M¨ obius Diagrams

1 · · · 1 c₀ · · · c_d+1 c²₀−r²₀ · · · c²_d+1−r²_d+1

whereci andri are respectively the center and the radius ofσi.

2.4 Voronoi Diagrams with Algebraic Bisectors

In this section, we introduce a first class of non-affine diagrams, namely the class of diagrams whose bisectors are algebraic hypersurfaces. We first consider the case of M¨obius diagrams whose bisectors are hyperspheres and the case of anisotropicdiagrams whose bisectors are quadratic hypersurfaces. These dia-grams can be computed through linearization, a technique to be described in full generality in Sect. 2.5. Apollonius (or Johnson-Mehl) diagrams, although semi-algebraic and not algebraic, are also described in this section since they are closely related to M¨obius diagrams and can also be linearized.

2.4.1 M¨obius Diagrams

In this section, we introduce a class of non-affine Voronoi diagrams, the so-called M¨obius diagrams, introduced by Boissonnat and Karavelas [63].

The class of M¨obius diagrams includes affine diagrams. In fact, as we will see, the class of M¨obius diagrams is identical to the class of diagrams whose bisectors are hyperspheres (or hyperplanes).

Definition of M¨obius Diagrams

Let

ω

={ω1, . . . , ωn} be a set of so-calledM¨obius sites of R^d, where ωi is a triple (pi, λi, µi) formed of a point pi ofR^d, and two real numbersλi andµi.

For a pointx∈R^d, the distanceδ_i(x) fromxto the M¨obius siteω_i is defined as

δi(x) =λi(x−pi)²−µi.

Observe that the graph ofδiis a paraboloid of revolution whose axis is vertical.

The M¨obius region of the M¨obius siteωi,i= 1, . . . , n, is M(ωi) ={x∈R^d:δi(x)≤δj(x),1≤j≤n}.

Observe that a M¨obius region may be non-contractible and even disconnected.

The minimization diagram of theδiis called theM¨obius diagramof

ω

and noted M¨ob (

ω

) (see Fig. 2.4.1).

Fig. 2.7.A M¨obius diagram

M¨obius diagrams are generalizations of Euclidean Voronoi and power dia-grams. In particular, if allλiare equal to some positiveλ, the M¨obius diagram coincides with the power diagram of a set of spheres{σi, i= 1, . . . , n}, where σi is the sphere centered at pi of squared radius µi/λ. If all µi are equal and allλi are positive, then the M¨obius diagram coincides with the so-called multiplicatively weighted Voronoi diagramof the weighted points (pi,√

λi).

The following lemma states that the bisector of two M¨obius sites is a hypersphere (possibly degenerated in a point or in a hyperplane). Its proof is straightforward.

Lemma 1.Letω_i={pi, λ_i, µ_i}andω_j={pj, λ_j, µ_j},ω_i=ω_j be two M¨obius sites. The bisector σij of ωi andωj is the empty set, a single point, a hyper-sphere or a hyperplane.

M¨obius Diagrams and Power Diagrams

We now present an equivalence between M¨obius diagrams in R^d and power diagrams in R^d+1. This result is a direct generalization of a similar result for multiplicatively weighted diagrams [35]. Given a cell complex C covering a subspace X, we call restriction of C toX the subdivision of X whose faces are the intersections of the faces of C with X. The restriction of C to X is denoted byCX. Note that the restrictionCX is not, in general, a cell complex and that its faces may be non-contractible and even non-connected.

We associate to

ω

⁼{ω1, . . . , ω_n}the set of hyperspheresΣ={Σ1, . . . , Σ_n} ofR^d+1 of equations

Σi(X) =X²−2Ci·X+si = 0,

where Ci = (λipi,−^λ₂ⁱ) andsi =λi p²_i −µi. We denote byQ the paraboloid ofR^d+1 of equationxd+1−x²= 0 .

Theorem 6 (Linearization). The M¨obius diagram M¨ob(

ω

) of

ω

is ob-tained by projecting vertically the faces of the restriction Pow_Q(Σ) of the power diagram ofΣ toQ.

Proof. Ifx∈R^d is closer toω_i than toω_j with respect toδ_M, we have for all j= 1, . . . , n,

λ_i(x−p_i)²−µ_i≤λ_j(x−p_j)²−µ_j

⇐⇒ λ_ix²−2λ_ip_i·x+λ_ip²_i −µ_i≤λ_jx²−2λ_jp_j·x+λ_jp²_j−µ_j

⇐⇒ (x²+^λ₂ⁱ)²+ (x−λ_ip_i)²−^λ₄²ⁱ −λ²_ip²_i +λ_ip²_i −µ_i

≤(x²+^λ₂^j)²+ (x−λjpj)²−^λ₄^2j −λ²_jp²_j+λjp²_j−µj

⇐⇒ (X−Ci)²−r_i²≤(X−Cj)²−r²_j

⇐⇒ Σi(X)≤Σj(X)

where X = (x, x²)∈ Q ⊂R^d+1, Ci = (λipi,−^λ₂ⁱ) ∈R^d+1 and r_i² =λ²_ip²_i +

λ²_i

4 −λ_ip²_i +µ_i. The above inequality shows thatxis closer to ω_i than to ω_j if and only if X belongs to the power region of Σ_i in the power diagram of the hyperspheresΣ_j,j= 1, . . . , n. AsX belongs toQand projects vertically ontox, we have proved the result.

Corollary 1.LetΣbe a finite set of hyperspheres ofR^d+1,Pow(Σ)its power diagram andPow_Q(Σ)the restriction ofPow(Σ)toQ. The vertical projection of Pow_Q(Σ) is the M¨obius diagramM¨ob(

ω

⁾of a set of M¨obius sites ofR^d.

Easy computations give

ω

^.

Combinatorial and Algorithmic Properties

It follows from Theorem 6 that the combinatorial complexity of the M¨obius diagram of n M¨obius sites in R^d is O(n^d2+1). This bound is tight since Aurenhammer [35] has shown that it is tight for multiplicatively weighted Voronoi diagrams.

We easily deduce from the proof of the Linearization Theorem 6 an algo-rithm for constructing M¨obius diagrams. First, we compute the power diagram of the hyperspheresΣiofR^d+1, intersect each of the faces of this diagram with the paraboloidQand then project the result on R^d.

Theorem 7.Let

ω

be a set ofnM¨obius sites inR^d,d≥2. The M¨obius dia-gramM¨ob(

ω

)of

ω

can be constructed in worst-case optimal timeΘ(nlogn+ n^d2+1).

Another consequence of the linearization theorem is the fact that any M¨obius diagram can be represented as a simplicial complexT_Qembedded in R^d+1. T_Q is a sub-complex of the regular triangulationT dual to the power diagram Pow(Σi) of the hyperspheresΣi. SinceT is embedded in R^d+1, T_Q is a simplicial complex ofR^d+1. More precisely,T_Q consists of the faces ofT that are dual to the faces of Pow_Q(Σ), i.e. the faces of the power diagram that intersectQ. We will callT_Q the dual of Pow_Q(Σ). Observe that since, in general, no vertex of Pow(Σ) lies on Q, T_Q is a d-dimensional simplicial complex (embedded in R^d+1).

Moreover, if the faces of Pow(Σ) intersectQtransversally and along topo-logical balls, then, by a result of Edelsbrunner and Shah [138],T_Q is homeo-morphic toQ and therefore toR^d. It should be noted that this result states that the simplicial complex T_Q has the topology ofR^d. This result, however, is mainly combinatorial, and does not imply that the embedding of T_Q into R^d+1 as a sub-complex of the regular triangulation T may be projected in a 1-1 manner ontoR^d.

Spherical Voronoi Diagrams

Lemma 1 states that the bisectors of two M¨obius sites is a hypersphere (possibly degenerated in a hyperplane). More generally, let us consider the Voronoi diagrams such that, for any two objects o_i ando_j ofO, the bisector σij = {x ∈ R^d, δi(x) = δj(x)} is a hypersphere. Such a diagram is called a spherical Voronoi diagram.

In Sect. 2.5, we will prove that any spherical Voronoi diagram ofR^d is a M¨obius diagram (Theorem 15).

M¨obius transformations are the transformations that preserve hyper-spheres. An example of a M¨obius transformation is the inversion with respect to a hypersphere. If the hypersphere is centered at c and has radius r, the inversion associates to a pointx∈R^d its image

x =c+r(x−c) (x−c)².

Moreover, it is known that any M¨obius transformation is the composition of up to four inversions [108]. An immediate consequence of Theorem 15 is that the set of M¨obius diagrams in R^d is stable under M¨obius transformations, hence their name.

M¨obius Diagrams on Spheres

Given a set

ω

^{of n M¨}obius sites of R^d+1, the restriction of their M¨obius diagram to a hypersphereS^dis called a M¨obius diagram onS^d. It can be shown that such a diagram is also the restriction of a power diagram of hyperspheres ofR^d+1 toS^d (Exercise 14) .

We define spherical diagrams onS^d as the diagrams onS^dwhose bisectors are hyperspheres ofS^d and that satisfy two properties detailed in Sect. 2.5.1 and 2.5.2. See Exercise 16 for more details on these conditions. This exercise proves that the restriction of a M¨obius diagram, i.e. a M¨obius diagram onS^d, is a spherical diagram.

Let us now prove the converse: any spherical diagram onS^d is a M¨obius diagram onS^d. Lethbe a hyperplane ofR^d+1. The stereographic projection that maps S^d to h maps any spherical diagram D on S^d to some spherical diagramDonh. Theorem 15 implies that thisDis in fact a M¨obius diagram.

Exercise 12 shows that D, which is the image of D by the inverse of the stereographic projection, is the restriction of some power diagram ofR^d+1 to S^d. Exercise 14 then proves that it is indeed a M¨obius diagram.

Exercise 12.Show that the linearization theorem and its corollary still hold if one replaces the paraboloidQby any hypersphere ofR^d+1 and the vertical projection by the corresponding stereographic projection.

Exercise 13.Show that the intersection of a M¨obius diagram in R^d with a k-flat or a k-sphereσis a M¨obius diagram inσ.

Exercise 14.Show that the restriction of a M¨obius diagram of n M¨obius sites to a hypersphere Σ ⊂R^d (i.e. a M¨obius diagram onΣ) is identical to the restriction of a power diagram ofn hyperspheres ofR^d withΣ, and vice versa.

Exercise 15.The predicates needed for constructing a M¨obius diagram are those needed to construct Pow(Σ) and those that decide whether a face of Pow(Σ) intersectsQ or not. Write the corresponding algebraic expressions.

Exercise 16.Explain how the two conditions A.C. and L.C.C. presented in Sect. 2.5.1 and 2.5.2 are to be adapted to the case of spherical diagrams on a sphere (Hint: consider the L.C.C. condition as a pencil condition, and define

a pencil of circles on a sphere as the intersection of a pencil of hyperplanes with this sphere).

Note that the restriction of a Voronoi diagram (affine or not) toS^dalways satisfies this adapted version of A.C. and prove that the restriction of an affine Voronoi diagram toS^d satisfies L.C.C. so that Exercise 14 allows to conclude that the restriction of a M¨obius diagram toS^d, i.e. M¨obius diagram onS^d, is a spherical diagram.

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