An Introduction to An Introduction to Computational Geometry:Computational Geometry:

Voronoi and Delaunay DiagramsVoronoi and Delaunay Diagrams

Joseph S. B. MitchellStony Brook University

Chapters 3,4: Devadoss-O’Rourke

Voronoi DiagramsVoronoi Diagrams

Georgy Voronoi1868-1908

Historical Origins and Diagrams in Historical Origins and Diagrams in NatureNature

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René Descartes1596-1650 1644: GravitationalInfluence of stars

Dragonfly wing

Honeycomb

Constrained soap bubbles

Giraffe pigmentation

Ack: Streinu&Brock

Craigslist MapCraigslist Map

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“A geometric pass at delineating areas within the United States potentially covered by each craigslist site -the United States of Craigslist.”

Source: idvsolutions.com

http://uxblog.idvsolutions.com/2011/07/chalkboard-maps-united-states-of.html

Voronoi USAVoronoi USA

55(source: http://www.reddit.com/r/MapPorn/comments/1hi07c/voronoi_map_thiessen_polygons_of_the_usa_based_on/)

Voronoi ApplicationsVoronoi Applications

Voronoi, together with point location Voronoi, together with point location search: nearest neighbor queriessearch: nearest neighbor queries

Facility location: Largest empty disk Facility location: Largest empty disk (centered at a Voronoi vertex)(centered at a Voronoi vertex)

Shape description/approximation: Shape description/approximation: medial axismedial axis

66[Lecture 12, David Mount notes]

Post Office ProblemPost Office ProblemPost officesQuery point

Starbucks

Voronoi DiagramVoronoi Diagram

Partition the plane according to the Partition the plane according to the equivalence classes: equivalence classes: V(Q) = { (x,y) : the V(Q) = { (x,y) : the closest sites of S to (x,y) is exactly the set closest sites of S to (x,y) is exactly the set Q Q S } S }• ||QQ| = 1 Voronoi cells (2-faces)| = 1 Voronoi cells (2-faces)• ||QQ| = 2 Voronoi edges (1-faces)| = 2 Voronoi edges (1-faces)• ||QQ| | 3 Voronoi vertices (0-faces) 3 Voronoi vertices (0-faces)

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Q cocircular

Voronoi cell of pi is open, convex

“cell complex”

ExampleExample

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Voronoi cell of p

pp

Join Join ppii to to ppjj iff there is an “empty circle” iff there is an “empty circle” through through ppii and and ppjj

Equivalent definition: “Dual” of the Equivalent definition: “Dual” of the Voronoi diagramVoronoi diagram

Applet [Chew] [Chew]

Delaunay DiagramDelaunay Diagram

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pi

pj

A witness to the Delaunayhood of (pi , pj)

If no 4 points cocircular (degenerate), then Delaunay diagram is a (very special) triangulation.

Voronoi/DelaunayVoronoi/Delaunay

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Delaunay: An Important, Delaunay: An Important, Practical TriangulationPractical Triangulation

Application: Piecewise-linear terrain Application: Piecewise-linear terrain surface interpolationsurface interpolation

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Terrain Interpolation from Terrain Interpolation from Point Sample DataPoint Sample Data

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Delaunay: An Important, Delaunay: An Important, Practical TriangulationPractical Triangulation

Application: Piecewise-linear terrain Application: Piecewise-linear terrain surface interpolationsurface interpolation

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Delaunay: An Important, Delaunay: An Important, Practical TriangulationPractical Triangulation

Application: Piecewise-linear terrain Application: Piecewise-linear terrain surface interpolationsurface interpolation

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Delaunay for Terrain Delaunay for Terrain InterpolationInterpolation

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Voronoi and DelaunayVoronoi and Delaunay

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Voronoi and DelaunayVoronoi and Delaunay

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Delaunay’s TheoremDelaunay’s Theorem

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Delaunay’s TheoremDelaunay’s Theorem

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Voronoi and Delaunay Voronoi and Delaunay PropertiesProperties

The planar dual of Voronoi, drawn with nodes at the The planar dual of Voronoi, drawn with nodes at the sites, edges straight segments, has no crossing sites, edges straight segments, has no crossing edges edges [Delaunay]. [Delaunay]. It is the Delaunay diagram, It is the Delaunay diagram, DD((SS) ) (defined by empty circle property).(defined by empty circle property).

Combinatorial size: Combinatorial size: 3 3nn-6 Voronoi/Delaunay edges; -6 Voronoi/Delaunay edges; 2 2nn-5 Voronoi vertices (Delaunay faces) -5 Voronoi vertices (Delaunay faces) O(O(nn))

A Voronoi cell is unbounded iff its site is on the A Voronoi cell is unbounded iff its site is on the boundary of CH(boundary of CH(SS))

CH(CH(SS) = boundary of unbounded face of ) = boundary of unbounded face of DD((SS)) Delaunay triangulation lexico-maximizes the angle Delaunay triangulation lexico-maximizes the angle

vector among all triangulationsvector among all triangulations• In particular, maximizes the min angleIn particular, maximizes the min angle

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Angle Sequence of a Angle Sequence of a TriangulationTriangulation

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Angle Sequence of a Angle Sequence of a TriangulationTriangulation

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Legal Edges, TriangulationsLegal Edges, Triangulations

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Edge Flips (Swaps)Edge Flips (Swaps)

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Geometry of Geometry of Circles/TrianglesCircles/Triangles

Thale’s TheoremThale’s Theorem

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Legal/Illegal EdgesLegal/Illegal Edges

Equivalent:Equivalent:• alphaalpha22 + alpha + alpha66 > pi > pi

• InCircle(pInCircle(pii ,p ,pkk ,p ,pjj ,p ,pll ) )

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Legal/Illegal EdgesLegal/Illegal Edges

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Algorithm: Legalize Algorithm: Legalize TriangulationTriangulation

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Example: 4 pointsExample: 4 points

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Voronoi and Delaunay Voronoi and Delaunay PropertiesProperties

In any partition of In any partition of SS, , S=B S=B RR, into blue/red , into blue/red points, any blue-red pair that is shortest points, any blue-red pair that is shortest BB--RR bridge is a Delaunay edge.bridge is a Delaunay edge.

Corollaries:Corollaries:• DD((SS) is connected) is connected• MST MST D D((SS) ) (MST=“min spanning tree”)(MST=“min spanning tree”)

• NNG NNG D D((SS) ) (NNG=“nearest neighbor graph”)(NNG=“nearest neighbor graph”)

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Voronoi and Delaunay Voronoi and Delaunay PropertiesProperties

In any partition of In any partition of SS, , S=B S=B RR, into blue/red , into blue/red points, any blue-red pair that is shortest points, any blue-red pair that is shortest BB--RR bridge is a Delaunay edge.bridge is a Delaunay edge.• DD((SS) is connected) is connected• MST MST D D((SS) ) (MST=“min spanning tree”)(MST=“min spanning tree”)

• NNG NNG D D((SS) ) (NNG=“nearest neighbor graph”)(NNG=“nearest neighbor graph”)

Voronoi/Delaunay can be built in time O(Voronoi/Delaunay can be built in time O(n log nn log n))• Divide and conquerDivide and conquer• SweepSweep• Randomized incrementalRandomized incremental

VoroGlide VoroGlide applet

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Nearest Neighbor Graph: Nearest Neighbor Graph: NNGNNG

Directed graph NNG(S):Directed graph NNG(S):

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Minimum Spanning Tree; Minimum Spanning Tree; MSTMST

Greedy algorithms: Greedy algorithms: • Prim: grows a treePrim: grows a tree• Kruskal: grows a forest: O(e log v)Kruskal: grows a forest: O(e log v)

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Euclidean MSTEuclidean MST

One way to compute: Apply Prim or One way to compute: Apply Prim or Kruskal to the complete graph, with Kruskal to the complete graph, with edges weighted by their Euclidean edges weighted by their Euclidean lengths: O(nlengths: O(n22 log n) log n)

Better: Exploit the fact that MST is a Better: Exploit the fact that MST is a subgraph of the Delaunay (which has subgraph of the Delaunay (which has size O(n)):size O(n)):• Apply Prim/Kruskal to the Delaunay: Apply Prim/Kruskal to the Delaunay:

O(n log n), since e = O(n)O(n log n), since e = O(n)3636

Euclidean MSTEuclidean MST

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Traveling Salesperson Traveling Salesperson Problem: TSPProblem: TSP

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Approximation AlgorithmsApproximation Algorithms

PTAS for Geometric TSPPTAS for Geometric TSP

Voronoi AppletVoronoi Applet VoroGlide VoroGlide applet

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Voronoi and Delaunay: 1DVoronoi and Delaunay: 1D

Easy: Just sort the sites along the lineEasy: Just sort the sites along the line

Delaunay is a pathDelaunay is a path

Voronoi is a partition of the line into Voronoi is a partition of the line into intervalsintervals

Time: O(n log n)Time: O(n log n)

Lower bound: Lower bound: (n log n), since the Delaunay (n log n), since the Delaunay gives the sorted order of the input points, gives the sorted order of the input points, {x{x11 , x , x22 , x , x33 ,…} ,…}

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Voronoi and Delaunay: 2D Voronoi and Delaunay: 2D AlgorithmsAlgorithms

Naïve O(nNaïve O(n22 log n) algorithm: log n) algorithm: Build Voronoi cells one Build Voronoi cells one by one, solving n halfplane intersection problems (each in O(n by one, solving n halfplane intersection problems (each in O(n log n), using divide-and-conquer) log n), using divide-and-conquer)

Incremental: worst-case O(nIncremental: worst-case O(n22)) Divide and conquer (first O(Divide and conquer (first O(n log nn log n)))) Lawson Edge Swap (“Legalize”): DelaunayLawson Edge Swap (“Legalize”): Delaunay Randomized Incremental Randomized Incremental (expected O(n log n))(expected O(n log n)) Sweep [Fortune]Sweep [Fortune] Any algorithm that computes CH in Any algorithm that computes CH in RR3 3 , , e.g., e.g.,

QHullQHull4444

Qhull website

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Delaunay: Edge Flip AlgorithmDelaunay: Edge Flip Algorithm Assume: No 4 co-circular points, for simplicity.Assume: No 4 co-circular points, for simplicity. Start with any triangulationStart with any triangulation Keep a list (stack) of “illegal” edges:Keep a list (stack) of “illegal” edges:

• abab is is illegalillegal if InCircle( if InCircle(a,c,b,da,c,b,d))• iff the smallest of the 6 angles goes up if flipiff the smallest of the 6 angles goes up if flip

Flip any illegal edge; update legality status of Flip any illegal edge; update legality status of neighboring edgesneighboring edges

Continue until no illegal edgesContinue until no illegal edges TheoremTheorem: A triangulation is Delaunay iff there : A triangulation is Delaunay iff there

are no illegal edges (i.e., it is “locally Delaunay”)are no illegal edges (i.e., it is “locally Delaunay”) Only O(Only O(nn22 ) flips needed.) flips needed.

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Lawson Edge Swap“Legalize” [BKOS]

a

b

c

d

Locally non-Delauany

Connection to Convex Hulls in Connection to Convex Hulls in 3D3D

Delaunay diagram Delaunay diagram lower convex hull lower convex hull of the of the liftedlifted sites, ( sites, (xxi i , y , yii , x , xii 22 +y +yii 22 ), on ), on the paraboloid of revolution, the paraboloid of revolution, z=xz=x22 + y + y22

Upper hull Upper hull “furthest site Delaunay” “furthest site Delaunay” 3D CH 3D CH applet

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Delaunay in 2D is Convex Hull in Delaunay in 2D is Convex Hull in 3D3D

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Delaunay in 2D is Convex Hull in Delaunay in 2D is Convex Hull in 3D3D

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Voronoi and DelaunayVoronoi and Delaunay

Algorithms:Algorithms:• Straightforward incremental: O(Straightforward incremental: O(nn22 ))• Divide and conquer (first O(Divide and conquer (first O(n log nn log n))))• SweepSweep• Randomized incrementalRandomized incremental• Any algorithm that computes CH in Any algorithm that computes CH in RR3 3 , ,

e.g., QHulle.g., QHull

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Qhull website

Straightforward Incremental Straightforward Incremental Algorithm: VoronoiAlgorithm: Voronoi

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Straightforward Incremental Straightforward Incremental Algorithm: VoronoiAlgorithm: Voronoi

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Fortune’s Sweep AlgorithmFortune’s Sweep Algorithm

Ack: Guibas&StolfiApplet

AMS 545 / CSE 555

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Parabolic FrontParabolic Front

Ack: Guibas&StolfiApplet

AMS 545 / CSE 555

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Site EventsSite Events

a) b) c)

Ack: Guibas&StolfiApplet

AMS 545 / CSE 555

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Circle EventsCircle Events

Ack: Guibas&StolfiApplet

AMS 545 / CSE 555

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Scheduling Circle EventsScheduling Circle Events

Ack: Guibas&StolfiApplet

AMS 545 / CSE 555

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Incremental ConstructionIncremental Construction

Add sites one by one, modifying the Add sites one by one, modifying the Delaunay (Voronoi) as we go:Delaunay (Voronoi) as we go:• Join Join vvi i to 3 corners of triangle containing itto 3 corners of triangle containing it

• Do edge flips to restore local Do edge flips to restore local DelaunayhoodDelaunayhood

If added in random order, simple If added in random order, simple “Backwards analysis” shows expected “Backwards analysis” shows expected time O(time O(n log nn log n) ) [Guibas, Knuth, Sharir][Guibas, Knuth, Sharir]

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AMS 545 / CSE 555

ExampleExample

CS691G Computational Geometry - UMass CS691G Computational Geometry - UMass AmherstAmherst

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Voronoi ExtensionsVoronoi Extensions

Numerous ! Numerous ! [see Okabe, Boots, Sugihara, Chiu][see Okabe, Boots, Sugihara, Chiu]

Different metricsDifferent metrics Higher dimensions: Higher dimensions:

• Delaunay in Delaunay in RRd d lower CH in lower CH in RRd+1d+1 • O( O( n log n +n log n + nn (d+1)/2(d+1)/2 ) )

Order-Order-k, k, furthest-sitefurthest-site Other sites: Voronoi of polygons, “medial axis”Other sites: Voronoi of polygons, “medial axis” Additive/multiplicative weights; power diagramAdditive/multiplicative weights; power diagram -shapes: a powerful shape representation-shapes: a powerful shape representation

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GeoMagic, biogeometry at Duke

Furthest Site VoronoiFurthest Site Voronoi

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Medial Axis: Convex PolygonMedial Axis: Convex Polygon

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Medial Axis: Convex PolygonMedial Axis: Convex Polygon

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Medial Axis: Nonconvex Medial Axis: Nonconvex PolygonPolygon

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Medial Axis: Curved DomainMedial Axis: Curved Domain

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Straight SkeletonStraight Skeleton

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VRONI: Fast, robust Voronoi of polygonal domains

Incremental algorithm

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Computing offsets with a Voronoi diagram

Alpha Shapes, HullsAlpha Shapes, Hulls

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“Erase” with a ball of radius to get -shape. Straighten edges to get -hull

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Applet

-Shapes-Shapes

TheoremTheorem: For each Delaunay edge, : For each Delaunay edge, ee=(=(ppii,p,pjj), there exists ), there exists minmin((ee)>0 and )>0 and maxmax((ee)>0 such that )>0 such that e e --shape of shape of SS iff iff minmin((ee) ) maxmax((ee).).

Thus, every alpha-hull edge is in the Thus, every alpha-hull edge is in the Delaunay, and every Delaunay edge Delaunay, and every Delaunay edge is in some alpha-shape.is in some alpha-shape.