Cours de Visualisation d'Information InfoVis Lecture !

Maps & Voronoï diagrams Frédéric Vernier Maître de conférence / Lecturer Univ. Paris Sud

[https://twitter.com/RodBogart]

Outline • Voronoï & Delaunay • Usages • Algorithms • Use case

Voronoï Diagrams & Delaunay triangulation

• Partitioning of a plane • List of non overlapping polygons • List of edges at equal distance of 2 pts • List of points at equal distance of 3 pts • Delaunay triangulation = dual • List of triangles

Properties and variations

[Review by A. Dobrin]

[Wikipedia]

[Manhattan dist]

• Weighted • Other distances, • Loyds relaxation • nD, shaped, #order

[3 steps of Loyds algo.]

[Voronoï height maps]

[Voro++] [Segment Voronoï]

Usages [Voronoi Treemaps, 
 Balzer&Deussen]

J. Snow map Again ! [Vorographs, IBM
 Dunne, Muller, Perra,& Martino]

• Delaunay = points2Triangles • Voronoï Treemaps Scatterplot Voronoï word clouds • • Picking plot picking • Vorographs

foamTree carrot2.org

Art &

Computer Graphics

Voronoï + map

[McDo Voronoï Paris]

[Airport Voronoï]

• Highlight natural region of influence

[Capital Voronoï]

[Youtube Voronoï soccer]

[Beer brewery 
 Voronoï]

More complex [adaptive crystal-growth Voronoï diagrams]

[the way european french people name a pencil] [Bikeshare Voronoï] 1 representation

2 presentations

[http://roadstorome.moovellab.com/]

Algorithms • 3 families • Bad n , n or n algos • Steven Fortune n.log(n) ! • Incremental algos (may be as good as n.log(n)) 4

3

2

Fortune’s algorithm

[http://www.ams.org/samplings/feature-column/fcarc-voronoi] [https://people.csail.mit.edu/indyk/6.838-old/handouts/lec8.pdf]

• Sort all the points according to one direction (ie y) • process point one by one and consider parabolas • equal dist. point & line d • n points => n parabolas d ! !

Beach line & Sweep line Sites/points already processed Beach line = n parabolas Breakpoint Sweep line Direction of the process Sites/points to process

New point event • Sweepline cross a point(site) • Parabola is first just a vertical segment • 2 new break points at the same location • When sweep line moves further away • parabola get wider and 2 breakpoints slides

parabola disappearing event Notice the 
 small hyperbola ?

connected to 3 (sliding) points it now disappears ! a new point found 
 its definitive position 
 in the Voronoï diagram

parabola disappearing • When does it happen ? • very likely between 2 points to process • if we process the next point … it’s too late • Let’s introduce a fake point in the ordered list

3 consecutive points in the beach line make
 a circle bottom of the circle = WHEN middle 
 parabola disappear

fortune algorithm • Ordered list of points (real sites and fake pt) • Ordered list of parabolas (act. parabola arcs) • use a (balanced) binary tree • leaf = parabola = site • node = intersection = point

2 Use case • 36000 towns of france • in OpenGL ! • Regional languages of France • from ALF points

let’s see how to do it • Read files as input => csv + geojson • Use a lib to compute Voronoï =>openCV • Attach info to sites/points • Parse results (read openCV refs !!!) • Build the triangles and the colors (+holes + blue sea) • Tweak, adjust, set parameters, test and debug

Conclusion • Voronoï diagrams look cool • Voronoï diagrams are useful • Voronoï diagrams complement maps • Voronoï diagrams are fast to compute • Voronoï diagrams are complex to implement • Voronoï diagrams involve complex data structures • Voronoï diagrams are another tool in your toolbox