Kinetic Voronoi Diagrams and Delaunay Triangulations under Polygonal Distance Functions

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1 Kinetic Voronoi Diagrams and Delaunay Triangulations under Polygonal Distance Functions Pankaj K. Agarwal Haim Kalan Natan Rubin Micha Sharir May 13, 2015 Abstract Let P be a set of n oints and Q a convex k-gon in R 2. We analyze in detail the toological (or discrete) changes in the structure of the Voronoi diagram and the Delaunay triangulation of P, under the convex distance function defined by Q, as the oints of P move along resecified continuous trajectories. Assuming that each oint of P moves along an algebraic trajectory of bounded degree, we establish an uer bound of O(k 4 nλ r (n)) on the number of toological changes exerienced by the diagrams throughout the motion; here λ r (n) is the maximum length of an (n, r)-davenort-schinzel seuence, and r is a constant deending on the algebraic degree of the motion of the oints. Finally, we describe an algorithm for efficiently maintaining the above structures, using the kinetic data structure (KDS) framework. Work by P.A. and M.S. was suorted by Grant 2012/229 from the U.S.-Israel Binational Science Foundation. Work by P.A. was also suorted by NSF under grants CCF , CCF , and CCF , by an ARO contract W911NF-13-P-0018, and by an ERDC contract W9132V-11-C Work by H.K. has been suorted 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) rogram (center no. 4/11). Work by N.R. was artially suorted by Grants 975/06 and 338/09 from the Israel Science Fund, by Minerva Fellowshi Program of the Max Planck Society, by the Fondation Sciences Mathématiues de Paris (FSMP), and by a ublic grant overseen by the French National Research Agency (ANR) as art of the Investissements d Avenir rogram (reference: ANR-10-LABX-0098). Work by M. S. has also been suorted by NSF Grant CCF , by Grants 338/09 and 892/13 from the Israel Science Foundation, by the Israeli Centers for Research Excellence (I-CORE) rogram (center no. 4/11), and by the Hermann Minkowski MINERVA Center for Geometry at Tel Aviv University. Deartment of Comuter Science, Duke University, Durham, NC , USA; ankaj@cs.duke.edu. School of Comuter Science, Tel Aviv University, Tel Aviv 69978, Israel; haimk@tau.ac.il. Deartment of Comuter Science, Ben-Gurion University of the Negev, Beer-Sheva, Israel 84105, rubinnat.ac@gmail.com. School of Comuter Science, Tel Aviv University, Tel Aviv 69978, Israel; michas@tau.ac.il. 1

2 1 Introduction Let P be a set of n oints in R 2, and let Q be a comact convex (not necessarily olygonal) set in R 2 with nonemty interior and with the origin lying in its interior. For an ordered air of oints x, y R 2, the Q-distance from x to y is defined as d Q (x, y) = min{λ y x + λq}; d Q is a metric if and only if Q is centrally symmetric with resect to the origin (otherwise d Q need not be symmetric). For a oint of P, the Q-Voronoi cell of is defined as Vor Q () = {x R 2 d Q (x, ) d Q (x, ) P}. If the oints of P are in general osition with resect to Q (see Section 2 for the definition), the Voronoi cells of oints in P are nonemty, have airwise-disjoint interiors, and artition the lane (see Figure 1(b)). The lanar subdivision induced by these Voronoi cells is referred to as the Q-Voronoi diagram of P and we denote it as VD Q (P). (a) (b) (c) Figure 1. (a) The Euclidean Voronoi diagram (dotted) and Delaunay triangulation (solid). (b) VD Q (P) and DT Q (P) for an axis-arallel suare Q, i.e., the diagrams VD Q and DT Q under the L -metric. (c) VD Q (P) and DT Q (P), for the same Q as in (b), with an emty-interior suort hull (VD Q (P) has no vertices in this case). The Q-Delaunay triangulation of P, denoted by DT Q (P), is the dual structure of VD Q (P). Namely, a air of oints, P are connected by an edge in DT Q (P) if and only if the boundaries of their resective Q-Voronoi cells Vor Q () and Vor Q () share a Q-Voronoi edge, given by e = {x R 2 d Q (x, ) = d Q (x, ) d Q (x, ) P}. DT Q (P) can be defined directly as well: it is comosed of all edges, with, P, for which there exists a homothetic lacement of Q whose boundary touches and and whose interior contains no other oints of P. 1 Placements of Q whose interior contains no oint of P are called 1 We remark that Vor Q () is often defined in the literature as the set Vor Q () = {x R 2 d Q (, x) d Q (, x) P} [4, 9, 26]. If Q is not centrally symmetric, then this definition of Vor Q () is not the same as the one given above. Furthermore, under this definition, is an edge of DT Q (P) if there exists a P-emty homothetic lacement of Q (and not of Q) whose boundary touches and. 1

3 P-emty. If Q is a circular disk then DT Q (P) (res., VD Q (P)) is the well-known Euclidean Delaunay triangulation (res., Voronoi diagram) of P. If P is in general osition with resect to Q, then DT Q (P) is sanned by so called Q-Delaunay triangles. Each of these triangles r corresonds to the (uniue) P-emty homothetic lacement Q r of Q whose boundary touches,, and r. That is, r corresonds to a Q-Voronoi vertex v r that lies at eual Q-distances to,, and r, so that v r is the center of Q r (that is, v r is the image of the origin under the homothetic maing of Q into Q r ). If Q is smooth (e.g., as in the Euclidean case), then DT Q (P) is a triangulation of the convex hull of P; otherwise it is a triangulation of a simly-connected olygonal subregion of conv(p), sometimes referred to as the suort hull of P with resect to Q (see [26] and Figure 1 (b)). The interior of the suort hull may be emty, as shown in Figure 1 (c). In many alications of Delaunay/Voronoi methods (e.g., mesh generation and kinetic collision detection), the oints in P move continuously, so these structures need to be udated efficiently as motion takes lace. Even though the motion of the oints of P is continuous, the toological structures of VD Q (P) and DT Q (P) change only at discrete times when certain events occur. 2 Assume that each oint of P moves indeendently along some known trajectory. Let i (t) = (x i (t), y i (t)) denote the osition of oint i at time t, and set P(t) = { 1 (t),..., n (t)}. We call the motion of P algebraic if each x i (t), y i (t) is a olynomial function of t, and the degree of the motion of P is the maximum degree of these olynomials. 3 In this aer we focus on the case when Q is a convex k-gon and study the resulting Q-Voronoi and Q-Delaunay structures as each oint of P moves continuously along an algebraic trajectory whose degree is bounded by a constant. Since Q will be either fixed or obvious from the context, we will use the simlified notations Vor(), VD(P), and DT(P) to denote Vor Q (), VD Q (P), and DT Q (P), resectively. Related work. There has been extensive work on studying the geometric and toological structure of Voronoi diagrams and Delauany triangulations under convex distance functions; see e.g. [4] and the references therein. In the late 1970s, O(n log n)-time algorithms were roosed for comuting the Voronoi diagram of a set of n oints in R 2 under any L -metric [14, 18, 19]. In the mid 1980s, Chew and Drsydale [9] and Widmayer et al. [26] showed that if Q is a convex k-gon, VD(P) has O(nk) size and that it can be comuted in O(kn log n) time. Motivated by a motion-lanning alication, Leven and Sharir [20] studied Voronoi diagrams under a convex olygonal distance function for the case where the inut sites are convex olygons. Efficient divide-and-conuer, swee-line, and edge-fli based incremental algorithms have been roosed to comute DT(P) directly [10, 21, 25]. Several recent works study the structure of VD(P) under a convex olyhedral distance function in R 3 [7, 15, 17]. One of the hardest and best-known oen roblems in discrete and comutational geometry is to determine the asymtotic behavior of the maximum ossible number of discrete changes 2 The toological structures of DT Q (P) and VD Q (P) are the grahs that they define. More secifically, the toological structure of DT Q (P) and VD Q (P) consists of the set of triles of oints of P defining the Voronoi vertices, and the sets of Voronoi and Delaunay edges and faces, where each edge is associated with a air of oints of P and with features of Q that define it. As we will see later each Voronoi edge is a seuence of one or more edgelets. Each such edgelet is defined by a air of edges of Q. The seuences of airs of edges of Q defining the edgelet structures of the Voronoi edges are also art of the toological structure of VD Q (P). 3 This assumtion can be somewhat relaxed to allow more general motions, as can be inferred from the analysis in the aer. 2

4 exerienced by the Euclidean Delaunay triangulation during an algebraic motion of constant degree of the oints of P, where the revailing conjecture is that this number is nearly uadratic in n. A near-cubic bound was roved in [13]. After almost 25 years of no real rogress, two recent works by one of the authors [22, 23] substantiate this conjecture, and establish an almost tight uer bound of O(n 2+ε ), for any ε > 0, for restricted motions where any four oints of P can become cocircular at most twice (in [22]) or at most three times (in [23]). In articular, the latter result [23], involving at most three cocircularities of any uadrule, alies to the case of oints moving along lines at common (unit) seed. Still, only near-cubic bounds are known so far for more general motions. Chew [8] showed that the number of toological changes in the Delaunay triangulation under the L 1 or L metric is O(nλ r (n)), where λ r (n) is the almost-linear maximum length of a Davenort- Schinzel seuence of order r on n symbols, and r is a constant that deends on the algebraic degree of the motions of the oints. Chew s result also holds for any convex uadrilateral Q. 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 toological structure of VD(P) are articularly subtle. Chew remarks, without sulying any details, that his techniue can be extended to general convex olygons. 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 toological structures of mobile data. Several algorithms have been develoed in this framework for kinetically maintaining various geometric and toological 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 inexensive to maintain as the oints move. When a certificate fails during the motion of the objects, the KDS fixes the configuration, relaces the failing certificate(s) by new valid ones, and comutes their failure times. The failure times, called events, are stored in a riority ueue, to kee track of the next event that the KDS needs to rocess. The erformance of a KDS is measured by the number of events that it rocesses, the time taken to rocess each event, and the total sace used. If these arameters are small (in a sense that may be roblem deendent and has to be made recise), the KDS is called, resectively, efficient, resonsive, and comact. 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 toological structure of the diagrams changes. The resulting KDS is comact (O(n) certificates suffice) and resonsive (each udate takes O(log n) time, mainly to udate the event riority ueue), but its efficiency, namely, the number of events that it has to rocess, deends on the number of toological changes in DT(P), so a near uadratic bound on the number of events for the Euclidean case holds only when each oint moves along some line with unit seed (or in similar situations when only three co-circularities can exist for any uadrule of oints). A KDS for DT(P) when Q is a convex uadrilateral was resented by Abam and de Berg [1], but it is not straightforward to extend their KDS for the case where Q is a general convex k-gon. Furthermore, it is not clear how to use their KDS for maintaining VD(P). Our contribution. First, we establish a few key toological roerties of VD(P) and DT(P) when P is a set of n stationary oints in R 2 and Q is a convex k-gon (Section 2). Although these roerties follow from earlier work on this toic (see [4, Chater 7]), we include them here because they are 3

5 imortant for the kinetic setting and most of them have not been stated in earlier work in exactly the same general and detailed form as here. Next, we characterize the toological changes that VD(P) and DT(P) can undergo when the oints of P move along continuous trajectories (Section 3). These changes occur at critical moments when the oints of P are not in Q-general osition, so that some O(1) oints of P are involved in a degenerate configuration with resect to Q. The most ubiuitous tye of such events is when four oints of P become Q-cocircular, in the sense that there exists a P-emty homothetic lacement of Q whose boundary touches those four oints. We rovide the first comrehensive and rigorous asymtotic analysis of the maximum number of toological changes that VD(P) and DT(P) can undergo during the motion of the oints of P (Section 4). Secifically, if Q has k vertices, then VD(P) and DT(P) exerience O(k 4 nλ r (n)) such changes, where λ r (n) is the almost-linear maximum length of a Davenort-Schinzel seuence of order r on n symbols, and r is some constant that deends on the algebraic degree of the motions of the oints. Some of these changes occur as comonents of so-called singular seuences, in which several events that affect the structure of VD(P) and DT(P) occur simultaneously, and their collective effect might involve a massive change in the toological structures of these diagrams. These comound effects are a conseuence of the non-strict convexity of Q, and their analysis reuires extra care. Nevertheless, the above near-uadratic bound on the number of changes also holds when we count each of the individual critical events in any such seuence searately. Finally, we describe an efficient algorithm for maintaining VD(P) and DT(P) during an algebraic motion of P, within the standard KDS framework (Section 5). Here we assume an algebraic model of comutation, in which algebraic comutations, including solving a olynomial euation of constant degree, can be erformed in an exact manner, in constant time. The recise sense of this assumtion is that comarisons between algebraic uantities that are defined in this manner can be erformed exactly in constant time. This is a standard model used widely in theory [24, Section 6.1] and nowadays also in ractice (see, e.g., [11]). This model allows us to erform in constant time the various comutations that are needed by our KDS, the most ubiuitous of which are the calculation of the failure times of the various certificates being maintained; see Section 5 for details. This aer is a stand-alone study of a fairly basic roblem in the theory and ractice of Delaunay and Voronoi diagrams. In contrast with earlier studies, it analyzes the roblem in full generality, and addresses all the intricate asects of the kinetic changes that the diagrams undergo. In addition, the results of this aer form a major ingredient of the analysis in the comanion aer [3] (see also [2]), concerning stable Delaunay grahs. Details concerning this construct, its roerties, and its connection to the results of this aer are resented in the the comanion aer. 2 The toology of VD(P) In this section we state and rove a few geometric and toological roerties of the Q-Voronoi diagram of a set of stationary oints when Q is a convex olygon. Some notations. Let Q be a convex k-gon with vertices v 0,..., v k 1 in clockwise order, whose interior contains the origin. For each 0 i < k, let e i denote the edge v i v i+1 of Q, where index addition is modulo k (so v k = v 0 ). We refer to the origin as the center of Q and denote it by o. A homothetic lacement (or lacement for short) Q of Q is reresented by a air (, λ), with R 2 and 4

6 λ R +, so that Q = + λq; is the location of the center of Q, and λ is the scaling factor of Q (about its center). The homothets of Q thus have three degrees of freedom. There is an obvious bijection between the edges (and vertices) of Q and of Q, 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 Q and to the corresonding edge or vertex of Q. For a oint u R 2, let Q[u] denote the homothetic coy of Q centered at u such that its boundary touches the d Q -nearest neighbor(s) of u in P, i.e., Q[u] is reresented by the air (u, λ) where λ = min P d Q (u, ). In other words, Q[u] is the largest homothetic coy of Q that is centered at u whose interior is P-emty. Q-general osition. To simlify the resentation, we assume our oint set P to be in general osition with resect to the underlying olygon Q. Secifically, this means that (Q1) no air of oints of P lie on a line arallel to a boundary edge or a diagonal of Q, (Q2) no four oints of P lie on the boundary of the same homothetic coy Q of Q, and (Q3) if some three oints in P lie on the boundary of the same homothetic coy Q of Q, then each of them is incident to a relatively oen edge of Q (and all the three edges are distinct, due to (Q1)), as oosed to one or more of these oints touching a vertex of Q. The above conditions can be enforced by an infinitesimally small rotation of Q or of P. Bisectors, corner lacements, and edgelets. The bisector between two oints and, with resect to the distance function d Q induced by Q, denoted by b or b, is the set of all oints x R 2 that satisfy d Q (x, ) = d Q (x, ). Euivalently, b is the locus of the centers of all homothetic lacements Q of Q that touch and on their boundaries; Q does not have to be P-emty, so it may contain additional oints of P \ {, }. If and are not arallel to an edge of Q (assumtion (Q1)), then b is a one-dimensional olygonal curve, whose structure will be described in detail momentarily. A homothetic lacement Q centered along b 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 osition) is called a corner lacement at ; see Figure 2 (a). Note that a corner lacement at which a vertex v i of (a coy Q of) Q touches has the roerty that the center o of Q lies on the fixed ray emanating from in direction v i o. Q Q e j o b o ξ ij v i e i (a) (b) Figure 2. Possible lacements of Q on b. (a) A corner lacement at. The center of Q lies on the fixed ray emanating from in direction v i o. (b) A lacement of Q with edge contacts of e i and e j at and, resectively. (The centers of) such lacements trace an edgelet of b with label (e i, e j ). 5

7 A non-corner lacement Q centered on b can be classified according to the air of edges of Q, say, e i and e j, that touch and, resectively. We may assume (by (Q1)) that e i = e j. Slide Q so that its center o moves along b and its size exands or shrinks to kee it touching and at the edges e i and e j, resectively. If e i and e j are arallel, then the center o of Q traces a line segment in the direction arallel to e i and e j ; otherwise o traces a segment in the direction that connects it to the intersection oint ξ ij of the lines containing (the coies on Q of) e i and e j. See Figure 2 (b) for the latter scenario. We refer to such a segment g as an edgelet of b, and label it by the air ψ(g) = (e i, e j ) (or by (i, j) for brevity). The orientation of the edgelet deends only on the corresonding edges e i, e j, and is indeendent of and. The structure of b is fully determined by the following roosition, with a fairly straightforward roof that is omitted from here. Lemma 2.1. An edgelet g with the label ψ(g) = (e i, e j ) aears on b if and only if there is an oriented line arallel to that crosses Q at (the relative interiors of) e i and e j, in this order. C b 4 (1,5) 3 (1,4) 5 2 (2,4) C 1 (2,3) Figure 3. The edgelets of b. The breakoints of b corresond to corner lacements of Q. We have C = 2, 1 and C = 3, 4, 5. The terminal edgelets of b are the rays with labels (1, 5) and (2, 3). See Figure 3 for an illustration. The endoints of edgelets are called the breakoints of b. Each breakoint is the center of a corner lacement of Q; If e i and e j are adjacent, then the edgelet labeled (i, j) is a ray and the common endoint of e i, e j is one of the two vertices of Q extremal in the direction orthogonal to (i.e., these vertices have a suorting line arallel to ). Assuming Q-general osition of P, Lemma 2.2 below imlies that b is the concatenation of exactly k 1 edgelets. Let C and C denote the two chains of Q, delimited by the vertices that are extremal in the direction orthogonal to, such that lies on an edge of C and on an edge of C at all lacements of Q touching and and centered along the bisector b. We orient both C, C so that they start (res., terminate) at the vertex of Q that is furthest to the left (res., to the right) of ; see Figure 3. Our characterization of b is comleted by the following lemma, which follows from Lemma 2.1 and the receding discussion. Lemma 2.2. Let (e 11, e 21 ),..., (e 1h, e 2h ) be the seuence of labels of the edgelets of b in their order along b when we trace it so that lies on its left side and on its right side. Then e 11,..., e 1h aear (with ossible reetitions as consecutive elements) in this order along C, and e 21,..., e 2h aear (again, with ossible reetitions) this order along C. Furthermore, the following additional roerties also hold: (a) All the edges of C (res., C ) aear, ossibly with reetitions, in the first (res., second) seuence. (b) The elements of C aear in clockwise order and the elements of C in counterclockwise order along Q. 6

8 (c) Assuming general osition, the assage from a label (e 1i, e 2i ) to the next label (e 1(i+1), e 2(i+1) ) is effected by either relacing e 1i by the following edge on C or by relacing e 2i by the following edge on C. In the former (res., latter) case, the common endoint of the two edgelets corresonds to the corner lacement of (res., ) at the common vertex of e 1i and e 1(i+1) (res., e 2i and e 2(i+1) ). The roof of the lemma, whose details are omitted, roceeds by sweeing a line arallel to, and keeing track of the airs of edges of Q that are crossed by the line, maing each osition of the line to a homothetic lacement of Q that touches and at the images of the two intersection oints. For 0 i < j < k, let θ ij be the orientation of the line assing through the vertices v i and v j of Q, and let Θ be the set of these orientations. Θ artitions the unit circle S 1 into a collection I of O(k 2 ) angular intervals (for a regular k-gon, the number of intervals is only Θ(k)). Lemmas 2.1 and 2.2 imlies the following corollary: Corollary 2.3. The seuence of edgelet labels along b is the same for all the ordered airs of oints, such that the orientation of the vector lies in the same interval of I. The following additional roerty of bisectors is crucial for understanding the toological structure of VD(P). Lemma 2.4. Let, 1, 2 be three distinct oints of P. The bisectors b 1, b 2 can intersect at most once, assuming that, 1 and 2 are in Q-general osition. Proof. Suose to the contrary that b 1, b 2 intersect at two oints. Then there exist two homothetic coies Q 1 and Q 2 of Q such that, 1, 2 Q 1 Q 2. However, it is well known that homothetic lacements of Q behave like seudo-disks, in the sense that the ortion of the boundary of each of them outside the other homothetic lacement is connected; see, e.g., [16]. Therefore, Q 1 and Q 2 intersect in at most two connected ortions, each of which is either a oint or a segment arallel to some edge of Q. Clearly, one of these connected comonents of Q 1 Q 2 must contains two out of the three oints, 1, and 2, in contradiction to the fact that the oints are in Q-general osition. The following lemma rovides additional details concerning the structure of the breakoints of the bisectors in case Q is a regular k-gon. Lemma 2.5. Let Q be a regular k-gon, and let and be two oints in general osition with resect to Q. The breakoints along the bisector b corresond alternatingly to corner lacements at and corner lacements at. Proof. Refer to Figure 4. Suose that two consecutive breakoints of b corresond to corner lacements at. From Lemmas 2.1 and 2.2, we obtain that these corner lacements are formed by two adjacent vertices, say v 0 and v 1 of Q, and lies in the relative interior of (the homothetic coies of) the same edge e of Q at these lacements. This imlies that the rojections of v 0 and v 1 in direction lie in the interior of the rojection of the edge e of Q in direction, which is imossible if Q is a regular k-gon. Indeed, the convex hull of e 0 = v 0 v 1 and of e is an isosceles traezoid τ, which imlies that, for any other stri σ bounded by two arallel lines through v 0 and v 1, e cannot cross both boundary lines of σ. We note that if σ is the stri sanned by τ, then e touches both lines bounding σ but does not cross any of them. This comletes the roof of the lemma. 7

9 b Figure 4. The bisector b for a regular octagon Q; it has seven edgelets and the centers of the corner lacements along b alternate between (hollow circles) and (filled circles). Voronoi cells, edges, and vertices. Each bisector b artitions the lane into oen regions H = {x d Q (, x) < d Q (, x)} and H = {x d Q (, x) < d Q (, x)}. Hence, for each oint P, its Q-Voronoi cell Vor() can be described as P\{} H. By Q-general osition of P, for any P, Vor() is comosed of Q-Voronoi edges, where each such edge is a maximal connected ortion of the bisector b, for some other oint P, that lies within Vor() Vor(). The ortion of b within this common boundary can be described as b r =, H r = b That is, this ortion is the locus of all centers x of lacements of Q for which the eual distances d Q (x, ) = d Q (x, ) are the smallest among the distances from x to the oints of P. Note that the homothetic coy x + d Q (x, )Q of Q touches and and is P-emty. Since P is in Q-general osition, Lemma 2.4 guarantees that this ortion of b is either connected or emty. Therefore, any bisector b contains at most one Q-Voronoi edge, which we denote by e. This edge is called a corner edge if it contains a breakoint (i.e., a center of a corner lacement); otherwise it is a non-corner edge a line segment. The endoints of Q-Voronoi edges e are called Q-Voronoi vertices. By the Q-general osition of P, each such vertex is incident on exactly three Voronoi cells Vor(), Vor(), and Vor(r). This vertex, denoted by ν r, can be described as the center of the uniue homothetic P-emty lacement Q = Q[ν r ] of Q, whose boundary contains only the three oints,, and r of P. From the Delaunay oint of view, DT(P) contains the triangle r. We say that g is an edgelet of e if (i) g is an edgelet of b, and (ii) the Voronoi edge e either contains or, at least, overlas g. We refer to an edgelet g of e as external if it contains one of the endoints of e, namely, a vertex of VD(P), and as internal otherwise. In general osition, an external edgelet of e is always roerly contained in an edgelet of b. We conclude this section by making the following remarks. If assumtion (Q3) does not hold, then a Voronoi vertex may coincide with a breakoint 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 air, P is arallel to a diagonal of Q, then an edgelet of a Voronoi edge may degenerate to a single oint; and if such a segment is arallel to an edge of Q then b may be a two-dimensional region (Figure 7 middle). These degenerate configurations are discussed in detail in the next section. r =, H r. 8

10 3 Kinetic Voronoi and Delaunay diagrams As the oints of P move along continuous trajectories, VD(P) also changes continuously, namely, vertices of VD(P) and breakoints of edgelets trace continuous trajectories, but, unless the motion is very degenerate, the toological structure of VD(P) changes only at discrete times, at which an edgelet in a Voronoi edge aears/disaears, a Voronoi vertex moves from one edgelet to another, or two adjacent Voronoi cells cease to be adjacent or vice versa (euivalently, an edge aears or disaears in DT(P)), because of a Q-cocircularity of four oints of P. In this section we discuss when do these changes occur and how does VD(P) change at such instances. To simlify the resentation, (i) we assume that the orientations of the edges and diagonals v i v j, for all airs of vertices v i, v j of Q, are distinct, and that they are different from those of ov i, for any vertex v i ; (ii) we make certain general-osition assumtions on the trajectories of P; and (iii) we augment P with some oints at infinity. At the end of the section, we remark what haens if we do not make these assumtions or do not augment P in this manner. Augmenting P. We add oints to P so that the convex hull of the augmented set does not change as the (original) oints move, and the boundary of DT(P) is this stationary convex hull at all times. Secifically, for each vertex v i of Q, we add a corresonding oint i at infinity, so that i lies in the direction v i o. Let P denote the set of these k new oints. We maintain VD(P P ) and DT(P P ). It can be checked that DT(P P ) contains all edges of DT(P), some unbounded edges (connecting oints of P to oints of P ), and k edges at infinity (forming the convex hull of P P ). Furthermore, every edge of DT(P P ) incident on at least one oint of P is adjacent to two triangles; only the edges at infinity are boundary edges of the triangulation. During the motion of the oints of P, the oints of P remain stationary. Let be a triangle of DT(P P ). There is a (P P )-emty homothetic coy Q of Q associated with, whose boundary touches the three vertices of. If two vertices of belong to P and one vertex of is a oint i at infinity, then Q is a wedge formed by the two corresonding consecutive edges e i 1 and e i of Q, each touching a vertex of not in P (e.g., C 1 in Figure 5). If only one vertex of, say, belongs to P, then is of the form i i+1 (for some 0 i < k), and there are arbitrarily large emty homothetic coies of Q incident on at the edge e i = v i v i+1 (e.g., C 2 in Figure 5). The number of triangles of the latter kind is only k, one for each edge of Q. Abusing the notation slightly, we will use P to denote P P from now on. Q-general osition for trajectories. We assume that the trajectories of the oints of P are in Q- general osition, which we define below. Informally, if the motion of each oint of P is algebraic of bounded degree, as we assume, then the time instances at which degenerate configurations occur, namely, configurations violating one of the assumtions (Q1) (Q3), can be reresented as the roots of certain constant-degree olynomials in t. The resent kinetic general-osition assumtion for the trajectories says that none of these olynomials is identically zero (so each of them has O(1) roots), that each root has multilicity one (so the sign of the olynomial changes in the neighborhood of each root), and that the roots of all olynomials are distinct. We now sell out these conditions in more detail and make them geometrically concrete. (T1) For any air of oints, P, (t) = (t) for all t, namely, and do not collide during the motion. 9

11 C C Figure 5. The extended DT(P) under the L -metric (where Q is an axis-arallel suare) for the set of oints in Figure 1, with four oints 1,..., 4 added at infinity in directions (±1, ±1). Each of the four shaded triangles has two vertices at infinity, and the unbounded half-stris between them reresent triangles with one vertex at infinity. The emty wedge C 1 corresonds to 12 1 (the half-stri right above the left shaded triangle), and the arbitrarily large emty suare C 2 (a halflane in the limit) corresonds to (the right shaded triangle). (T2) For any air of oints, P, there exist at most O(1) times when the segment is arallel to any given edge or diagonal v i v j of Q, and at each of these times roerly crosses the line through arallel to v i v j, which moves continuously, together with (and does the same for the arallel line through ). (T3) For any ordered set of three oints,, r P and for any vertex v and a air of edges e, e r of Q, there exist at most O(1) times when touches e and r touches e r at a corner lacement Q of Q at in which touches v. Furthermore, given that e is not adjacent to v, at each such time the oint r either enters or leaves the interior of the uniue P-emty homothetic coy of Q that touches at v and at e. (T4) For any four oints,, a, b P and any ordered uadrule e, e, e a, e b of edges of Q, such that at least three of these edges are distinct, there are only O(1) times at which there exists a lacement Q of Q such that,, a, b touch the resective relative interiors of e, e, e a, e b. We say that,, a, b are Q-cocircular at these O(1) times. At any such Q-cocircularity, the four oints,, a, b are artitioned into two airs, say, (, ) and (a, b), so that right before the cocircularity there exists a homothetic coy of Q that is disjoint from a and b and whose boundary touches and, and right afterwards there exists a homothetic coy of Q that is disjoint from and and whose boundary touches a and b. (T5) Events of tye (T2) (T4) do not occur simultaneously, excet when two oints and become arallel to an edge of Q. In this case there could be many events of tye (T3) and (T4) that occur simultaneously, each of which involves, ; see below for more details. Events. Since the motion of P is continuous, the toological changes in VD(P) occur only when some oints of P are involved in a degenerate configuration, i.e., they violate one of the assumtions (Q1) (Q3). However not every degenerate configuration causes a change in VD(P). We define an event to be the occurrence of a P-emty lacement of a homothetic coy of Q whose boundary contains two, three, or four oints of P that are in a Q-degenerate configuration. The center of such 10

12 a lacement lies on an edge or at a vertex of VD(P). The subset of oints involved in the degenerate configuration is referred to as the subset involved in the event. The event is called a bisector, corner, or fli event if assumtion (Q1), (Q2), or (Q3), resectively, is violated. An event is called singular if some air among the (constantly many) oints involved in the event san a line arallel to an edge of Q. Otherwise, we say that the event is generic. The Q- general-osition assumtion on the trajectories of the oints of P imlies, in articular, 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 air of oints, that san a line arallel to an edge of Q. The changes in VD(P) are simle and local at a generic event, but VD(P) can undergo a major change at a singular event. We therefore first discuss the changes at a generic event and then discuss singular events. 2 3 (1,5) (1,5) (1,5) (1,4) 4 (2,5) 5 (2,4) 1 (2,4) (2,4) r r r e c r e c r c r e (a) (b) (c) e wr w e w w r r r Figure 6. Different tyes of generic events for a entagon Q: (a) a bisector event, (b) a corner event, and (c) a fli event. Thick segments denote an aroriate ortion of DT(P) and the elements of VD(P) that change at the event. 3.1 Generic events Recall that the orientation of, for every air, P, at a generic event is different from that of any edge of Q, which imlies that no two oints of P lie on the same edge of a homothetic coy of Q at a generic event. Bisector event. A air of oints, P are incident to the vertices v i and v j, resectively, of a P-emty homothetic coy of Q so that the vertices v i and v j are not consecutive along Q. In articular, e is an edge in VD(P). 11

13 Recall that in our notation, v i is adjacent to the consecutive edges e i 1, e i, and v j is adjacent to the consecutive edges e j 1, e j ; in the resent scenario, these four edges are all distinct. Without loss of generality, we may assume that before the event, there is an oriented line arallel to that intersects e i 1 and e j 1 in this order, and there is no such line after the event. Similarly, after the event there is an oriented line arallel to that intersects e i and e j in this order, and there is no such line before the event. Hence, Lemma 2.1 and assumtion (T2) imly that e loses a bounded edgelet with label (i 1, j 1), which is relaced by a new bounded edgelet with label (i, j). Our assumtion (T5) imlies (i 1, j 1) cannot be an external edgelet of e. Hence, (i 1, j 1) aears shortly before the event as an internal edgelet of e, shrinks to a oint and is relaced by the new internal edgelet (i, j); see Figure 6 (a). This is the only toological change in VD(P) at this event. Notice that whenever the direction of coincides with that of a diagonal v i v j of Q, and are incident to the vertices v i and v j, resectively, of a uniue coy of Q. If this coy contains further oints of P, then the bisector b still loses an edgelet (i 1, j 1), which is relaced by a new edgelet (i, j). However, both of these edgelets now belong to the ortion of b outside e, so the discrete structure of VD(P) does not change (and, therefore, no bisector event is recorded). Corner event. A corner event occurs when there is a P-emty homothetic coy Q = Q[ν r ] of Q with a corner lacement of a vertex of v Q at and two other oints and r lie on two distinct edges of Q, none of which is incident to v. We refer to such an event as a generic corner event of. This event corresonds to a vertex ν r of VD(P), an endoint of an edge e, coinciding with a breakoint of b. Then ν r, also an endoint of the Voronoi edge e r, coincides with a breakoint of b r as well. By assumtion (T3), one of the Q-Voronoi edges e and e r gains a new edgelet and the other loses an edgelet at this event; see Figure 6 (b). Fli event. A fli event occurs when there is a P-emty homothetic coy Q of Q that touches four oints,,, at four distinct edges of Q, in this circular order along Q. By assumtion (T4), u to a cyclic relabeling of the oints, the Voronoi edge e flis to a new Voronoi edge e at this event; see Figure 6 (c). Note that e (res., e ) is a non-corner edge immediately before (res., after) the fli event, as both the vanishing edge e and the newly emerging edge e are too short to have breakoints near the event (this is a conseuence of the kinetic Q-general osition assumtions). This comletes the descrition of the changes in VD(P) at a generic event. We remark that a Voronoi edge newly aears or disaears only at a fli event, so, by definition, DT(P) changes only at a fli event. Suose the Voronoi edge e flis to the edge e at a fli event. Then,,, are vertices of two adjacent triangles and immediately before the event, the edge of DT(P) flis to at the event, and the Delaunay triangles, fli to, at the event; again, see Figure 6 (c). 3.2 Singular events Recall that a singular event occurs, at time t 0, if two oints, P lie on an edge e i = v i v i+1 of a P-emty homothetic coy of Q. Hence, a singular bisector event (involving and ) occurs at t 0. We may assume that neither nor is in P since in such a case the orientation of remains fixed 12

14 ρ + v i ρ ζ v i v i v i+1 v i+1 v i+1 Figure 7. A singular bisector event. The ray ρ of b is relaced instantly by the ray ρ +, and the entire shaded wedge is art of b at the event itself. throughout the motion (namely, it is ov i ) and, as we have assumed, different from the orientations of the edges of Q. Changes in bisectors. Assume that becomes arallel to the edge e i, and, without loss of generality, assume that and v i v i+1 have the same orientation, as in Figure 7 (center). When this occurs, the set of lacements Q of Q at which both and touch e i is a wedge W 0 whose boundary rays ρ and ρ + have resective directions v i o and v i+1 o, and whose aex ζ corresonds to the lacement at which and touch v i and v i+1 resectively; see Figure 7 (center). Let t 0 (res., t+ 0 ) denote an instance of time immediately before (res., after) t 0, so that no event occurs in the interval [t 0, t 0) (res., (t 0, t + 0 ]). Then the terminal ray of b that becomes the wedge W 0 at time t 0 is either in direction v i o or v i+1 o at time t 0. Without loss of generality, throughout the resent discussion of the singular event, we assume that this ray is in the direction v i o, i.e., it consists of all lacements with e i touching and e i 1, the other edge adjacent to v i, touching. This ray is arallel to ρ and aroaches ρ as time aroaches t 0 ; see Figure 7 (left). By assumtion (T2), the bisector b at time t + 0 contains a terminal ray arallel to ρ+, which consists of all lacements with e i touching, and with the other edge e i+1 adjacent to v i+1 touching. At time t 0 this ray coincides with ρ +, which is clearly different from ρ. See Figure 7 (right). That is, the terminal ray of b instantly switches from ρ to ρ + at time t 0. Changes in VD(P). All toological changes in VD(P) at the time t 0 of a singular event occur on the boundaries of the Q-Voronoi cells Vor() and Vor(). Since and are not in P, both of these cells are bounded. Refer to the state of VD(P) at times t 0, t 0, and t + 0, as illustrated in Figure 8. For t [t 0, t 0), let ρ (t) be the edgelet of the bisector b that is arallel to ρ at time t, and let η (t) be the Voronoi vertex which is incident to e ρ (t) and to some other air of edges e r and e r. Similarly, for t (t 0, t + 0 ], let ρ+ (t) be the edgelet of the bisector b that is arallel to ρ + at time t, and let η + (t) be the Voronoi vertex which is incident to e ρ + (t) and to some other air of edges e r + and e r +. Let η (res., η + ) be the limit of η (t) (res., of η + (t)) as t t 0 (res., t t 0 ). Alternatively, η is the center of a P-emty corner lacement of Q that touches, at time t 0, at v i, and also touches (at e i ) and r. As is easily checked, the edgelets e r and e r incident on η are collinear at time t 0, but just before that time they were searated by another short edgelet that has shrunk to a oint; see Figure 8 (a). Similarly, η + is the center of a P-emty corner lacement of Q that touches, at time t 0, at v i+1, and also touches (at e i ) and r +. Here too the edgelets of e r + and e r + incident on 13

15 r = r 0 e r1 e r0 η (t 0 ) g e r4 e r5 ρ (t 0 ) e Vor() Vor() (a): t = t 0 r 1 r 2 r 3 r 4 r + = r 5 r 1 r 2 r = r 0 e r0 e r4 Vor() (c): t = t + 0 r 3 r 4 e r1 r + = r 5 e r5 η + (t + 0 ) h ρ + (t + 0 ) e Vor() r = r 0 (b): t = t 0 Q[η ] r 1 r 2 r 3 r 4 r + = r 5 η ρ e W ζ η + ρ + e Figure 8. Changes in VD(P) at the time t 0 of a singular bisector event. (a): VD(P(t 0 )); (b):vd(p(t 0)): the terminal ray of b instantly switches from ρ to ρ + and the cell Vor() loses the entire region W to Vor(); (c): VD(P(t + 0 )). η + are collinear at t 0, and get searated from each other by a newly emerging short edgelet; see Figure 8 (c). Let γ be the olygonal chain connecting η and η + at time t 0, consisting of all centers of lacements touching and (at e i ) and some other oint r i of P. At time t 0 the degenerate Voronoi edge e includes a two-dimensional star-shaed olygonal region, denoted by W, bounded by ζη, ζη +, and γ. (It is a ortion of the wedge W 0 discussed earlier.) At time t 0 the terminal edgelet of e instantly switches from ζη to ζη +, the Voronoi cell Vor() loses the entire region W to Vor(). As a result, VD(P) can exerience Ω(n) toological changes at time t 0 (in addition to the obvious change of the edgelet structure of e, as just discussed). Secifically, let e r = e r0, e r1,..., e rs = e r + be the edges of Vor() at times t [t 0, t 0) that also overla γ at t 0 (listed in the order their aearance along Vor()). Right after time t 0, the oint η + = ρ + e r + becomes a new Voronoi vertex instead of η = ρ e r. Assuming s 1 (or alternatively, r = r + ), every old edge e rj of Vor(P), that is comletely 14

16 contained in γ (such edges exist only if s 2), is instantly relabeled as the new edge e rj of Vor(). The edge e r, which was incident at times t [t 0, t 0) to η (t), loses its ortion within γ to the adjacent edge e r. Symmetrically, the old edge e r + of Vor(), which is hit by ρ + (t), for t (t 0, t + 0 ] is slit at η + into e r + (its ortion within γ) and e r + (its ortion outside γ). Changes in DT(P). If s 1 then each of the old Delaunay edges r j, for j = 0,..., s 1, flis in DT(P) to the new edge r j+1 ; see Figure 9. These are the only changes that DT(P) exeriences at time t 0. Singular seuences While we can regard the overall change in VD(P) and DT(P) at a singular event as a single comound event, it is more convenient for our analysis, for the KDS imlementation resented in Section 5, and erhas also for a better intuitive ercetion of this change, to consider it as a seuence of individual searate singular corner and fli events, as we describe next. r = r 0 r 1 r 2 r 3 r 4 r = r 0 r 1 r 2 r 3 r4 r + = r 5 r + = r 5 Figure 9. Toological changes in DT(P) at the time t 0 of a singular bisector event; five singular fli events occur at t 0, where r 0,..., r 4 (left) fli to r 1,..., r 5 (right). To do so, we stretch the time t 0 and continuously rotate the terminal ray ρ of b from ρ to ρ +. Hence, the intersection oint η = ρ γ traces γ from η to η + ; see Figure 10. At any given moment during this virtual rotation, ρ hits some old Voronoi edge e rj, for 0 j s, which is slit by the current η into e rj (the ortion not yet swet by ρ) and e rj (the swet ortion). That is, we can interret η as an instantaneous Q-Voronoi vertex ν rj which is incident to three Voronoi edges, namely, e, e rj and e rj (since e is two-dimensional, this does not fix the vertex and indeed leaves it with one degree of freedom, of moving along the aroriate ortion of γ). The toological structure of VD(P) changes (during the above rotation) only at the following two kinds of events, which closely resemble their generic counterarts in Section 3.1. Singular corner event. This occurs when the Voronoi vertex η = ν rj coincides with a breakoint of γ b rj = b rj, occurring along the corresonding Voronoi edge. There are three tyes of singular corner events, deending on whether the corner lacement occurs at,, or r j for some 0 j s. Each of the first two tyes occurs just once at a singular bisector event, whereas the third tye may occur multile times. 15

17 r 1r2 r3 r = r 0 r 4 r + = r 5 η ρ Vor() e r4 η e η + r4 ρ ρ + e r5 ζ Vor() Figure 10. Stretching the time t 0. As the terminal ray ρ of b continuously rotates from ρ to ρ +, the vertex η = ρ γ traces γ from η to η +. We only show the intersection of the rays ρ, ρ, and ρ + with e. In the deicted snashot ρ hits γ within the old edge e r4, slitting it into e r4 and e r4. Hence, η is the Voronoi vertex adjacent to Vor(), Vor(), and Vor(r 4 ). (i) η = ν rj is the center of a corner lacement at along e rj. This occurs at η = η, i.e., at the starting oint of the rotation, for Q[η ] is indeed a corner lacement at. We refer to this event as the initial corner event of the singular seuence. As already discussed, Q[η ] touches at the vertex v i, on the edge v i v i+1, and the third oint r at some other edge e (see Figure 8 (a) the figure deicts what haens just before t 0 ; the limiting situation is deicted in Figure 8 (b)). Immediately after the singular corner event of (in the sense of stretching the time during the virtual rotation of ρ), the Q-Voronoi edge e r loses an edgelet (denoted as g in that figure), namely the edgelet corresonding to touching e i 1 and r touching e. It is interesting to note that, unlike at a generic corner event, e does not gain an edgelet. (The only edgelet that it could have gained is the one that encodes the double contact of and at e i, which is not a real edgelet.) Note also that the external edgelet of e r becomes aligned with the next edgelet of e r, and as η starts rotating, begins to annex it. (ii) η = ν rj is the center of a corner lacement at along e rj. This event occurs only at the end of the rotation, when ρ = ρ + and η coincides with the Voronoi vertex η + = ν r + newly created at this singular event. We refer to this event as the final corner event of the singular seuence. The situation is symmetric to that in (i). Secifically, Q[η + ] touches at the vertex v i+1, at the edge v i v i+1, and the third oint r + at some other edge e +. Immediately after the singular corner event at (as the real time t increases ast t 0 ), the Q-Voronoi edge e r + gains an edgelet (marked by h in Figure 8 (c)), but e does not lose an edgelet. The external edges of e r + and of e r +, which were aligned as η aroaches η +, begin to shift aart from each other, with h in between them. (iii) η = ν rj coincides with the center of a corner lacement at r j along the currently traced edge e rj (see Figure 11). Let Q[η] = Q[ν rj ] be the resulting corner lacement of Q at r j, which touches also and, both on the edge e i = v i v i+1 of Q[η]. We refer to this event as an intermediate corner event of the singular seuence. Immediately after a singular corner event of r j (again, in the sense of the stretched time), the Q-Voronoi edge e rj loses an edgelet, and the adjacent Q-Voronoi edge e rj gains an edgelet. Singular fli event. The moving vertex η = ν rj coincides with a Q-Voronoi vertex ν rj r j+1, which is the common endoint of the Voronoi edges e rj and e rj+1 along γ (thinking of the scenario just 16

18 r = r 0 Q[η] r 1 r 2 r 3 r 4 r + = r 5 η Vor() η ρ ρ + ζ e r4 e r4 η + Vor() Figure 11. A singular corner event at r 4. A corner of Q[η] touches r 4, so η coincides with a breakoint on each of the overlaing bisectors b r4 and b r4. before t 0 ; see Figure 12). In the stretched time during the rotation of ρ, the Q-Voronoi edge e rj shrinks (or, more recisely, overtaken by e rj ) and disaears from VD(P). The new edge e rj+1 is born, as η starts moving along the old edge e rj+1 of Vor(), annexing a ortion of it for Vor(). The growing ortion of that edge between ν rj r j+1 and η becomes e rj+1, and e rj+1 is the shrinking remainder of that edge. Accordingly, the edge r j of DT(P) flis to r j+1. Analogous to a generic fli event, e rj is a non-corner edge when the above fli event occurs. It shrinks to a oint at the event, and the four oints,, r j, r j+1, which are Q-cocircular, are vertices of the two adjacent triangles r j and r j r j+1 of DT(P) (sharing the edge r j ). The event is singular because and are on the same edge of Q[η] (with label e i ), and the remaining two oints r j and r j+1 are incident to some air of other distinct edges of Q[η]. r = r 0 Q[η] r 1 r 2 r 3 r 4 r + = r 5 η η η + e r3 e r4 r = r 0 r 1 r 2 r 3 r 4 r + = r 5 ρ ρ + ζ Vor() Vor() (a) (b) Figure 12. A singular fli event involving,, r 3 and r 4. (a): The Voronoi ersective. The vertex η coincides with the common endoint ν r3 r 4 of e r3 and e r4. The corresonding coy Q[η] touches,, r 3 and r 4 in this clockwise order. (b): The Delaunay ersective. The reviously Delaunay edge r 3 flis into r 4. We remark that, unless j = 0, the edge r j does not belong to DT(P) at time t 0. It becomes an edge of DT(P) only after executing the revious singular fli, which adds it to DT(P). In other words, the rotational order in which ρ generates these events is such that each fli event facilitates the next one, by introducing the aroriate edges r j into the diagram. We refer to the entire seuence of events that are triggered by a singular bisector event (including the singular bisector event itself) as a singular seuence. The above order, in which the events of 17