A Review of Properties and Variations of Voronoi Diagrams

A REVIEW OF PROPERTIES AND VARIATIONS OFVORONOI DIAGRAMSADAM DOBRIN1. IntroductionThis paper is a review of Voronoi diagrams, Delaunay triangulations,and many properties of specialized Voronoi diagrams. We willalso look at various algorithms for computing these diagrams.Themajority of the material covered is based on research compiled by AtsuyukiOkabe in Spatial Tessellations: Concepts and Applications ofVoronoi Diagrams [6].However, there will also be references to researchand results presented in many papers. Multiple algorithms forcomputing different diagrams were found and translated from, amongothers, Centroidal Voronoi Tessellations: Applications and Algorithms[3] and A Sweepline Algorithm for Voronoi Diagrams [4].Section 2 will introduce Voronoi diagrams and provide examples ofwhere they can be seen and how they are applied.Sections 3 and4 will discuss basic properties associated with Voronoi diagrams andtheir duals: Delaunay triangulations. These building blocks will allowthe progression into discussing more complex ideas regarding Voronoi1

2 ADAM DOBRINdiagrams. In Section 5, there will be an exploration of weighted Voronoidiagrams, followed by a study of different methods for constructingVoronoi diagrams in Section 6. The last topic covered, in Section 7,will be the idea of centroidal Voronoi diagrams and different algorithmsfor their construction.2. What is a Voronoi Diagram?First, it should be noted that for any positive integer n, there aren-dimensional Voronoi diagrams, but this paper will only be dealingwith two-dimensional Voronoi diagrams. The Voronoi diagram of a setof “sites” or “generators” (points) is a collection of regions that divideup the plane. Each region corresponds to one of the sites or generators,and all of the points in one region are closer to the corresponding sitethan to any other site. Where there is not one closest point, there is aboundary. Note that in Figure 1, the point p is closer to p 1 than to anyother enumerated points. Also note that p ′ , which is on the boundarybetween p 1 and p 3 , is equidistant from both of those points.As an analogy, imagine a Voronoi diagram in R 2 to contain a seriesof islands (our generator points). Suppose that each of these islandshas a boat, with each boat capable of going the same speed. Let everypoint in R that can be reached from the boat from island x before any

A REVIEW OF PROPERTIES AND VARIATIONS OF VORONOI DIAGRAMS 3Figure 1. A basic Voronoi diagram [6].other boat be associated with island x. The region of points associatedwith island x is called a Voronoi region.The basic idea of Voronoi diagrams has many applications in fieldsboth within and outside the math world. Voronoi diagrams can be usedas both a method of solving problems or as a model for examples thatalready exist. They are very useful in computational geometry, particularlyfor representation or quantization problems, and are used in thefield of robotics for creating a protocol for avoiding detected obstacles.For modeling natural occurrences, they are helpful in the studies ofplant competition (ecology and forestry), territories of animals (zoology)and neolithic clans and tribes (anthropology and archaelogy), andpatterns of urban settlements (geography) [2].

4 ADAM DOBRIN3. Basic Properties of the Voronoi Diagram3.1. Formal Definition of the Voronoi Diagram. We have defineda Voronoi diagram informally. Since we are going to be dealing withmathematical problems associated with and algorithms for computingthe Voronoi diagram, we must formally define the two-dimensional ordinaryVoronoi diagram.First, we shall denote the location of a point p i as (x i1 , x i2 ), and thecorresponding vector will be ⃗x. Let P = {p 1 , p 2 , .., p n } ∈ R 2 , where2 ≤ n < ∞ and p i ≠ p j , i ≠ j and ∀ i, j = 1, 2, . . . , n be the set ofgenerator points, or generators. We call the region given byV (p i ) = {⃗x ∣ ∣ ||⃗x − ⃗xi || ≤ ||⃗x − ⃗x j || ∀j ∋ i ≠ j}the Voronoi region of p i , where || · || is the usual Euclidean distance.V (p i ) may also be referred to as V i . All Voronoi regions in an ordinaryVoronoi are connected and convex. We call the set given byV = {V (p 1 ), V (p 2 ), . . . , V (p n )}the Voronoi diagram of P . Different notation for V is ⋃ ni=1 V i.3.2. Basic Components of the Voronoi Diagram. The Voronoidiagram is composed of three elements: generators, edges, and vertices.

A REVIEW OF PROPERTIES AND VARIATIONS OF VORONOI DIAGRAMS 5P is the set of generators. Every point on the plane that is not a vertexor part of an edge is a point in a distinct Voronoi region.⋂An edge between the Voronoi regions V i and V j is V i Vj = e(p i , p j ).If e(p i , p j ) ≠ ∅, V i and V jare considered adjacent. Any point ⃗x one(p i , p j ) has the property that ||⃗x − ⃗x i || = ||⃗x − ⃗x j ||. An edge can bedenoted as e i , where i is an index for the edges and does not have tobe related to the index of generator points. For example, we can labelour edges from 1 to n, n being the total number of edges, going topto bottom left to right. The labelling of edges is merely a convenience,and does not have to follow a pre-determined algorithm. It should bedecided per Voronoi diagram.Also, the set of edges surrounding aVoronoi region V i can be referred to as ∂V (p i ), ∂V i , or the boundaryof V i .A vertex is located at any point that is equidistant from the three(or more) nearest generator points on the plane. Vertices are denotedq i , and are the endpoints of edges. The number of edges that meet ata vertex is called the degree of the vertex. If ∀q i ∈ V, degree(q i ) = 3,then V is considered to be non-degenerate. Otherwise, V is consideredto be degenerate. Throughout this paper, we will, for the most part,assume non-degeneracy in our Voronoi diagrams.

6 ADAM DOBRINFigure 2. Voronoi Diagrams on Bounded Subsets of R 2 [6].3.3. Voronoi Diagrams on a Bounded Subset of R 2 . While mostof the time we will consider Voronoi diagrams on R 2 , we can also havethem on any set S ⊆ R 2 .We will assume S to be non-empty, fora Voronoi diagram on an empty set would be trivial.The boundedVoronoi diagram is defined by V ∩S= {V 1 ∩ S, V 2 ∩ S, . . . , V n ∩ S}.If, for any i, V i shares the boundary of S, we call V i ∩ S a boundaryVoronoi region.Unlike ordinary Voronoi regions, boundary Voronoiregions need not be connected or convex.In Figure 2, we see two Voronoi diagrams generated by the sameset P , but on different subsets of R 2 . In the left diagram, the shadedregion is not connected, and in both diagrams, many of the regions arenot convex. Note that the non-convex regions are boundary regions.3.4. Dominance Regions. Given any two generators, p i and p j , theperpendicular bisector of the line connecting p i and p j is b(p i , p j ) ={⃗x ∣ ∣ ||⃗x − ⃗xi || = ||⃗x − ⃗x j ||}, i ≠ j.H(p i , p j ) = {⃗x ∣ ∣ ||⃗x − ⃗xi || ≤||⃗x − ⃗x j ||}, i ≠ j is the dominance region of p i over p j , and consists of

A REVIEW OF PROPERTIES AND VARIATIONS OF VORONOI DIAGRAMS 7Figure 3. Construction of a Voronoi Region UsingHalf-Planes [6].every point of the plane that is closer to p i than p j or equidistant fromthe two. H(p j , p i ), or Dom(p j , p i ), is the dominance region of p j overp i . In the basic Voronoi diagram, H(p j , p i ) is a half-plane.From our definition of dominance regions, we can define Voronoiregions in yet another way.Let P = {p 1 , p 2 , . . . , p n } ∈ R 2 be a setof generator points.V i = ⋂ j∈Z + ≤n H(p i, p j ) is the ordinary Voronoiregion associated with p i . The set V = {V 1 , V 2 , . . . , V n } is the Voronoidiagram on R 2 generated by P .In Figure 3, we can see the construction of a Voronoi region usingdominance regions. By drawing in the half-planes associated with p 1 ,we can see how a Voronoi region is created using the method of finding⋂j∈Z + ≤n H(p i, p j ). Using this method for all of the points in a Voronoi

8 ADAM DOBRINFigure 4. A largest empty circle in a basic Voronoi diagram[6].diagram becomes overly complicated, and is generally bypassed in lieuof other algorithms, which will be discussed in Section 6.3.5. The Convex Hull of P . The convex hull of P is the smallestconvex set containing the generator set P , and is denoted as CH(P ).The boundary of this region is referred to as ∂ CH(P ). Given a Voronoidiagram V(P ), V i is unbounded iff p i ∈ ∂ CH(P ). With knowledge ofthe CH(P ), we know the following about a Voronoi diagram V(P ):(i) A Voronoi edge e(p i , p j )(≠ ∅) is a line segment iff the line connectingp i and p j (p i p j ) is not on ∂ CH(P ).(ii) A Voronoi edge e(p i , p j )(≠ ∅) is a half-line or ray iff P is noncollinearand p i and p j are consecutive generator points on ∂ CH(P ).(iii) All Voronoi edges are lines iff P is collinear.3.6. Empty Circles. With a Voronoi diagram V(P ), it is helpful toknow about empty circles. An empty circle is one with no generator

A REVIEW OF PROPERTIES AND VARIATIONS OF VORONOI DIAGRAMS 9points within its boundary. For each vertex q i ∈ V(P ), there exists aunique empty circle C i centered at q i that passes through at least threegenerator points, and it is the largest empty circle centered at q i . If weassume non-degeneracy, then C i passes through exactly three generatorpoints. Note that the largest empty circle represented in Figure 4 hasthree generator points on its boundary.3.7. Graph Theory and Voronoi Diagrams. Also interesting tonote are the correlations between basic graph theory and Voronoi diagrams.If we let n, n e , and n vbe the number of generator points,Voronoi edges, and Voronoi vertices of a Voronoi diagram, respectively.We find that(1) n v − n e + n = 1.If P has more than 3 elements, thenn e ≤ 3n − 6 and n v ≤ 2n − 5Furthermore, if we let n c be the number of unbounded Voronoi polygonsand continue to assume that P has more than 3 elements, thenn v ≥ 1 2 (n − n c) + 1 and n e ≤ 3n v − n c − 3

A REVIEW OF PROPERTIES AND VARIATIONS OF VORONOI DIAGRAMS11of the infinite region. (Note: If we had no unbounded Voronoi regionsto begin with, we still have an infinite region not previously defined;thus n c = 1.) Since we have included an extra Voronoi region, we’vealtered our equation to be n v − n e + n = 2. Disregard our generatorset P . Our manipulated Voronoi diagram V has become a connectedplanar graph G. The number of regions n becomes the number of facesf in G. The number of vertices n v becomes the number of vertices v inG. The number of edges n e becomes the number of edges e in G. Ourequation becomes v − e + f = 2, Euler’s Theorem.4. Delaunay Tessellations4.1. Introduction. Continuing the theme of graph theory, we willnow discuss Delaunay tessellations, which are considered to be dualto Voronoi diagrams [7]. For all Delaunay tessellations, we will assumenon-collinearity. This means that for our generator set P , the points inP are not all on the same line. Given a Voronoi diagram V(P ), join allpairs of generator points whose Voronoi regions share an edge. Thus,in the Delaunay tessellation of P , or D(P ), there exists the edge p i p jif and only if e(p i , p j ) ∈ V(P ) ≠ ∅. If this tessellation consists of onlytriangles, we call it a Delaunay triangulation. If not, we call it a Delaunaypretriangulation. A Delaunay tessellation will be a triangulation if

12 ADAM DOBRINFigure 6. A Voronoi diagram and a Delaunay triangulation[6].and only if all vertices of V(P ) are non-degenerate. See Figure 6 for anillustration of the relation between a Voronoi diagram (dashed lines)and a Delaunay triangulation (solid lines) of the same generator set P(solid dots).In a Delaunay triangulation, regions are called Delaunay triangles.Edges in Delaunay tessellations are called Delaunay edges. If a Delaunayedge is shared by two Delaunay triangles, then we call it an internalDelaunay edge. Otherwise, we call it an external Delaunay edge. Theexternal Delaunay edges in D(P ) constitute ∂ CH(P ).Interesting to note is the fact that (assuming non-degeneracy), Voronoiedges and Delaunay edges are orthogonal. With a little thought, thisis fairly obvious, because a Voronoi edge e(p i , p j ) lies on the perpendicularbisector of p i p j , a Delaunay edge.This means that Voronoidiagrams and their Delaunay tessellations are not only dual graphs,but are reciprocal figures [7].

A REVIEW OF PROPERTIES AND VARIATIONS OF VORONOI DIAGRAMS13Figure 7. Delaunay triangulations: (a) is not a Pittewaytriangulation, while (b) is a Pitteway triangulation[6].4.2. The Pitteway Triangulation. The Delaunay triangulation of Pis a Pitteway triangulation of P if and only if every internal Delaunayedge p i p j crosses the associated Voronoi edge e(p i , p j ) of V(P ).4.3. Correspondence Between Voronoi and Delaunay. We mentionedthat Delaunay triangulations are non-degenerate duals to Voronoidiagrams, but have not yet discussed the meaning and results of thisclaim. Given a non-degenerate Voronoi diagram V(P ) and a Delaunaytriangulation D(P ), let Q and Q d be the sets of Voronoi vertices andDelaunay vertices, respectively. Let E and E d be the sets of Voronoiedges and Delaunay edges, respectively. Let C d be the set of circumcentersof Delaunay triangles. Then the following are true:(i) Q d = P(ii) C d = Q

14 ADAM DOBRINFigure 8. Empty circumcircles of Delaunay triangles(i.e. Delaunay circles) in a Delaunay triangulation [6].(iii)|E d | = |E|Statement (i) says that each generator point p i is a vertex of a Delaunaytriangle. Statement (ii) says that the circumcenter of each Delaunaytriangle corresponds to a Voronoi vertex. Statement (iii) says that thenumber of Delaunay edges is equal to the number of Voronoi edges.On a side note, the circumcenter of a Delaunay triangle, is the pointequidistant from the three vertices. It is the center of the circumcircle,or Delaunay circle, which is the largest empty circle contained in aDelaunay triangle.The circumcircle of the Delaunay triangle is anempty circle only if the triangulation of P is a Delaunay triangulation.The points on a Delaunay circle, which are Delaunay vertices, are callednatural neighbors. Two vertices are natural neighbors if and only if theyare connected by a Delaunay edge.

A REVIEW OF PROPERTIES AND VARIATIONS OF VORONOI DIAGRAMS15To close, we will look at another interesting property of Delaunaytriangulations. Let D(p i , p j ) be the shortest path along the Delaunayedges of D(P ) from p i to p j . Then D(p i , p j ) ≤ c ∗ d(p i , p j ), whered(p i , p j ) is the Euclidean distance from p i to p j and c =2π ≈ 2.42.3cos( π 6 )5. The Weighted Voronoi Diagram5.1. Introduction. So far, in our discussion of Voronoi diagrams, wehave assumed that our generator points, besides their location, haveequal value, or weight. The idea of assigning distinct weight to generatorpoints can be more useful than having uniformly weighted points insome scenarios. Weighted generator points are sometimes more applicablewhen looking at, for example, the population size of a settlement,the number of stores in a shopping center, or the size of an atom in acrystal structure [6].Recall from our definition of the Voronoi diagram from Section 3.4.that a Voronoi region V i is the intersection of the dominance regions ofp i over every other generator point in P . While we will have differentformulae for dominance regions in weighted voronoi diagrams, the idearemains the same. The dominance region of a generator point p i overanother, p j , where i ≠ j and d w (p x , p y ) is the weighted distance between

16 ADAM DOBRINpoints x and y, is written asDom w (p i , p j ) = {p|d w (p, p i ) ≤ d w (p, p j )}.LetV w (p i ) =⋂Dom w (p i , p j ).p j ∈P \{p i }V w (p i ), or V w (i), is called a weighted Voronoi region, andV w = {V w (p 1 ), V w (p 2 ), . . . , V w (p n )} is called the weighted Voronoi diagram.Another way to denote V w is V(P, d w ), where P is the generatorset with weights W = {W 1 , W 2 , . . . , W n } and d w is the weighted distance.We will discuss d w more in depth in the following sections.5.2. The Multiplicatively Weighted Voronoi Diagram. Recallthe analogy of generator points to islands with boats from Section 2.The multiplicatively-weighted Voronoi diagram assigns a speed to eachboat. Therefore, a faster boat will reach boats previously outside of itsregion.This weighted Voronoi diagram has its weighted distance given byd mw (p, p i ) = ||⃗x − ⃗x i||w i, where w i > 0.This is called the multiplicatively weighted distance or MW-distance.There are many names for the associated Voronoi diagram: V mw , the

A REVIEW OF PROPERTIES AND VARIATIONS OF VORONOI DIAGRAMS17multiplicatively-weighted Voronoi diagram, the MW-Voronoi diagram,the circular Dirichlet tessellation, or the Apollonius model. We willnot be using the last two, as they are highly specialized and this paperis a more general survey of ideas and concepts associated with Voronoidiagrams.A MW-Voronoi region is a non-empty set, it does not have to beconvex or connected, and it can have a hole or holes. A MW-Voronoiregion V w (p i ) is convex if and only if the weights of adjacent MW-Voronoi regions are not smaller than the weight of p i . Another wayto denote “the weight of p i ” is w i .Also, two MW-Voronoi regionsmay share disconnected edges. Edges in V mw are circular arcs if andif only if the weights of the two affective regions are not equal. Edgesin V mw are straight lines if and only if the weights of the two affectiveregions are equal. See Figure 9 for a diagram of the bisectors betweenpoints p i and p j with multiplicatively weighted distance for severalratios α = w iw j= 1, 2, 3, 4, 5.Let w max = max{w j , p j ∈ P } and P max = {p j |w j = w max }. A MW-Voronoi region V mw (p i ) is unbounded if and only if p i ∈ P max and p i ison ∂ CH(P max ).5.3. The Additively Weighted Voronoi Diagram. Continuing theboat/island analogy, imagine that boats in the additively weighted

18 ADAM DOBRINFigure 9. Multiplicatively weighted Voronoi diagramsfor two generator points [6].Figure 10. A multiplicately weighted Voronoi diagram(the numbers in parentheses represent weights associatedwith the generators) [6].

A REVIEW OF PROPERTIES AND VARIATIONS OF VORONOI DIAGRAMS19Voronoi diagram start a certain distance away from their respectiveislands. They all still travel at the same speed, but some boats begincloser to points than they would in the ordinary Voronoi diagram.This type of weighted Voronoi diagram has its weighted distancegiven byd aw (p, p i ) = ||⃗x − ⃗x i || − w i .The dominance region of the additively-weighted Voronoi diagram isgiven byDom aw (p i , p j ) = {⃗x ∣ ∣ ||⃗x − ⃗xi || − ||⃗x − ⃗x j || ≤ w i − w j }, where i ≠ j.If we let α = ||⃗x i − ⃗x j ||, and β = w i − w j , we get the followingresults. If α = β, then the dominance region of p j over p i is a half-lineradiating from p j directly away from p i . This result is impossible withthe multiplicatively-weighted Voronoi diagram. If 0 < α < β, then p icompletely dominates p j , and V aw (p j ) = ∅, another result not possiblewith MW-Voronoi diagrams.With these properties in mind, we find the following statements tobe true. The set V aw (p i ) = ∅ if and only ifmin{||x − x i || − w j , ∀p j ∈ P ∋ i ≠ j} < −w i .

20 ADAM DOBRINFigure 11. Additively weighted Voronoi diagrams fortwo generator points [6].The set V aw (p i ) is a half-line if and only ifmin{||x − x i || − w j , ∀p j ∈ P ∋ i ≠ j} = −w i .The set V aw (p i ) has positive area if and only ifmin{||x − x i || − w j , ∀p j ∈ P ∋ i ≠ j} > −w i .Every edge in V aw is either a hyperbolic arc, line segment, half-line,or infinite line. See Figure 11 for a diagram of the bisectors betweenpoints p i and p j with additively weighted distance for several parametervalues α = ||⃗x i −⃗x j || and β = |w i −w j | = 0, 1, 2, 3, 4, 5, 6, 8, 9, 9.8, 10.

A REVIEW OF PROPERTIES AND VARIATIONS OF VORONOI DIAGRAMS21Figure 12. An additively weighted Voronoi diagram(the numbers indicate weights) [6].If at least one weight, w i , is different from another, and V aw (p i )has positive area, then there exists at least one non-convex additivelyweightedVoronoi region. Every non-convex AW-Voronoi region is starshapedwith respect to its generator point. This means that from p i ,we can draw a line to any point in the region V aw (p i ), and the line willbe contained entirely within V aw (p i ).6. Algorithms for Constructing the Voronoi Diagram6.1. Introduction. There are many different ways to construct theordinary Voronoi diagram (defined by d w = ||⃗x − ⃗x i ||, i ≠ j). In thissection, we will look at two of these: the Plane-Sweep and the TreeExpansion and Deletion methods.6.2. Plane-Sweep Method. For the plane-sweep method of constructingthe Voronoi diagram, we draw all generator points, Voronoi vertices,

22 ADAM DOBRINand Voronoi edges in a non-Cartesian plane. There is a one-to-one correspondencebetween the non-Cartesian plane and the Cartesian plane,given some initial assumptions. Therefore, we can copy our Voronoi diagramin the non-Cartesian plane to the Voronoi diagram in the Cartesianplane, which is what we ultimately want.Let P = {p 1 , p 2 , . . . , p n } be our generator set. For any point p, wedefine r(p) = min{d(p, p i ) ∣ 1 ≤ i ≤ n}. By the properties of Voronoidiagrams, r(p) is the radius of the largest empty circle centered at p.We shall denote the location of p as (x(p), y(p)). Letφ(p) = (x(p) + r(p), y(p)).In other words, for any point p ∈ V i , φ(p) = (x(p) + d(p, p i ), y(p)).φ(p) = p if and only if p is a generator point. Any q ∈ V, e ∈ V,V i ∈ V, and V have images φ(q), φ(e), φ(V i ), and φ(V), respectively.Let p i and p j be two generator points such that x(p i ) > x(p j ) wheree(p i , p j ) ≠ ∅.φ maps e(p i , p j ) to part of a hyperbola with leftmostpoint p i . We will cut this hyperbola at p i into h + (p i , p j ) and h − (p i , p j ).h + (p i , p j ) = h + (p j , p i ) and h − (p i , p j ) = h − (p j , p i ).To help the reader better understand φ(R 2 ), we can draw an analogybetween it and R 2 . Recall the boat/island analogy from before. Imaginethat there is a current, moving directly to the right, that is faster

A REVIEW OF PROPERTIES AND VARIATIONS OF VORONOI DIAGRAMS23than the uniform-velocity abled boats. Therefore, no boat can reach apoint to the left of its island. Voronoi regions in φ(R 2 ) are containedwithin open-right hyperbolas.For the plane-sweep method, we will need to make four initial assumptions.First, numerical computation will be carried out in precisearithmetic. Second, no four generator points align on a common circle.This is the same as assuming non-degeneracy. Third, no two generatorpoints align vertically. Lastly, any generator point and any of thatgenerator point’s Voronoi vertices do not align horizontally.In this method, we take a vertical line and move it from left to rightover the plane. We update our data everytime there is an event, whichwill be defined as::the sweepline hits a generator point.:the sweepline hits a Voronoi vertex in φ(V).To represent the structure of φ(V) along the sweepline, we will usean alternating list L of regions and boundary edges which appear onthe line, from bottom to top. We will also use Q, a set of points inthe plane. It is a list of all possible points where events may occur.To begin with, Q = P , the set of all generator points. Candidates forVoronoi vertices φ(q) are added to Q as they are found by the sweepline.

24 ADAM DOBRINFigure 13. Voronoi diagram and its image: (a) Voronoidiagram V for five generators and (b) its image φ(V) [6].Given a set P = {p 1 , p 2 , . . . , p n } of n generator points, using the followingalgorithm, we will end up with the transformed Voronoi diagramφ(V).(1) Let Q = P .(2) Choose and delete the leftmost point, say p i , from Q.(3) Let L be a list consisting of a single region, φ(V (p i )).(4) While Q is non-empty, repeat 4.a, 4.b, and 4.c.(a) Choose and delete the leftmost point w from Q.(b) If w is a generator point, say w = p i , do 4.b.i, 4.b.ii, and4.b.iii.(i) Find the region φ(V (p j )) on L containing p i .

A REVIEW OF PROPERTIES AND VARIATIONS OF VORONOI DIAGRAMS25(ii) Replace (φ(V (p j ))) on L with(φ(V (p j )), h − (p i , p j ), φ(V (p i )), h + (p i , p j ), φ(V (p j ))).(iii) Add to Q the intersection of h − (p i , p j ) with the immediatelower-half hyperbola on L and the intersectionof h + (p i , p j ) with the immediate upper-half hyperbolaon L.(c) If w is an intersection, say w = φ(q t ), do 4.c.i, 4.c.ii, 4.c.iii,and 4.c.iv.(i) Replace the subsequence (h ± (p i , p j ), φ(V (p j )), h ± (p j , p k ))on L with h = h − (p i , p k ) or h = h + (p i , p k ).(ii) Delete from Q any intersections of h ± (p i , p j ) or h ± (p j , p k )with other half-hyperbolas.(iii) Add to Q any intersections of h with its immediateupper- and lower-half hyperbola on L.(iv) Mark φ(q t ) as a Voronoi vertex incident to h ± (p i , p j ),h ± (p j , p k ), and h.(5) Report all half-hyperbolas that were ever listed on L, all theVoronoi vertices marked in 4.c.iv, and the incidence relationsamong them. [6]

26 ADAM DOBRINFrom this algorithm, we have all edges, vertices, and generator pointsof φ(V) on φ(R 2 ). We now need to be able to convert a point φ(p) top ∈ R 2 .We know to which generator points each edge and vertexis associated. Call one of these generator points p i . We know thatφ(p) = (x(p) + r(p), y(p)). Let x i = x(p i ), y i = y(p i ), x φ = x(φ(p)),and y = φ = y(φ(p)). Let m be the slope of the line between p i andφ(p) and d be the Euclidean distance between p i and φ(p). m = y φ−y ix φ −x iand d = √ (y φ − y i ) 2 + (x φ − x i ) 2 . From this, we find that(2) r(p) =d2 cos(tan −1 (m)) .Our point,(3) p = (x φ − r(p), y φ ),can now be written with Eq. 2 and Eq. 3:(4) p = (x φ −d2 cos(tan −1 (m)) , y φ).Using the sweepline algorithm and Eq. 4, we can graph a Voronoidiagram in φ(R 2 ), then translate each point on an edge or vertex backto R 2 , giving us our Voronoi diagram V.

A REVIEW OF PROPERTIES AND VARIATIONS OF VORONOI DIAGRAMS276.3. Tree Expansion and Deletion Algorithm. While the Plane-Sweep Method is a good system to construct the Voronoi diagram,it is more useful as a programmable way to do so. Carrying out themethod by hand, or even with Maple or Mathematica, is a laborioustask, involving much more work than is actually necessary. For simplerVoronoi diagrams (those with a relatively small generator set P ), wecan utilize a simpler algorithm. This algorithm takes a Voronoi diagramon l − 1 generator points and another generator point in the plane, andgives a Voronoi diagram on l generator points.To add p l , the l th generator point, we need to know the vertices ofV l−1 that will be affected by V (p l ). We can do so with the followinginformation. Let q ijk denote the vertex incident to V (p i ), V (p j ), andV (p k ) in that order. For some point p = (x, y), we let H(p i , p j , p k , p) =0 be the circle that passes through p i , p j , p k , and p. The circle containsp if and only if H(p i , p j , p k , p) < 0. Consequently, q ijk is in V (p l ) if andonly if H(p i , p j , p k , p l ) < 0. We will let∣ ∣∣∣∣∣∣∣∣∣∣∣∣∣∣ 1 x i y i x 2 i + yi2 1 x j y j x 2 j(5) H(p i , p j , p k , p l ) =+ y2 j.1 x k y k x 2 k + y2 k∣ 1 x y x 2 + y 2

28 ADAM DOBRIN(a)(b)(c)(d)Figure 14. Implementation of the Tree Expansion andDeletion Algorithm.Like the Plane-Sweep Method, we will need to make some basicassumptions to begin. We will need non-degeneracy, so we will assumethat the degree of any vertex in V l−1 is 3. A problem arises, though,when H(p i , p j , p k , p l ) = 0. When we add p l to V l−1 , we will have avertex of degree 4. To avoid this, we can change our inequality to saythat H(p i , p j , p k , p l ) ≥ 0 if and only if q ijk is outside of V (p l ). We willalso assume that V l−1 divides the plane into i + 1 regions; call themcells. Assume that every cell, besides infinite ones, is simply connectedwith no holes. Also, two cells share at most one edge.

A REVIEW OF PROPERTIES AND VARIATIONS OF VORONOI DIAGRAMS29We will let T be the set of vertices of V l−1 that will be in V (p l ). Assumethat T is non-empty. Also, assume that V l−1 (T ) (the componentsof V l−1 in T ) is a tree. Following the aforementioned assumptions andgiven P = (p 1 , p 2 , . . . , p l−1 , p l ) and V l−1 , we will implement the followingprocedure to compute a new Voronoi diagram V l :(1) Find the generator p i (1 ≤ i ≤ l − 1) such that d(p i , p l ) is minimized.(2) Among the vertices q ijk on ∂V (p i ), find the one that gives thesmallest value of H(p i , p j , p k , p l ). Let T be the set consisting ofonly this vertex.(3) Repeat 3.a until T cannot be further augmented.(a) For each vertex q ijk connected by an edge to an existingelement of T , add q ijkto T if H(p i , p j , p k , p l ) < 0 andif the resultant T is a tree (satisfying one of our initialassumptions).(4) For every generator p i associated with a vertex q ijk that satisfiesassumption 4.6.7., draw the perpendicular bisector betweenp i and p l from e(p i , p j ) to e(p i , p k ). Let this line segment bee(p i , p l ), the interesection of e(p i , p l ) and e(p i , p j ) be the vertexq ijl , and the intersection of e(p i , p l ) and e(p i , p k ) be the vertexq ikl . In the case that q ijk lies on the outer circuit of V l−1 , draw

30 ADAM DOBRINthe perpendicular bisector between p i and p l . When applicable,the bisector is a segment from e(p i , p j ) to e(p i , p k ). In this case,let this line segment be e(p i , p l ), the intersection of e(p i , p l ) ande(p i , p j ) be the vertex q ijl , and the intersection of e(p i , p l ) ande(p i , p k ) be the vertex q ikl . If the bisector does not intersect eithere(p i , p j ) or e(p i , p k ), it will be a ray originating on e(p i , p m ),where p m is either p j or p k , whichever is associated edge intersectswith the perpendicular bisector. Let this ray be e(p i , p l ),and the intersection of e(p i , p l ) and e(p i , p m ) be the vertex q iml .(5) Remove all vertices in T , and the edges incident to them, fromV l−1 .(6) Consider the interior of the edges and vertices added in 4 to beV (p l ), and the resulting diagram to be V l .In Figure 14, we see the use of the Tree Expansion and DeletionAlgorithm on a Voronoi diagram with six generator points. The newgenerator point affects only the central vertex. For the other three vertices,H ≥ 0. In Figure 15, we see the implementation of the algorithmon a Voronoi diagram that, while symmetrical, much more complicatedthan in Figure 14. We find that H < 0 for four vertices.To keep our vertices and regions organized, we have terms by whichwe can refer to them. We call vertices in T in. Vertices outside of T

A REVIEW OF PROPERTIES AND VARIATIONS OF VORONOI DIAGRAMS31(a)(b)(c)Figure 15. A Second Implementation of the Tree Expansionand Deletion Algorithm.are out. During the algorithm, vertices that are as of yet unchecked areundecided. We call polygons with a vertex in T incident and all otherpolygons non-incident. All vertices and polygons are initially labelledundecided and non-incident, respectively.

32 ADAM DOBRIN7. The Centroidal Voronoi Diagram7.1. Introduction. A centroidal Voronoi diagram, or tessellation, isa Voronoi diagram of a given set such that every generator point isalso the centroid, or center of mass, of its Voronoi region. Typically,centroidal Voronoi tessellations (CVT’s) have an associated densityfunction, ρ(x, y), on Ω, a subset of R 2 .7.2. Real-World Applications and Observations. Applications forCVT’s cover a multitude of disciplines from computer science to urbanplanning.An interesting area in which CVT’s pop up is in the territorial behaviorof animals. For example, the male mouthbreeder fish, to establishdomain, will spit sand away from the center of its territory. For a highenough density of fish introduced simultaneously into a body of waterwith a uniform sandy floor, we find an interesting result. The spittingof sand results in raised bars of sand, and, when viewed from above,creates a pattern which very closely approximates a centroidal Voronoitessellation [3].Also in biology, CVT’s appear in the cell division of animal andplant cells. In a study of the development of starfish embryos, it wasfound that a view of a layer of columnar cells in an arrangement in a

A REVIEW OF PROPERTIES AND VARIATIONS OF VORONOI DIAGRAMS33hollow sphere showed the cells to closely match a centroidal Voronoitessellation [3].Looking at urban planning ideas, we can see close correlations tothose of the CVT. If we just look at the convenience of mailbox placementthroughout a population, we make the following assumptions:(1) A person will use the mailbox nearest to their home.(2) The cost to a person of using a mailbox is a function of thedistance from the person to the mailbox.(3) The cost to the general population is measured by the distanceto the nearest mailbox averaged over the population.(4) The optimal placement of mailboxes is one that minimizes cost,or the distance to the population in general.It makes sense, then, that the optimal placement of mailboxes is atthe centroids of a CVT, using the population density as a basis forgenerator points. We can use these ideas in many different aspects ofurban planning and the placement of resources. Schools, distributioncenters, mobile vendors, bus terminals, voting stations, and servicestops are some examples [3].7.3. Generator Sets. The idea of the centroidal Voronoi tessellationis somewhat limiting in terms of the various diagrams that we can

36 ADAM DOBRIN(2) Construct a Voronoi tessellation of Ω, V, associated with P .(3) Compute the mass centroids of the Voronoi regions in V(4) Let P be the set of mass centroids computed in Step 3.(5) If P meets some predetermined criteria, implement Step 2, thenterminate; otherwise, return to Step 2.7.6. MacQueen’s Method.(1) Select an initial set of n points, (P ) in Ω, by using a MonteCarlo method; let i = 1.(2) Determine a point y in Ω by using the same Monte Carlomethod as in Step 1.(3) Find the point p ∗ in P that is closest to y.(4) Setp ∗ = ip∗ + yi + 1and i = i + 1.(5) If P meets some predetermined criteria, go to Step 6; otherwise,return to Step 2.(6) Construct a Voronoi tessellation of Ω, V, associated with P .7.7. Similarities and Differences. Lloyd and MacQueen’s methodsare very similar, but also very different. They differ just in their method

A REVIEW OF PROPERTIES AND VARIATIONS OF VORONOI DIAGRAMS37of resetting P . For Lloyd’s method, every time we go through the cycleof steps (an iteration), we are constructing a new Voronoi diagram.Considering the fact that we probably must go through several hundrediterations, this is a very time-consuming method for computers. ForMacQueen’s method, instead of recomputing the Voronoi diagrams, weare simply moving a point, then checking our results with the “predeterminedcriteria.” With MacQueen’s method, though, we only moveone point at a time, instead of many points in the set. Lloyd’s methodgives a better approximation of a CVT, but the difference is not notableenough to justify its use over MacQueen’s method. Because ofthis, we will use MacQueen’s method for constructing the CVT.7.8. Stopping Criteria. Typically, we will have two different typesof criteria, but only use one for any given construction of a CVT. Onecriterion will be the total number of iterations, i. The higher the numberof iterations is, the closer V will be to an actual centroidal Voronoitessellation. The other type of criterion will be an error counter, ɛ. Ifour set P remains relatively unchanged for a certain number of iterations,then we consider MacQueen’s method completed. Because y ischosen using a Monte Carlo method, it is possible that p ∗ = y. Then,P would be unchanged. Since this could happen on any given iteration,we would want to set our error counter to cover multiple iterations. It

38 ADAM DOBRIN(a)(b)(c)(d)Figure 16. 256-Point CVT’s Using a Probability DensityFunction of ρ(x, y) = xy on [−1, 1] × [−1, 1] withLimits of: (a) i = 10 3 , (b) i = 10 5 , (c) ɛ = 10 −4 , and (d)ɛ = 10 −8 .is very rare that it would happen multiple times in a row. The lowerthe error counter is, the closer V will be to an actual centroidal Voronoitessellation.7.9. Imperfections in Constructing a CVT. It is important totake note of the fact that we do not get CVT’s with either Lloyd orMacQueen’s methods. We consider the approximations that we receive

A REVIEW OF PROPERTIES AND VARIATIONS OF VORONOI DIAGRAMS39(a)(b)(c)(d)Figure 17. 256-Point CVT’s Using a Probability DensityFunction of ρ(x, y) = e −10x2 y 2 on [−1, 1] × [−1, 1]with Limits of: (a) i = 10 3 , (b) i = 10 5 , (c) ɛ = 10 −4 ,and (d) ɛ = 10 −8 .using these methods close enough to true CVT’s because the differencesbetween them are minimal. The only way to guarantee that the Voronoidiagram of P that we get using Lloyd or MacQueen’s method is truly acentroidal Voronoi diagram is by carrying the method out indefinitely.This cannot be done for all practical purposes. Thus, we accept theapproximation that we make.

40 ADAM DOBRIN7.10. Figures 16 and 17. In both Figures 16 and 17, (a) and (b)are computed using an iteration limit, while (c) and (d) are computedusing an error counter. In Figure 16, note the difference between thetwo diagrams on the left and the two on the right. In (a) and (c), wesee Voronoi regions bunched together that are relatively far away fromthe points (−1, −1) and (1, 1). Some of these problems are corrected in(b) and (d), as we can observe a better order of smaller regions closerto those points than in (a) and (c). In Figure 17, we can see the samegeneral trends as in Figure 16. The smaller regions are closer to theorigin in (b) and (d) than in (a) and (c). These results are expectedbecause (b) uses a higher iteration limit than does (a), and (d) uses alower error counter than does (c). These results follow directly fromthe ideas put forth in Section 7.8.8. ConclusionThis paper is a survey of subjects closely related to Voronoi diagrams.There are many topics on Voronoi diagrams that were not coveredin this paper. Future studies may include an analysis of problemsabout properties of the types of Voronoi diagrams that are discussedin this paper. A seemingly simple, and particularly interesting problemis, given the Voronoi edges of a non-degenerate, ordinary Voronoi

A REVIEW OF PROPERTIES AND VARIATIONS OF VORONOI DIAGRAMS41diagram, to find the locations of the generator set P [6]. Also fascinatingare “post-office” problems, ones that, given a set of locations(generators) and mail centers, finding an optimal and efficient algorithmfor which mail centers deliver to which location, and in whatorder. Problems involving biological modeling using Voronoi diagramsare very interesting as well. Charting patterns of crystal growth usingadditively- and multiplicatively-weighted Voronoi diagrams is aninteresting application of Voronoi diagrams to biological studies [2].There are also many types of Voronoi diagrams with interestingproperties that are not included in this paper.For example, thereare many other metrics to use when computing weighted Voronoi diagrams.Compoundly-weighted Voronoi diagrams use a combinationof multiplicatively- and additively-weighted diagrams. Edges of CWweighteddiagrams are fairly complex, and are either part of a fourthorderpolynomial curve, a hyperbolic arc, a circular arc, or a line segment,half-line, or line. The (additively-weighted) power distance toa point is characterized by subtracting the weight of a generator fromthe square of the Euclidean distance between the generator and thepoint. Power diagrams contain only line segments, half-lines, and/orlines. These two metrics of Voronoi diagrams are very useful in modelingreal-world occurrences, as are the two covered in the paper, because

42 ADAM DOBRINthe introduction of weighted distances account for properties inherentin generators. These properties are akin to those of the boats describedin the boats/island analogies.We can also apply weighted distances in the field of centroidal Voronoidiagrams. Another possible future project is to use weighted CVT’s toreassociate congressional districts. Instead of creating districts basedon many different subjects (including politics or geography), using populationdensity to redistrict a state could result in a more balancedsystem.

A REVIEW OF PROPERTIES AND VARIATIONS OF VORONOI DIAGRAMS43References[1] Aurenhammer, F. and Edelsbrunner, H. “An Optimal Algorithm for Constructingthe Weighted Voronoi Diagram in the Plane.” Pattern Recognition17, 2 (1984): 251-257.[2] Geometry in Action. Drysdale, Scot. 14 Nov 1996. Donald Bren School ofInformation and Computer Sciences at University of California, Irvine. http://www.ics.uci.edu/~eppstein/gina/scot.drysdale.html.[3] Du, Qiang, Vance Faber, and Max Gunzburger. “Centroidal Voronoi Tessellations:Applications and Algorithms.” SIAM Review 21 (1999): 637-676.[4] Fortune, Steven. “A Sweepline Algorithm for Voronoi Diagrams.” Proceedingsof the Second Annual ACM Symposium on Computational GeometryYorktown Heights, New York: Association for Computing Machinery, 1986:313-322.[5] Ju, Lili, Qiang Du, and Max Gunzburger. “Probabilistic Methods for CentroidalVoronoi Tessellations and Their Parallel Implementations.” ParallelComputer 28 (2002):1477-1500.[6] Okabe, Atsuyuki, Barry Boots, Kokichi Sugihara, and Sung Nok Chiu. SpatialTessellations: Concepts and Applications of Voronoi Diagrams, 2nd Ed..Chichester, West Sussex, England: John Wiley & Sons, 1999.[7] Wilson, Robin J. Introduction to Graph Theory. Boston: Addison Wesley,1996.