tesselations, an unbiased way of dividing a space into random regions.

Road selection based on Voronoi diagrams and “strokes” in map generalization

Author links open overlay panel XingjianLiuaF. BenjaminZhanab1TinghuaAibc

Texas Center for Geographic Information Science, Department of Geography, Texas State University – San Marcos, 601 University Drive, San Marcos, TX 78666, United States

School of Resources and Environment Science, Wuhan University, 129 Luoyu Road, Wuhan 430079, China

Key Laboratory of Geographic Information System, Ministry of Education, Wuhan University, 129 Luoyu Road, Wuhan 430079, China

Received 29 March 2009, Accepted 20 October 2009, Available online 19 November 2009.

This article proposes a road selection algorithm based on Voronoi diagrams and “strokes”. We argue that selection of road segments from road networks during generalization should utilize metric, thematic, topological, and statistical road network information at macro-, mezzo-, and micro-spatial levels. This approach focuses on Voronoi-based local road density at the mezzo-spatial level and a more comprehensive and scale-dependent “stroke” generation procedure.


Road selection is a prerequisite to effective road network generalization. This article introduces a novel algorithm for road network selection in map generalization, which take four types of information into consideration: statistical, metric, topological, and thematic at three spatial scales: macro-scale which describes the general pattern of networks, mezzo-scale that handles relationships among road segments, and micro-scale that focuses on individual roads’ properties. A set of measures is selected to quantify these different types of information at various spatial levels. An algorithm is then developed with the extraction of these measures based on Voronoi diagrams and a perceptual grouping method called “stroke”. The selection process consists of three consecutive steps: measuring network information based on Voronoi partitioning and stroke generation, selecting roads based on information extraction in the first step with strokes as selection unit, and assessing selection results. The algorithm is further tested with a real-world dataset: road network map at 1:10,000 scale and its generalized version at 1:50,000 scale in Wuhan, China. The result reveals that the algorithm can produce reasonable selection results and thus has the potential to be adopted in road selection in map generalization.


The name Voronoi comes from Voronoi diagrams: a kind of mathematical mosaic known for their artistic and scientific value. Founding partners Josh and Tim have been working together now for over a decade in a variety of contexts: from building video analysis mobile apps to university application management systems. Josh’s visual expertise and Tim’s development experience bring together the aesthetic and the technical, a combination that is the cornerstone of Voronoi LLC.