Kinetic Voronoi Diagrams and Delaunay Triangulations under Polygonal Distance FunctionsWork by P.A. and M.S. was supported by Grant 2012/229 from the U.S.-Israel Binational Science Foundation. Work by P.A. was also supported by NSF under grants CCF-09-40671, CCF-10-12254, and CCF-11-61359, by an ARO contract W911NF-13-P-0018, and by an ERDC contract W9132V-11-C-0003. Work by Haim Kaplan has been supported by grant 822/10 from the Israel Science Foundation, grant 1161/2011 from the German-Israeli Science Foundation, and by the Israeli Centers for Research Excellence (I-CORE) program (center no. 4/11). Work by N.R. was partially supported by Grants 975/06 and 338/09 from the Israel Science Fund, by Minerva Fellowship Program of the Max Planck Society, by the Fondation Sciences Mathématiques de Paris (FSMP), and by a public grant overseen by the French National Research Agency (ANR) as part of the Investissements d’Avenir program (reference: ANR-10-LABX-0098). Work by Micha Sharir has also been supported by NSF Grant CCF-08-30272, by Grants 338/09 and 892/13 from the Israel Science Foundation, by the Israeli Centers for Research Excellence (I-CORE) program (center no. 4/11), and by the Hermann Minkowski–MINERVA Center for Geometry at Tel Aviv University.

Let be a set of points and a convex -gon in . We analyze in detail the topological (or discrete) changes in the structure of the Voronoi diagram and the Delaunay triangulation of , under the convex distance function defined by , as the points of move along prespecified continuous trajectories. Assuming that each point of moves along an algebraic trajectory of bounded degree, we establish an upper bound of on the number of topological changes experienced by the diagrams throughout the motion; here is the maximum length of an -Davenport-Schinzel sequence, and is a constant depending on the algebraic degree of the motion of the points. Finally, we describe an algorithm for efficiently maintaining the above structures, using the kinetic data structure (KDS) framework.

1 Introduction

Let be a set of points in , and let be a compact convex (not necessarily polygonal) set in with nonempty interior and with the origin lying in its interior. For an ordered pair of points , the -distance from to is defined as

 dQ(x,y)=min{λ∣y∈x+λQ};

is a metric if and only if is centrally symmetric with respect to the origin (otherwise need not be symmetric). For a point of , the -Voronoi cell of is defined as

 VorQ(p)={x∈R2∣dQ(x,p)≤dQ(x,p′)∀p′∈P}.

If the points of are in general position with respect to (see Section 2 for the definition), the Voronoi cells of points in are nonempty, have pairwise-disjoint interiors, and partition the plane (see Figure 1(b)). The planar subdivision induced by these Voronoi cells is referred to as the -Voronoi diagram of and we denote it as .

The -Delaunay triangulation of , denoted by , is the dual structure of . Namely, a pair of points are connected by an edge in if and only if the boundaries of their respective -Voronoi cells and share a -Voronoi edge, given by

 epq={x∈R2∣dQ(x,p)=dQ(x,q)≤dQ(x,p′)∀p′∈P}.

can be defined directly as well: it is composed of all edges , with , for which there exists a homothetic placement of whose boundary touches and and whose interior contains no other points of .111We remark that is often defined in the literature as the set  [4, 9, 26]. If is not centrally symmetric, then this definition of is not the same as the one given above. Furthermore, under this definition, is an edge of if there exists a -empty homothetic placement of (and not of ) whose boundary touches and . Placements of with this latter property are called -empty. If is a circular disk then (resp., ) is the well-known Euclidean Delaunay triangulation (resp., Voronoi diagram) of .

If is in general position with respect to , then is spanned by so called -Delaunay triangles. Each of these triangles corresponds to the (unique) -empty homothetic placement of whose boundary touches and . That is, corresponds to a -Voronoi vertex that lies at equal -distances from , , and , so that is the center of (that is, is the image of the origin under the homothetic mapping of into ). If is smooth (e.g., as in the Euclidean case), then is a triangulation of the convex hull of ; otherwise it is a triangulation of a simply-connected polygonal subregion of , sometimes referred to as the support hull of (see [26] and Figure 1 (b)). The interior of the support hull may be empty, as shown in Figure 1 (c).

In many applications of Delaunay/Voronoi methods (e.g., mesh generation and kinetic collision detection), the points in move continuously, so these structures need to be updated efficiently as motion takes place. Even though the motion of the points of is continuous, the topological structures of and change only at discrete times when certain events occur.222The topological structures of and are the graphs that they define. More specifically, the topological structure of and consists of the set of triples of points defining the Voronoi vertices, and the sets of Voronoi and Delaunay edges. As we will see later each Voronoi edge is a sequence of one or more edgelets. Each such edgelet is defined by a pair of edges of . The sequences of pairs of edges of defining the edgelet structures of the Voronoi edges are also part of the topological structure of . Assume that each point of moves independently along some known trajectory. Let denote the position of point at time , and set . We call the motion of algebraic if each is a polynomial function of , and the degree of the motion of is the maximum degree of these polynomials.333This assumption can be somewhat relaxed to allow more general motions, as can be inferred from the analysis in the paper.

In this paper we focus on the case when is a convex -gon and study the resulting -Voronoi and -Delaunay structures as each point of moves continuously along an algebraic trajectory whose degree is bounded by a constant. Since will be either fixed or obvious from the context, we will use the simplified notations , , and to denote , , and , respectively.

Related work.

There has been extensive work on studying the geometric and topological structure of Voronoi diagrams and Delauany triangulations under convex distance functions; see e.g. [4] and the references therein. In the late 1970s, -time algorithms were proposed for computing the Voronoi diagram of a set of points in under any -metric [14, 18, 19]. In the mid 1980s, Chew and Drsydale [9] and Widmayer et al. [26] showed that if is a convex -gon, has size and that it can be computed in time. Motivated by a motion-planning application, Leven and Sharir [20] studied Voronoi diagrams under a convex polygonal distance function for the case where the input sites are convex polygons. Efficient divide-and-conquer, sweep-line, and edge-flip based incremental algorithms have been proposed to compute directly [10, 21, 25]. Several recent works study the structure of under a convex polyhedral distance function in  [7, 15, 17].

One of the hardest and best-known open problems in discrete and computational geometry is to determine the asymptotic behavior of the maximum possible number of discrete changes experienced by the Euclidean Delaunay triangulation during an algebraic motion of constant degree of the points of , where the prevailing conjecture is that this number is nearly quadratic in . A near-cubic bound was proved in [13]. After almost 25 years of no real progress, two recent works by one of the authors [22, 23] substantiate this conjecture, and establish an almost tight upper bound of , for any , for restricted motions where any four points of can become cocircular at most two times (in [22]) or at most three times (in [23]). In particular, the latter result [23], involving at most three cocircularities of any quadruple, applies to the case of points moving along lines at common (unit) speed. Only near-cubic bounds are known so far for more general motions. Chew [8] showed that the number of topological changes in the Delaunay triangulation under the or metric is , where is the almost-linear maximum length of a Davenport-Schinzel sequence of order on symbols, and is a constant that depends on the algebraic degree of the motions of the points. Chew’s result also holds for any convex quadrilateral . He focuses on bounding the number of changes in the Delaunay triangulation and not how it changes at each “event,” so his analysis omits some critical details of how the Delaunay triangulation and the Voronoi diagram change at an event; changes in the topological structure of are particularly subtle. Chew remarks, without supplying any details, that his technique can be extended to general convex polygons.

Later, Basch et al. [5] introduced the kinetic data structure (KDS in short) framework for designing efficient algorithms for maintaining a variety of geometric and topological structures of mobile data. Several algorithms have been developed in this framework for kinetically maintaining various geometric and topological structures; see [12]. The crux in designing an efficient KDS is finding a set of certificates that, on one hand, ensure the correctness of the configuration currently being maintained, and, on the other hand, are inexpensive to maintain as the points move. When a certificate fails during the motion of the objects, the KDS fixes the configuration, replaces the failing certificate(s) by new valid ones, and computes their failure times. The failure times, called events, are stored in a priority queue, to keep track of the next event that the KDS needs to process. The performance of a KDS is measured by the number of events that it processes, the time taken to process each event, and the total space used. If these parameters are small (in a sense that may be problem dependent and has to be made precise), the KDS is called, respectively, efficient, responsive, and compact. See [5, 12] for details.

Delaunay triangulations and Voronoi diagrams are well suited for the KDS framework because they admit local certifications associated with their individual features. These certifications fail only at the events when the topological structure of the diagrams changes. The resulting KDS is compact ( certificates suffice) and responsive (each update takes time, mainly to update the event priority queue), but its efficiency, namely, the number of events that it has to process, depends on the number of topological changes in , so a near quadratic bound on the number of events for the Euclidean case holds only when each point moves along some line with unit speed (or in similar situations when only three co-circularities can exist for any quadruple of points). A KDS for when is a convex quadrilateral was presented by Abam and de Berg [2], but it is not straightforward to extend their KDS for the case where is a general convex -gon. Furthermore, it is not clear how to use their KDS for maintaining .

Our contribution.

First, we establish a few key topological properties of and when is a set of stationary points in and is a convex -gon (Section 2). Although these properties follow from earlier work on this topic (see [4, Chapter 7]), we include them here because they are important for the kinetic setting and most of them have not been stated in earlier work in exactly the same form as here.

Next, we characterize the topological changes that and can undergo when the points of move along continuous trajectories (Section 3). These changes occur at critical moments when the points of are not in -general position, so that some points of are involved in a degenerate configuration with respect to . The most ubiquitous type of such events is when four points of become -cocircular, in the sense that there exists a -empty homothetic placement of whose boundary touches those four points.

We provide the first comprehensive and rigorous asymptotic analysis of the maximum number of topological changes that and can undergo during the motion of the points of (Section 4). Specifically, if has vertices, then and experience such changes, where is the almost-linear maximum length of a Davenport-Schinzel sequence of order on symbols, and is some constant that depends on the algebraic degree of the motions of the points. Some of these changes occur as components of so-called singular sequences, in which several events that affect the structure of and occur simultaneously, and their collective effect might involve a massive change in the topological structures of these diagrams. These compound effects are a consequence of the non-strict convexity of , and their analysis requires extra care. Nevertheless, the above near-quadratic bound on the number of changes also holds when we count each of the individual critical events in any such sequence separately.

Finally, we describe an efficient algorithm for maintaining and during an algebraic motion of , within the standard KDS framework (Section 5). Here we assume an algebraic model of computation, in which algebraic computations, including solving a polynomial equation of constant degree, can be performed in an exact manner, in constant time. The precise sense of this assumption is that comparisons between algebraic quantities that are defined in this manner can be performed exactly in constant time. This is a standard model used widely in theory [24, Section 6.1] and nowadays also in practice (see, e.g., [11]). This model allows us to perform in constant time the various computations that are needed by our KDS, the most ubiquitous of which are the calculation of the failure times of the various certificates being maintained; see Section 5 for details.

Stable Delaunay edges.

Our study of Voronoi diagrams under a convex polygonal distance function, to a large extent, is motivated by the notion of stable Delaunay edges, introduced by the authors in a companion paper [3], and defined as follows: Let be a Delaunay edge under the Euclidean norm, and let and be the two Delaunay triangles adjacent to . For a fixed parameter , is called an -stable (Euclidean) Delaunay edge if its opposite angles in these triangles satisfy . An equivalent and more useful definition, in terms of the Voronoi diagram, is that is -stable if the equal angles at which and see their common (Euclidean) Voronoi edge are at least each. It is shown in [3] that if is -stable in the Euclidean Delaunay triangulation, then it also appears, and at least -stable, in the -Delaunay triangulation for any shape that is sufficiently close to (in terms of its Hausdorff distance from) the unit disk. The results in this paper, along with the aforementioned result, imply that by maintaining , where is a regular (convex) -gon, for , we can maintain (a superset of) the stable edges of the Euclidean Delaunay triangulation, as a subgraph of , and that we have to handle only a nearly quadratic number of topological changes if the motion of the points of is algebraic of degree bounded by a constant. See [3] for details.

2 The topology of VD(P)

In this section we state and prove a few geometric and topological properties of the -Voronoi diagram of a set of stationary points when is a convex polygon.

Some notations.

Let be a convex -gon with vertices in clockwise order, whose interior contains the origin. For each , let denote the edge of , where index addition is modulo (so ). We refer to the origin as the center of and denote it by . A homothetic placement (or placement for short) of is represented by a pair , with and , so that ; is the location of the center of , and is the scaling factor of (about its center). The homothets of thus have three degrees of freedom.

There is an obvious bijection between the edges (and vertices) of and of , so, with a slight abuse of notation, we will not distinguish between them and use the same notation to refer to an edge or vertex of and to the corresponding edge or vertex of . For a point , let denote the homothetic copy of centered at such that its boundary touches the -nearest neighbor(s) of in , i.e., is represented by the pair where . In other words, is the largest homothetic copy of that is centered at whose interior is -empty.

Q-general position.

To simplify the presentation, we assume our point set to be in general position with respect to the underlying polygon . Specifically, this means that

• no pair of points of lie on a line parallel to a boundary edge or a diagonal of ,

• no four points of lie on the boundary of the same homothetic copy of , and

• if some three points in lie on the boundary of the same homothetic copy of , then each of them is incident to a relatively open edge of (and all the three edges are distinct, due to (Q1)), as opposed to one or more of these points touching a vertex of .

The above conditions can be enforced by an infinitesimally small rotation of or of .

Bisectors, corner placements, and edgelets.

The bisector between two points and , with respect to the distance function induced by , denoted by or , is the set of all points that satisfy . Equivalently, is the locus of the centers of all homothetic placements of that touch and on their boundaries; does not have to be -empty, so it may contain additional points of . If and are not parallel to an edge of (assumption (Q1)), then is a one-dimensional polygonal curve, whose structure will be described in detail momentarily.

A homothetic placement centered along that touches one of and , say, , at a vertex, and touches at the relative interior of an edge (as must be the case in general position) is called a corner placement at ; see Figure 2 (a). Note that a corner placement at which a vertex of (a copy of) touches has the property that the center of lies on the fixed ray emanating from in direction .

A non-corner placement centered on can be classified according to the pair of edges of , say, and , that touch and , respectively. We may assume (by (Q1)) that . Slide so that its center moves along and its size expands or shrinks to keep it touching and at the edges and , respectively. If and are parallel, then the center of traces a line segment in the direction parallel to and ; otherwise traces a segment in the direction that connects it to the intersection point of the lines containing (the copies on of) and . See Figure 2 (b) for the latter scenario. We refer to such a segment as an edgelet of , and label it by the pair (or by for brevity). The orientation of the edgelet depends only on the corresponding edges , and is independent of and . The structure of is fully determined by the following proposition, with a fairly straightforward proof that is omitted from here.

Lemma 2.1.

An edgelet with the label appears on if and only if there is an oriented line parallel to that crosses at (the relative interiors of) and , in this order.

See Figure 3 for an illustration. The endpoints of edgelets are called the breakpoints of . Each breakpoint is the center of a corner placement of ; If and are adjacent, then the edgelet labeled is a ray and the common endpoint of is one of the two vertices of extremal in the direction orthogonal to (i.e., these vertices have a supporting line parallel to ).

Assuming -general position of , Lemma 2.2 below implies that is the concatenation of exactly edgelets. Let and denote the two chains of , delimited by the vertices that are extremal in the direction orthogonal to , such that lies on an edge of and on an edge of at all placements of touching and and centered along the bisector . We orient both , so that they start (resp., terminate) at the vertex of that is furthest to the left (resp., to the right) of ; see Figure 3.

Our characterization of is completed by the following lemma, which follows from Lemma 2.1 and the preceding discussion.

Lemma 2.2.

Let be the sequence of labels of the edgelets of in their order along when we trace it so that lies on its left side and on its right side. Then appear (with possible repetitions as consecutive elements) in this order along , and appear (again, with possible repetitions) this order along . Furthermore, the following additional properties also hold:

• All the edges of (resp., ) appear, possibly with repetitions, in the first (resp., second) sequence.

• The elements of appear in clockwise order and the elements of in counterclockwise order along .

• Assuming general position, the passage from a label to the next label is effected by either replacing by the following edge on or by replacing by the following edge on . In the former (resp., latter) case, the common endpoint of the two edgelets corresponds to the corner placement of (resp., ) at the common vertex of and (resp., and ).

The proof of the lemma, whose details are omitted, proceeds by sweeping a line parallel to , and keeping track of the pairs of edges of that are crossed by the line, mapping each position of the line to a homothetic placement of that touches and at the images of the two intersection points.

For , let be the orientation of the line passing through the vertices and of , and let be the set of these orientations. partitions the unit circle into a collection of angular intervals (for a regular -gon, the number of intervals is only ). Lemmas 2.1 and 2.2 implies the following corollary:

Corollary 2.3.

The sequence of edgelet labels along is the same for all the ordered pairs of points such that the orientation of the vector lies in the same interval of .

The following additional property of bisectors is crucial for understanding the topological structure of .

Lemma 2.4.

Let be three distinct points of . The bisectors , can intersect at most once, assuming that and are in -general position.

Proof.

Suppose to the contrary that , intersect at two points. Then there exist two homothetic copies and of such that . However, it is well known that homothetic placements of behave like pseudo-disks, in the sense that the portion of the boundary of each of them outside the other homothetic placement is connected; see, e.g., [16]. Therefore, and intersect in at most two connected portions, each of which is either a point or a segment parallel to some edge of . Clearly, one of these connected components of must contains two out of the three points , , and , in contradiction to the fact that the points are in -general position. ∎

The following lemma provides additional details concerning the structure of the breakpoints of the bisectors in case is a regular -gon.

Lemma 2.5.

Let be a regular -gon, and let and be two points in general position with respect to . The breakpoints along the bisector correspond alternatingly to corner placements at and corner placements at .

Proof.

Refer to Figure 4. Suppose that two consecutive breakpoints of correspond to corner placements at . From Lemmas 2.1 and 2.2, we obtain that these corner placements are formed by two adjacent vertices, say and of , and lies in the relative interior of (the homothetic copies of) the same edge of at these placements. This implies that the projections of and in direction lie in the interior of the projection of the edge of in direction , which is impossible if is a regular -gon. Indeed, the convex hull of and of is an isosceles trapezoid , which implies that, for any other strip bounded by two parallel lines through and , cannot cross both boundary lines of . We note that if is the strip spanned by , then touches both lines bounding but does not cross any of them. This completes the proof of the lemma. ∎

Voronoi cells, edges, and vertices.

Each bisector partitions the plane into open regions and . Hence, for each point , its -Voronoi cell can be described as .

By -general position of , for any , is composed of -Voronoi edges, where each such edge is a maximal connected portion of the bisector , for some other point , that lies within . The portion of within this common boundary can be described as

 bpq∩⋂r≠p,qHpr=bpq∩⋂r≠p,qHqr.

That is, this portion is the locus of all centers of placements of for which the equal distances are the smallest among the distances from to the points of . Note that the homothetic copy of touches and and is -empty.

Since is in -general position, Lemma 2.4 guarantees that this portion of is either connected or empty. Therefore, any bisector contains at most one -Voronoi edge, which we denote by . This edge is called a corner edge if it contains a breakpoint (i.e., a center of a corner placement); otherwise it is a non-corner edge—a line segment.

The endpoints of -Voronoi edges are called -Voronoi vertices. By the -general position of , each such vertex is incident on exactly three Voronoi cells , , and . This vertex, denoted by , can be described as the center of the unique homothetic -empty placement of , whose boundary contains only the three points , , and of . ¿From the Delaunay point of view, contains the triangle .

We say that is an edgelet of if (i) is an edgelet of , and (ii) the Voronoi edge either contains or, at least, overlaps . We refer to an edgelet of as external if it contains one of the endpoints of , namely, a vertex of , and as internal otherwise. In general position, an external edgelet of is always properly contained in an edgelet of .

We conclude this section by making the following remarks. If assumption (Q3) does not hold, then a Voronoi vertex may coincide with a breakpoint of an edge adjacent to it; if (Q2) does not hold then a Voronoi vertex may have degree larger than three; if the segment connecting a pair is parallel to a diagonal of , then an edgelet of a Voronoi edge may degenerate to a single point; and if such a segment is parallel to an edge of then may be a two-dimensional region (Figure 7 middle). These degenerate configurations are discussed in detail in the next section.

3 Kinetic Voronoi and Delaunay diagrams

As the points of move along continuous trajectories, also changes continuously, namely, vertices of and breakpoints of edgelets trace continuous trajectories, but, unless the motion is very degenerate, the topological structure of changes only at discrete times, at which an edgelet in a Voronoi edge appears/disappears, a Voronoi vertex moves from one edgelet to another, or two adjacent Voronoi cells cease to be adjacent or vice versa (equivalently, an edge appears or disappears in ), because of a -cocircularity of four points of . In this section we discuss when do these changes occur and how does change at such instances. To simplify the presentation, (i) we assume that the orientations of the edges and diagonals , for all pairs of vertices of , are distinct, and that they are different from those of , for any vertex ; (ii) we make certain general-position assumptions on the trajectories of ; and (iii) we augment with some points at infinity. At the end of the section, we remark what happens if we do not make these assumptions or do not augment in this manner.

Augmenting P.

We add points to so that the convex hull of the augmented set does not change as the (original) points move, and the boundary of is this stationary convex hull at all times. Specifically, for each vertex of , we add a corresponding point at infinity, so that lies in the direction . Let denote the set of these new points. We maintain and . It can be checked that contains all edges of , some “unbounded” edges (connecting points of to points of ), and edges at infinity (forming the convex hull of ). Furthermore, every edge of incident on at least one point of is adjacent to two triangles; only the edges at infinity are “boundary” edges of the triangulation. During the motion of the points of , the points of remain stationary.

Let be a triangle of . There is a ()-empty homothetic copy of associated with , whose boundary touches the three vertices of . If two vertices of belong to and one vertex of is a point at infinity, then is a wedge formed by the two corresponding consecutive edges and of , each touching a vertex of not in (e.g., in Figure 5). If only one vertex of , say , belongs to , then is of the form (for some ), and there are arbitrarily large empty homothetic copies of incident on at the edge (e.g., in Figure 5). The number of triangles of the latter kind is only , one for each edge of . Abusing the notation slightly, we will use to denote from now on.

Q-general position for trajectories.

We assume that the trajectories of the points of are in -general position, which we define below. Informally, if the motion of each point of is algebraic of bounded degree, as we assume, then the time instances at which degenerate configurations occur, namely, configurations violating one of the assumptions (Q1)–(Q3), can be represented as the roots of certain constant-degree polynomials in . The present “kinetic” general-position assumption for the trajectories says that none of these polynomials is identically zero (so each of them has roots), that each root has multiplicity one (so the sign of the polynomial changes in the neighborhood of each root), and that the roots of all polynomials are distinct. We now spell out these conditions in more detail and make them geometrically concrete.

(T1) For any pair of points , for all , namely, and do not collide during the motion.

(T2) For any pair of points , there exist at most times when the segment is parallel to any given edge or diagonal of , and at each of these times properly crosses the line through parallel to , which moves continuously, together with (and does the same for the parallel line through ).

(T3) For any ordered set of three points and for any vertex and a pair of edges , of , there exist at most times when touches and touches at a corner placement of at in which touches . Furthermore, given that is not adjacent to , at each such time the point either enters or leaves the interior of the unique -empty homothetic copy of that touches at and at .

(T4) For any four points and any ordered quadruple , , , of edges of , such that at least three of these edges are distinct, there are only times at which there exists a placement of such that touch the respective relative interiors of , , , . We say that are -cocircular at these times. At any such -cocircularity, the four points are partitioned into two pairs, say, and , so that right before the cocircularity there exists a homothetic copy of that is disjoint from and and whose boundary touches and , and right afterwards there exists a homothetic copy of that is disjoint from and and whose boundary touches and .

(T5) Events of type (T2)–(T4) do not occur simultaneously, except when two points and become parallel to an edge of . In this case there could be many events of type (T3) and (T4) that occur simultaneously, each of which involves ; see below for more details.

Events.

Since the motion of is continuous, the topological changes in occur only when some points of are involved in a degenerate configuration, i.e., they violate one of the assumptions (Q1)–(Q3). However not every degenerate configuration causes a change in . We define an event to be the occurrence of a -empty placement of a homothetic copy of whose boundary contains two, three, or four points of that are in a -degenerate configuration. The center of such a placement lies on an edge or at a vertex of . The subset of points involved in the degenerate configuration is referred to as the subset involved in the event. The event is called a bisector, corner, or flip event if assumption (Q1), (Q2), or (Q3), respectively, is violated.

An event is called singular if some pair among the (constantly many) points involved in the event span a line parallel to an edge of . Otherwise, we say that the event is generic. The -general-position assumption on the trajectories of the points of implies, in particular, that (i) no generic event can occur simultaneously with any other event, and (ii) all singular events that occur at a given time, must involve the same pair of points that span a line parallel to an edge of .

The changes in are simple and local at a generic event, but can undergo a major change at a singular event. We therefore first discuss the changes at a generic event and then discuss singular events.

3.1 Generic events

Recall that the orientation of , for every pair , at a generic event is different from that of any edge of , which implies that no two points of lie on the same edge of a homothetic copy of at a generic event.

Bisector event.

A pair of points are incident to the vertices and , respectively, of a -empty homothetic copy of so that the vertices and are not consecutive along . In particular, is an edge in .

Recall that in our notation, is adjacent to the consecutive edges , and is adjacent to the consecutive edges ; in the present scenario, these four edges are all distinct. Without loss of generality, we may assume that before the event, there is an oriented line parallel to that intersects and in this order, and there is no such line after the event. Similarly, after the event there is an oriented line parallel to that intersects and in this order, and there is no such line before the event. Hence, Lemma 2.1 and assumption (T2) imply that loses a bounded edgelet with label , which is replaced by a new bounded edgelet with label . Our assumption (T5) implies cannot be an external edgelet of . Hence, appears shortly before the event as an internal edgelet of , shrinks to a point and is replaced by the new internal edgelet ; see Figure 6 (a). This is the only topological change in at this event.

Notice that whenever the direction of coincides with that of a diagonal of , and are incident to the vertices and , respectively, of a unique copy of . If this copy contains further points of , then the bisector still loses an edgelet , which is replaced by a new edgelet . However, both of these edgelets now belong to the portion of outside , so the discrete structure of does not change (and, therefore, no bisector event is recorded).

Corner event.

A corner event occurs when there is a -empty homothetic copy of with a corner placement of a vertex of at and two other points and lie on two distinct edges of , none of which is incident to . We refer to such an event as a generic corner event of .

This event corresponds to a vertex of , an endpoint of an edge , coinciding with a breakpoint of . Then , also an endpoint of the Voronoi edge , coincides with a breakpoint of as well. By assumption (T3), one of the -Voronoi edges and gains a new edgelet and the other loses an edgelet at this event; see Figure 6 (b).

Flip event.

A flip event occurs when there is a -empty homothetic copy of that touches four points at four distinct edges of , in this circular order along . By assumption (T4), up to a cyclic relabeling of the points, the Voronoi edge flips to a new Voronoi edge at this event; see Figure 6 (c). Note that (resp., ) is a non-corner edge immediately before (resp., after) the flip event, as both the vanishing edge and the newly emerging edge are “too short” to have breakpoints near the event (this is a consequence of the kinetic -general position assumptions).

This completes the description of the changes in at a generic event. We remark that a Voronoi edge newly appears or disappears only at a flip event, so, by definition, changes only at a flip event. Suppose the Voronoi edge flips to the edge at a flip event. Then are vertices of two adjacent triangles and immediately before the event, the edge of flips to at the event, and the Delaunay triangles flip to at the event; again, see Figure 6 (c).

3.2 Singular events

Recall that a singular event occurs, at time , if two points lie on an edge of a -empty homothetic copy of . Hence, a singular bisector event (involving and ) occurs at . We may assume that neither nor is in since in such a case the orientation of remains fixed throughout the motion (namely, it is ) and, as we have assumed, different from the orientations of the edges of .

Changes in bisectors.

Assume that becomes parallel to the edge , and, without loss of generality, assume that and have the same orientation, as in Figure 7 (center). When this occurs, the set of placements of at which both and touch is a wedge whose boundary rays and have respective directions and , and whose apex corresponds to the placement at which and touch and respectively; see Figure 7 (center).

Let (resp., ) denote an instance of time immediately before (resp., after) , so that no event occurs in the interval (resp., ). Then the terminal ray of that becomes the wedge at time is either in direction or at time . Without loss of generality, throughout the present discussion of the singular event, we assume that this ray is in the direction , i.e., it consists of all placements with touching and , the other edge adjacent to , touching . This ray is parallel to and approaches as time approaches ; see Figure 7 (left). By assumption (T2), the bisector at time contains a terminal ray parallel to , which consists of all placements with touching , and with the other edge adjacent to touching . At time this ray coincides with , which is clearly different from . See Figure 7 (right). That is, the terminal ray of instantly switches from to at time .

Changes in VD(P).

All topological changes in at the time of a singular event occur on the boundaries of the -Voronoi cells and . Since and are not in , both of these cells are bounded.

Refer to the state of at times , , and , as illustrated in Figure 8. For , let be the edgelet of the bisector that is parallel to at time , and let be the Voronoi vertex which is incident to and to some other pair of edges and . Similarly, for , let be the edgelet of the bisector