Voronoi Diagram and Delaunay Triangulation

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Voronoi Diagram and Delaunay Triangulation

Chih-Hung Liu

October 21

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Outline

1 Voronoi Diagrams and Delaunay Triangulations Properties and Duality

2 3D geometric transformation

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Voronoi Diagram

Given a setSofnpoint sites, Voronoi DiagramV(S)is a planar subdivision

1 Each region contains exactly one sitep∈Sand is denoted byVR(p,S).

2 For each pointx ∈VR(p,S),pis its closest site inS. VR(p,S)is the locus of points closer topthan any other site.

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Voronoi Diagram

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Voronoi Diagram

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Voronoi Diagram

p

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Voronoi Diagram

p

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Voronoi Diagram

2 For each pointx ∈VR(p,S),pis its closest site inS.

VR(p,S)is the locus of points closer topthan any other site.

p

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Voronoi Diagram

2 For each pointx ∈VR(p,S),pis its closest site inS.

VR(p,S)is the locus of points closer topthan any other site.

p x

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Voronoi Region

BisectorB(p,q)={x ∈R²|d(x,p) =d(x,q)}.

D(p,q)={x ∈R²|d(x,p)<d(x,q)}.

Two half-planesD(p,q)andD(q,p)separated byB(p,q).

VR(p,S) = \

q∈S,q6=p

D(p,q).

p

B(p, q)

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Voronoi Region

D(p,q)={x ∈R²|d(x,p)<d(x,q)}.

VR(p,S) = \

q∈S,q6=p

D(p,q).

p

q

B(p, q) D(p, q)

D(q, p)

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Voronoi Region

D(p,q)={x ∈R²|d(x,p)<d(x,q)}.

VR(p,S) = \

q∈S,q6=p

D(p,q).

p q

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Voronoi Region

D(p,q)={x ∈R²|d(x,p)<d(x,q)}.

VR(p,S) = \

q∈S,q6=p

D(p,q).

p

q

D(p, q) D(q, p)

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Voronoi Region

D(p,q)={x ∈R²|d(x,p)<d(x,q)}.

VR(p,S) = \

q∈S,q6=p

D(p,q).

p

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Voronoi Region

D(p,q)={x ∈R²|d(x,p)<d(x,q)}.

VR(p,S) = \

q∈S,q6=p

D(p,q).

p

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Voronoi Edge and Vertex

Voronoi Edge

Common intersection between twoadjacentVoronoi regionsVR(p,S)andVR(q,S)

A piece ofB(p,q) Voronoi Vertex

Common intersection among more than two Voronoi regionsVR(p,S),VR(q,S),VR(r,S), and so on.

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Voronoi Edge and Vertex

Voronoi Edge

A piece ofB(p,q)

Voronoi Vertex

p

q

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Voronoi Edge and Vertex

Voronoi Edge

A piece ofB(p,q)

Voronoi Vertex

p q

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Voronoi Edge and Vertex

Voronoi Edge

A piece ofB(p,q) Voronoi Vertex

p

q r

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Growing Circle

Grow a circle from a pointx on the plane

Hit one sitep∈S→x belongs toVR(p,S)

Hit two sitesp,q∈S→x belongs to the Voronoi edge betweenVR(p,S)andVR(q,S)

Hit more than two sitesp,q,r, . . .∈S→x is the Voronoi vertex amongVR(p,S),VR(q,S),VR(r,S),. . .

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Growing Circle

Grow a circle from a pointx on the plane

Hit one sitep∈S→x belongs toVR(p,S)

x

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Growing Circle

Grow a circle from a pointx on the plane Hit one sitep∈S→x belongs toVR(p,S)

p

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Growing Circle

x p

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Growing Circle

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Growing Circle

x p

q

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Growing Circle

p x

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Growing Circle

x

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Growing Circle

x p

r

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Growing Circle

x p

r

q

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Wavefront Model (Growth Model)

Grow circles from∀p∈S at unit speed

x ∈R²is first hit by a circle fromp→x belongs toVR(p,S) x ∈R²is first hit by two circles frompandq→x belongs to a Voronoi edge betweenVR(p,S)andVR(q,S) x ∈R²is first hit by three circles fromp,q, andr →x is a Voronoi vertex amongVR(p,S),VR(q,S)andVR(r,S)

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Wavefront Model (Growth Model)

x p

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Wavefront Model (Growth Model)

x ∈R²is first hit by a circle fromp→x belongs toVR(p,S)

x ∈R²is first hit by two circles frompandq→x belongs to a Voronoi edge betweenVR(p,S)andVR(q,S) x ∈R²is first hit by three circles fromp,q, andr →x is a Voronoi vertex amongVR(p,S),VR(q,S)andVR(r,S)

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Wavefront Model (Growth Model)

x ∈R²is first hit by a circle fromp→x belongs toVR(p,S)

x ∈R²is first hit by two circles frompandq→x belongs to a Voronoi edge betweenVR(p,S)andVR(q,S) x ∈R²is first hit by three circles fromp,q, andr →x is a Voronoi vertex amongVR(p,S),VR(q,S)andVR(r,S)

x p

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Wavefront Model (Growth Model)

x ∈R²is first hit by a circle fromp→x belongs toVR(p,S) x ∈R²is first hit by two circles frompandq→x belongs to a Voronoi edge betweenVR(p,S)andVR(q,S)

x ∈R²is first hit by three circles fromp,q, andr →x is a Voronoi vertex amongVR(p,S),VR(q,S)andVR(r,S)

p q x

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Wavefront Model (Growth Model)

x ∈R²is first hit by a circle fromp→x belongs toVR(p,S) x ∈R²is first hit by two circles frompandq→x belongs to a Voronoi edge betweenVR(p,S)andVR(q,S)

x ∈R²is first hit by three circles fromp,q, andr →x is a Voronoi vertex amongVR(p,S),VR(q,S)andVR(r,S)

p q x

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Wavefront Model (Growth Model)

p q

r x

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Wavefront Model (Growth Model)

p q

r x

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Unbounded Region

VR(p,S)isunboundedif and only ifp is a vertex of the convex hull ofS.

Select a pointcin the convex hull Shoot a ray−→

cpfromc top For any pointx ∈−→

cp\cp,x belongs toVR(p,S)

−→

cpextends to the infinity.

IfSis in convex position,V(S)is a tree.

An unbounded Voronoi edge corresponds to a hull edge.

p

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Unbounded Region

Select a pointcin the convex hull

Shoot a ray−→

−→

c p

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Unbounded Region

cpfromc top

For any pointx ∈−→

−→

p

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Unbounded Region

−→

c p

x

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Unbounded Region

−→

p x

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Unbounded Region

−→

c p

x

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Unbounded Region

−→

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Unbounded Region

−→

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Voronoi Diagram (Mathematic Definition)

Voronoi DiagramV(S) V(S) =R²\([

p∈S

VR(p,S)) = [

p∈S

∂VR(p,S)

∂VR(p,S)is the boundary ofVR(p,S)

∂VR(p,S)6⊂VR(p,S)

V(S)is the union of all the Voronoi edges Voronoi EdgeebetweenVR(p,S)andVR(q,S)

e=∂VR(p,S)∩∂VR(q,S)

Voronoi Vertexv amongVR(p,S),VR(q,S), andVR(r,S)

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Complexity of V (S)

Theorem

V(S)hasO(n)edges and vertices. The average number of edges of a Voronoi region is less than6.

Add a large curveΓ

Γonly passes through unbounded edges ofV(S) Cut unbounded pieces outsideΓ

One additional face and several edges and vertices.

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Complexity of V (S)

Theorem

V(S)hasO(n)edges and vertices. The average number of edges of a Voronoi region is less than6.

Euler’s Polyhedron Formula:v −e+f =1+c v: # of vertices,e: # of edges,f: # of faces, andc: # number of connected components.

An edge hastwoendpoints, and a vertex is incident to at leastthreeedges.

3v ≤2e→v ≤2e/3 f =n+1andc=1

v =1+c+e−f =e+1−n≤2e/3→e≤3n−3 e=v+f−1−c=v+n−1≥3v/2→v ≤2n−2

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Triangulation

Definition

Given a setS of points on the plane, atriangulationis maximal collection ofnon-crossingline segments amongS.

Crossing (pq) p

q

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Triangulation

Definition

p

q

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Triangulation

Definition

Triangulation

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Delaunay Edge

Definition

An edgepq is calledDelaunayif there exists a circle passing throughpandqand containingnoother point in its interior.

p

q

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Delaunay Edge

Definition

pqisDelaunay p

q

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Delaunay Edge

Definition

p q

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Delaunay Triangulation

Definition

ADelaunay Triangulationis a triangulation whose edges are allDelaunay.

For each face, there exists a circle passing all its vertices and containing no other point.

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Delaunay Triangulation

Definition

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Delaunay Triangulation

Definition

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Delaunay Triangulation

Definition

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General Position Assumption

1 No more thantwopoint sites arecolinear

V(S)isconnected

2 No more thanthreepoint sites arecocircular (At mostthreepoints are on the same circle)

degreeof each Voronoi vertex is exactly3.

Each face of the Delaunay triangulation is atriangle.

There is auniqueDelaunay triangulation.

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General Position Assumption

1 No more thantwopoint sites arecolinear V(S)isconnected

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General Position Assumption

Each face of the Delaunay triangulation is atriangle. There is auniqueDelaunay triangulation.

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General Position Assumption

2 No more thanthreepoint sites arecocircular (At mostthreepoints are on the same circle) degreeof each Voronoi vertex is exactly3.

Each face of the Delaunay triangulation is atriangle. There is auniqueDelaunay triangulation.

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General Position Assumption

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General Position Assumption

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Duality

Theorem

Under the general position assumption, the Delaunay triangulation isa dual graphof the Voronoi diagram.

A sitep↔a Voronoi regionVR(p,S)

A Delaunay edgepq↔a Voronoi edge betweenVR(p,S) andVR(q,S)

A Delaunay triangle∆pqr ↔a Voronoi vertex among VR(p,S),VR(q,S)andVR(r,S)

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Duality

Theorem

p

q

p

x x

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Duality

Theorem

p

q q

r

p

r

x x

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Geometric Transformation from 2D to 3D

A paraboloidP ={(x₁,x₂,x₃)|x₁²+x₂²=x₃}in 3D For a pointx = (x₁,x₂)in 2D,x⁰= (x₁,x₂,x₁²+x₂²)is its lifted image in 3D

x⁰←vertical projection fromx toP For a setAof points in 2D, its lifted image A⁰={x⁰= (x₁,x₂,x₁²+x₂²)|x = (x₁,x₂)∈A}

E Z

C’

p p’

q q’

r r’

x₃=x₁²+x₂²

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Circle in 2D ↔ Planar Curve in P

Lemma

LetCbe a circle in the plane. ThenC⁰ is a planar curve on the paraboloidP

C is given byr²= (x₁−c₁)²+ (x₂−c₂)² r²=x₁²+x₂²−2x₁c₁−2x₂c₂+c₁²+c₂² C⁰ satisfiesx₁²+x₂²=x₃

Substitutingx₁²+x₂²byx₃, we obtain a planeE x₃−2x₁c₁−2x₂c₂+c₁²+c₂²−r²=0 C⁰ =P∩E

Intersection betweenE andP is a planar curve

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Lower Convex Hull

S⁰ onP →S⁰in convex position

Each point ofS⁰ is a vertex ofconv(S⁰)

Lower convex hulllconv(S⁰)ofS⁰is the part ofconv(S⁰) visible fromx₃=−∞

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Duality between DT(S ) and lconv (S

⁰

) (1)

Theorem

The Delaunay triangulationDT(S)equals to the vertical projection onto thex₁x₂-planeof the lower convex hull ofS⁰

p,q,r ∈S.C: circumcircle ofp,q,r C⁰ lies on a planeE defined byp⁰,q⁰,r⁰ a pointx insideC ↔lifted imagex⁰belowE

E Z

C’

C p p’

q q’

r r’

x₃=x₁²+x₂²

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Duality between DT(S ) and lconv (S

⁰

) (2)

Theorem

The Delaunay triangulationDT(S)equals to the vertical projection onto thex₁x₂-planeof the lower convex hull ofS⁰

p,q,r defines a triangle ofDT(S)

↔no point ofSinCdefined byp,q,r

↔no point ofS⁰ belowE defined byp⁰,q⁰,r⁰

↔p⁰,q⁰,r⁰ defines a facet oflconv(S⁰)

Computing a convex hull in 3D takesO(nlogn)time

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Another Viewpoint of paraboloid

For eachs= (s₁,s₂)∈S, a paraboloid

P_s={(x₁,x₂,x₃)|x₃= (x₁−s₁)²+ (x₂−s₂)²}

For eachx = (σ1, σ2)inx₁x₂plane, vertical distance fromx toP_sisd(x,s)²

Opaqueand of pairwisedifferentcolors Looking fromx₃=−∞upward→V(S)

Vertical fromx upward first hitsPs →x ∈VR(p,S) P_s∩P_t →B(s,t)

Lower envelope ofS

s∈SP_s→V(S)

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Wavefront model revisited

P_s ={(x₁,x₂,x₃)|x₃=f((x₁−s₁)²+ (x₂−s₂)²)}

f is astrictly increasing function Lower envelope ofS

s∈SP_s→V(S)

f(x) =√ x =p

(x₁−s₁)²+ (x₂−s₂)²

Cones of slope45^◦ with apices at sitess∈S

Expanding circlesCsfrom sitess∈Sat equalunitspeed timet = radiusr

r²= (x₁−s₁)²+ (x₂−s₂)² x3=p

(x1−s1)²+ (x2−s2)²=radius=time

x first hit byCs↔upward vertical projection fromx first hit P_s

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Wavefront model revisited

P_s ={(x₁,x₂,x₃)|x₃=f((x₁−s₁)²+ (x₂−s₂)²)}

s∈SP_s→V(S) f(x) =√

x =p

(x₁−s₁)²+ (x₂−s₂)²

Cones of slope45^◦with apices at sitess∈S

r²= (x₁−s₁)²+ (x₂−s₂)² x3=p

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Wavefront model revisited

P_s ={(x₁,x₂,x₃)|x₃=f((x₁−s₁)²+ (x₂−s₂)²)}

x =p

(x₁−s₁)²+ (x₂−s₂)²

r²= (x₁−s₁)²+ (x₂−s₂)² x3=p

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Wavefront model revisited

P_s ={(x₁,x₂,x₃)|x₃=f((x₁−s₁)²+ (x₂−s₂)²)}

x =p

(x₁−s₁)²+ (x₂−s₂)²

r²= (x1−s1)²+ (x2−s2)² x3=p

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Wavefront model revisited

P_s ={(x₁,x₂,x₃)|x₃=f((x₁−s₁)²+ (x₂−s₂)²)}

x =p

(x₁−s₁)²+ (x₂−s₂)²

r²= (x1−s1)²+ (x2−s2)² x3=p

x first hit byCs↔upward vertical projection fromx first hit P

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