On 2Site Voronoi Diagrams under Geometric Distance Functions ^{†}^{†}thanks: ©2011 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.
Abstract
We revisit a new type of a Voronoi diagram, in which distance is measured from a point to a pair of points. We consider a few more such distance functions, based on geometric primitives, and analyze the structure and complexity of the nearest and furthestneighbor Voronoi diagrams of a point set with respect to these distance functions.
I Introduction
The Voronoi diagram is one of the most fundamental concepts in computational geometry, which has plenty of applications in science and industry. Much information in this respect can be found in [4] and [13]; for important recent achievements, see [8].
The basic definition of the Voronoi diagram applies to a set of points (also called sites) in the plane: its nearestneighbor Voronoi diagram is a partition of the plane into regions, each corresponding to a distinct site , and consisting of all the points being closer to than to any other site from . Similarly, the furthestneighbor Voronoi diagram of is obtained by assigning each point in the plane to the region of the most remote site. These notions can be generalized to higherdimensional spaces, different types of sites, and in other ways.
One of the recent generalizations of this concept is a family of socalled 2site Voronoi diagrams [5], which are based on distance functions that define a distance from a point in the plane to a pair of sites from a given set . Consequently, each Voronoi region corresponds to an (unordered) pair of sites from . The original motivation for the study [5] was the famous Heilbronn’s triangle problem [14]. Other motivations are mentioned therein.
For being a set of points, Voronoi diagrams under a number of 2site distance functions have been investigated, which include arithmetic combinations of pointtopoint distances [5, 17] and certain geometric distance functions [5, 7, 9]. In this work, we develop further the latter direction.
Let , and consider and a point in the plane. We shall focus our attention on a few circlebased distance functions:

radius of circumscribing circle: , where is the circle defined by and is the radius of the circle ;

radius of containing circle: , where is the minimum circle containing ;^{1}^{1}1 Obviously, if any of the three points is properly contained in the circle whose diameter is defined by the two other points.

view angle: , or, equivalently, half of the angular measure of the arc of that the angle subtends;

radius of inscribed circle: is the radius of the circle inscribed in ;

centerofcircumscribingcirclebased functions: let denote the center of the circle ; then , , and are the distance from to the segment , the area of , and the perimeter of , respectively;
and on a parameterized perimeter distance function:

parameterized perimeter: , where .
The first and third circlebased distance functions were first mentioned in [10]. The last function generalizes the perimeter distance function introduced in [5], and later addressed in [7, 9].
Since two points define a segment, any 2point site distance function provides a distance between the point and the segment , and vice versa. Consequently, geometric structures akin to 2site Voronoi diagrams can arise as Voronoi diagrams of segments. This alternative approach was independently undertaken by Asano et al., and the “view angle” and “radius of circumscribing circle” distance functions reappeared in their works [2, 3] on Voronoi diagrams for segments soon after they had been proposed by Hodorkovsky [10] in the context of 2site Voronoi diagrams. However, as Asano’s et al. research was originally motivated by mesh generation and improvement tasks, they were mostly interested in sets of segments representing edges of a simple polygon, and thus, nonintersecting (except, possibly, at the endpoints), what significantly alters the essence of the problem.
In this paper, we analyze the structure and complexity of 2site Voronoi diagrams under the distance functions listed above. Our obtained results are mostly of theoretical interest. The method used to derive an upper bound on the complexity of the nearestneighbor 2site Voronoi diagram under the “parameterized perimeter” distance function is first developed for the case of , yielding a much simpler proof for the “perimeter” function than the one developed in [9], and then generalized to any . We summarize our new results in Table I.
,  ,  
,  ,  , 
,  ,  ,  
,  ,  , 
Throughout the paper we use the notation (resp., ) for denoting the nearest (resp., furthest) 2site Voronoi diagram, under the distance function , of a point set . The set is always assumed to contain points.
Ii Circumscribing Circle
Let denote the unique circle defined by three distinct points , , and in the plane. We now define the 2site circumscribingcircle distance function:
Definition 1
Given two points in the plane, the “circumcircle distance” from a point in the plane to the unordered pair is defined as .
For a fixed pair of points and , the curve is the line . This implies that all the points on belong to the region of in . In this section we assume that the points in are in general position, i.e., there are no three collinear points, and no three pairs of points define three distinct lines that intersect at one point. The given sites are singular points, that is, for any two sites , the function is not defined at or .
Theorem 1
Let be a set of points in the plane. The combinatorial complexity of is .
The points of define lines, which always have intersection points. All these intersection points are features of , and hence the lower bound.
Theorem 2
Let be a set of points in the plane. The combinatorial complexity of both and is (for any ).
Clearly, the combinatorial complexity of or is identical to that of the respective diagram of the 2site distance function . It is known that The respective collection of Voronoi surfaces fulfills Assumptions 7.1 of [16, p. 188]:

Each surface is an algebraic surface of maximum constant degree;

Each surface is totally defined (this is stronger than needed); and

Each triple of surfaces intersects in at most a constant number of points.
Hence, we may apply Theorem 7.7 of [ibid., p. 191] and obtain the claimed bound on the complexity of .
Iii Containing Circle
Let denote the minimumradius circle containing three points , , and in the plane. (That it, is the minimum circle containing the triangle .) We now define the 2site containingcircle distance function:
Definition 2
Given two points in the plane, the “containingcircle distance” from a point in the plane to the unordered pair is defined as .
In our context we have that . Assume first that . Observe that if all angles of are acute (or is rightangled), then is identical to . Otherwise, if one of the angles of is obtuse, then is the circle whose diameter is the longest edge of , that is, the edge opposite to the obtuse angle. If coincides with either or , then is the circle whose diameter is the line segment .
Theorem 3
Let be a set of points in the plane. The combinatorial complexity of is .
For simplicity assume that each point from has a unique closest neighbor in . For each point , consider its closest neighbor . Then, the points on the line segment lying sufficiently close to belong to the region of in , which is thus nonempty. Since no region is thereby encountered more than twice, has at least nonempty regions. The claim follows.
Theorem 4
Let be a set of points in the plane. The combinatorial complexity of is (for any ).
Let a point belong to a nonempty region of . No matter if the triangle is acute (Figure 1(a)),


(a) Acute triangle  (b) Obtuse 

(c) Obtuse 
is obtuse with being the obtuse vertex (Figure 1(b)), or is obtuse with or being the obtuse vertex (Figure 1(c)), the circle cannot contain any other point . Otherwise, regardless of the location of in , we will always have , which is a contradiction. This follows from the fact (see [6, Lemma 4.14]) that given a point set and its minimum enclosing circle , where is defined by three points (resp., two diametrical points ), removing from one of (resp., one of ) will result in a point set with a smaller minimum enclosing circle. Thus, there is a circle containing that is empty of any other site from . This immediately implies that is an edge of the Delaunay triangulation of . Consequently, there are pairs of sites in that have nonempty regions in . Furthermore, it follows from the definition of that the respective Voronoi surface of is made of a constant number of patches, each of which is a “wellbehaved” function in the sense discussed in the proof of Theorem 2. Again, by standard DavenportSchinzel machinery, the combinatorial complexity of the lower envelope of these surfaces is (for any ), and the claim follows.
Theorem 5
Let be a set of points in the plane. The combinatorial complexity of is (for any ).
As in the proof of Theorem 2, we prove this claim by using the upper envelope of “wellbehaved” Voronoi surfaces.
Iv View Angle
We now define the 2site viewangle distance function:
Definition 3
Given two points in the plane, the “viewangle distance” from a point in the plane to the unordered pair is defined as .
Similarly to the circumcircleradius distance function, the viewangle function is undefined at the given points. For a fixed pair of points and , the curve is the open line segment connecting the two points and , while the curve is the line excluding the closed line segment . The curve is the circle whose diameter is the line segment (excluding, again, and ).
Theorem 6
Let be a set of points in the plane. The combinatorial complexity of is .
Consider a set of points in the plane. An example of the intersection of the complements of two segments defined by two pairs of points (with respect to the supporting lines) is shown in Figure 2(a).

(a) Intersection point 

(b) Graph 
These intersection points are features of ; we show that there are such points. To this aim we create a geometric graph whose vertices are the given points, in which each segment’s complement defines two edges. We add one additional point far away from the convex hull of , and connect it (without adding intersections) to all the rays as shown in Figure 2(b). We can now use the crossingnumber lemma for bounding from below the number of intersections of the original rays. The lemma tells us that every drawing of a graph with vertices and edges (without self or parallel edges) has crossing points [1, 12]. In our case , so the number of intersection points in is . All these intersection points are features of , and hence the lower bound.
Theorem 7
Let be a set of points in the plane. The combinatorial complexity of both and is (for any ).
For analyzing (S) and we consider the function instead of that of . This is permissible since the cosine function is strictly decreasing in the range . By the cosine law, we have . As we have already seen more than once in this paper, this means that the respective collection of Voronoi surfaces fulfills Assumptions 7.1 of [16, p. 188]. Hence, we may apply Theorem 7.7 of [ibid., p. 191] and obtain the claimed bound on the complexity of .
Theorem 8
Let be a set of points in the plane. The combinatorial complexity of is .
Given a set of points in the plane, we count the intersections of pairs of line segments, where each segment is defined by points of (see Figure 3(a)).

(a) Intersection point 

(b) Graph 
We create a geometric graph whose vertices are the given points, and the edges are the line segments connecting every pair of points (see Figure 3(b)). The intersections of the segments defined by all pairs of points define features of , because along these segments the viewangle function assumes its maximum possible value, . We can now use the crossingnumber lemma for counting these intersections. The graph with vertices and edges (without self or parallel edges) has crossing points [1, 12]. In this case , hence is a lower bound on the complexity of .
Results by Asano et al. [2] immediately imply that the edges of represent pieces of polynomial curves of degree at most three. However, the structure of the part of that lies outside the convex hull of is fairly simple: it is given by the arrangement of lines supporting the edges of . This arrangement can be computed by a standard incremental algorithm in optimal time and space, where denotes the number of vertices of . Each cell of the arrangement should then be labeled with a pair of sites from , to the Voronoi region of which it belongs; this extra task can be completed within the same complexity bounds.
V Radius of Inscribed Circle
We now define the 2site “radiusofinscribedcircle” distance function:
Definition 4
Given two points in the plane, the “inscribed radius distance” from a point in the plane to the unordered pair , denoted by , is defined as the radius of the circle inscribed in the triangle (Figure 4).
Theorem 9
Let be a set of points in the plane. The combinatorial complexity of is .
The intersection point of any two lines defined by the points from is a distinct feature of the Voronoi diagram under discussion. Thus, points in define lines, which have intersection points.
Theorem 10
Let be a set of points in the plane. The combinatorial complexity of both and is (for any ).
Let be two points in , and a point in the plane. It is a wellknown fact that , where and are the area and perimeter, respectively, of the triangle . Both the numerator and denominator of this fraction can be written as algebraic expressions using the coordinates of the points . Hence, as above, the standard DavenportSchinzel machinery can be applied for obtaining the claim bounds.
Theorem 11
Let be a set of points in the plane. The combinatorial complexity of is in the worst case.
The complexity of can be as high as . Let be a set of point in convex position with no three collinear points. Let and be two antipodal vertices of , the convex hull of , and consider two parallel lines and tangent to only at and , respectively. Next, consider any point , and let it move along in either direction. In the limit, the distance from to any pair of sites in equals the width of the infinite strip bounded by two lines parallel to and passing through and , respectively. Consequently, the points of lying sufficiently far from belong to the Voronoi region of . Since the number of pairs of antipodal vertices of is , the bound follows.
A similar reasoning leads to a conclusion that has at most a linear number of unbounded regions. To demonstrate this, consider any point in the plane, and a line . Observe that the points of lying sufficiently far from belong to the Voronoi region of the pair(s) of points from that define the width of in the direction orthogonal to , and, thus, represent a pair (pairs) of antipodal vertices of . Since the union of all such lines gives the whole plane, and the number of antipodal vertices of is at most linear, the claim follows.
Vi Distances Based on the Center of the Circumscribing Circle
Let be three points in the plane. Consider the circle passing through with center . We now define three more distance functions based on the above notation:
Definition 5
Given two points in the plane, the three distances, denoted by , , and , respectively, are the distance from to the line segment , the area of the triangle , and the perimeter of , respectively (Figure 5).
The upper bound of (for any ) on the complexity of the nearest and furthestneighbor Voronoi diagrams under each of these distance functions can be, again, derived by means of DavenportSchinzel machinery. Below we provide some lower bounds. First, we address the nearestneighbor case.
Theorem 12
Let be a set of points in the plane. The combinatorial complexity of and is in the worst case.
The key observation is the following. Consider a pair of sites, and let denote the circle with the diameter . Then, for any point , we have .
Consider two parallel lines and , and let denote the distance between them. For a given , let us construct a set of points as a union of two sets and consisting of and points, respectively, in the following way. The sets and are constructed iteratively; at each odd step, a new point is added to , and at each even one—to . For any : , let and denote the two sets constructed so far, and let denote the set of circles defined by pairs of points from different sets. We want each circle from to pass through precisely two points from (those defining it), each two circles from to intersect, and no three of them to pass through the same point not contained in . Then circles composing will give rise to distinct intersection points, each belonging to a separate feature of either Voronoi diagram under consideration, and the claim will follow.
To ensure the first property, we select the points so that the distance between each two points contained in the same set is much smaller than , where . To guarantee the second property, at each step : , when adding a new point to the respective set, we make sure that for any point from the other set, the circle passes neither through any point from nor through any intersection point of the circles from . This completes the proof.
Theorem 13
Let be a set of points in the plane. The combinatorial complexity of is in the worst case.
A linear lower bound in the worst case for can be obtained in the following way. Choose the set of points to lie on some line , so that the distance between any two consecutive points is 1. Then, the minimum possible value for the distance function is obviously 2, and can be achieved only for a pair of consecutive points. For each such pair , consider the circle with the diameter . Evidently, for any point , we have , and for any other pair of sites, . We conclude that each pair of consecutive points along has a nonempty region in . Since there are pairs of consecutive points, the bound follows.
Second, we address the furthestneighbor Voronoi diagrams.
Theorem 14
Let be a set of points in the plane. The combinatorial complexity of all of , , and is .
In each case, the proof is identical to that of Theorem 9.
Vii Parameterized Perimeter
Finally, we define the 2site parameterized perimeter distance function:
Definition 6
Given two points in the plane and a real constant , the “parameterized perimeter distance” from a point in the plane to the unordered pair is defined as .
We require that be greater than or equal to since allowing would result in negative distances. Letting results in a distance function that equals 0 for all the points on the line segment . If , we deal with the “sum of distances” distance function introduced in [5] and recently revisited in [17]. For , the above definition yields the “perimeter” distance function .
In [9] it was proven that the combinatorial complexity of the nearestneighbor 2site perimeter Voronoi diagram of a set of points is slightly superquadratic in . In a nutshell, the proof was based on the observation that any pair of sites that has a nonempty region in the perimeter diagram also has a nonempty region in the sumofdistances diagram. This immediately implies that the number of such pairs is linear in . (However, unlike in the sumofdistances diagram, a region in the perimeter diagram is not necessarily continuous. We were able to construct examples in which the number of connected components of a single region is comparable to the number of points!) Again, one can apply the standard DavenportSchinzel machinery and conclude the claimed upper bound on the complexity of the diagram. It remains unclear whether the worstcase complexity of the diagram is linear, quadratic, or in between. The proof in [9] of the main observation was extremely complex. We provide here an alternative and much simpler proof of the same bound, which generalizes to the case of “parameterized perimeter” distance function for any .
Theorem 15
Let be a set of points in the plane. The combinatorial complexity of is (for any ).
Refer to Figure 6.
Let be two sites which have a nonempty region in , and let be a point in this region, noncollinear with and . In addition, let be the perpendicular bisector of the line segment . Assume, without loss of generality, that .
Consider the ellipse passing through with and as foci. By definition, for any point inside this ellipse we have . Therefore,
This means that cannot be a site in , for otherwise would belong to the region of instead of to the region of . It follows that the ellipse is empty of any sites other than and .
Now consider the line that is tangent to at , and the ray perpendicular to at and passing through . It is a known property of ellipses that this ray bisects the angle , and, thus, it intersects the line segment , say, at point . The circle centered at and passing through is tangent to at (as well as at another point), and is entirely contained in . Since is closer to than to (by our assumption), it follows that the circle also contains . (If were on the extension of in the shaded area, a contradiction would easily be obtained by using the triangle inequality: , hence , contradicting the assumption that .) Since is empty of sites (except and ), so is the circle . Therefore, is an edge of the Delaunay triangulation of . The number of such edges is linear in , the cardinality of .
Hence, there are respective surfaces of these pairs of sites. One can now apply the standard DavenportSchinzel machinery (as in the proof of Theorem 2). The claim follows.
Finally, we state the following theorem.
Theorem 16
Let be a set of points in the plane.

The combinatorial complexity of is and (for any ).

If there is a unique closest pair , then when , the combinatorial complexity of is asymptotically 1.

For , the combinatorial complexity of is (for any ).

To see the lower bound on the complexity of , note that every point on the segment has distance zero to the pair , and therefore, the intersection of any pair of segments and defined by sites is a feature of . As is demonstrated in the proof of Theorem 8, the number of these features is . The upper bound is obtained by using the usual DavenportSchinzel machinery, as in the proof of Theorem 2.

It is easy to verify that as , the term dominates the distance , and, hence, every point in the plane is closer to the unique closest pair of sites than to any other pair in . Hence, the asymptotic diagram contains zero vertices, zero edges, and one face (the entire plane).

The proof is a generalized version of the proof of the special case . Refer to Figure 7.
As in the proof of Theorem 15, we assume that there is a point in the region of , such that , and is noncollinear with and . Our goal is to show that for any there exists a circle having on its boundary and containing , which is empty of any other site , implying that are Delaunay neighbors.
As in the proof of Theorem 15, let be the locus of points for which . Thus, is the Cartesian oval consisting of all points that satisfy , where is constant. (Unless , this oval has exactly one axis of symmetry: the line joining the two foci .) Then, if there were a site within , it would lead to a smaller value of , so must be empty of sites other than .
As before, let be the ray emanating from perpendicular to and pointing into , and let be the point where crosses the line .
Let us further suppose that . Without loss of generality, assume that is symmetric with respect to the axis of abscissas (see Figure 7); consequently, the points , , and belong to the latter. Let , , , and denote the corresponding coordinate of , , , and , respectively.
Consider a circle centered at of the radius . By construction, is tangent to at .
For any , such that the point lies inside , let denote the point of lying above . For any such , let
Since represents a square root of a linear function, it is concave on its domain. The same will hold for a function . Consequently, their weighted combination is also concave on the same domain, and, thus, has a single local maximum.
Recall that the circle is tangent to at by construction. It is easy to see that is tangent to from the inside: otherwise, would be a local minimum of achieved at an inner point of the domain, contradicting the concavity of . It follows that has a local maximum at . Together with the previous observation, this implies that has a global maximum at . This means that is the only common point of and the upper half of . By symmetry, we conclude that lies inside and touches it at and the point symmetric to . Thus, must be empty of sites other than .
It remains to demonstrate that lies inside . To this end, it is sufficient to show that the point lies between and ; then, as in the case of , the needed property can be easily derived using the triangle inequality.
Let us argue as follows. The above reasoning can be carried out for any point noncollinear with and , providing us with a maximum empty circle inscribed in , and tangent to it at precisely two points—namely, at and its symmetric point. It follows that the medial axis of is a segment of the line through and . Let and be the intersection points of and being closer to and , respectively (see Figure 7). Consider the circle with radius centered at . Obviously, is a common point of and , but any other point of lies strictly inside , since for any such point , we have and . This implies that the radius of curvature of at is greater than . A similar statement holds for . Consequently, the two endpoints of the medial axis must lie between and , and the same must hold for the point .
We conclude that is a circle containing both and and otherwise empty of sites, so and are Delaunay neighbors. Hence, there are pairs of sites that generate regions in the Voronoi diagram, and the claim follows from the standard DavenportSchinzel machinery.
Viii Conclusion
In this paper, we have investigated 2site Voronoi diagrams of point sets with respect to a few geometric distance functions. The Voronoi structures obtained in this way cannot be explained in terms of the previously known kinds of Voronoi diagrams (which is the case for the 2site distance functions thoroughly analyzed in [5]), what makes them particularly interesting. On the other hand, our results can be exploited to advance research on Voronoi diagram for segments. Potential directions for future work include consideration of other distance functions, and generalizations to higher dimensions and to site Voronoi diagrams.
Acknowledgments
Work on this paper by the first author was performed while he was affiliated with Tufts University in Medford, MA. Work by the last author was partially supported by Russian Foundation for Basic Research (grant 100700156a).
References
 [1] M. Ajtai, V. Chvátal, M. Newborn, and E. Szemerédi, Crossingfree subgraphs, Annals of Discrete Mathematics, 12 (1982), 9–12.
 [2] T. Asano, N. Katoh, H. Tamaki, and T. Tokuyama, Angular Voronoi diagram with applications, Proc. 3rd Int. Symp. on Voronoi Diagrams in Science and Engineering, Banff, Canada, 32–39, 2006.
 [3] T. Asano, N. Katoh, H. Tamaki, and T. Tokuyama, Voronoi diagrams with respect to criteria on vision information, Proc. 4th Int. Symp. on Voronoi Diagrams in Science and Engineering, Pontypridd, Wales, UK, 25–32, 2007.
 [4] F. Aurenhammer, Voronoi diagram—A survey of a fundamental geometric data structure, ACM Computing Surveys, 23 (1991), 345–405.
 [5] G. Barequet, M.T. Dickerson, and R.L.S. Drysdale, 2point site Voronoi diagrams, Discrete Applied Mathematics, 122 (2002), 37–54.
 [6] M. de Berg, M. van Kreveld, M. Overmars, and O. Schwarzkopf, Computational Geometry, Algorithms, and Applications (3rd ed.), SpringerVerlag, Berlin, 2008.
 [7] M.T. Dickerson and D. Eppstein, Animating a continuous family of twosite Voronoi diagrams (and a proof of a bound on the number of regions), Proc. 25th ACM Symp. on Computational Geometry, Aarhus, Denmark, 92–93, 2009.
 [8] M. Gavrilova (ed.), Generalized Voronoi Diagram: A GeometryBased Approach to Computational Intelligence. Springer, 2008.
 [9] I. Hanniel and G. Barequet, On the triangleperimeter twosite Voronoi diagram, Proc. 6th Int. Symp. on Voronoi Diagrams, Copenhagen, Denmark, 129–136, 2009.
 [10] D. Hodorkovsky, 2Point Site Voronoi Diagrams, M.Sc. Thesis, The Technion—Israel Inst. of Technology, Haifa, Israel, 2005.
 [11] D.T. Lee, On knearest neighbor Voronoi diagrams in the plane, IEEE Trans. on Computers, 31 (1982), 478–487.
 [12] F.T. Leighton, Complexity Issues in VLSI, MIT Press, Cambridge, MA, 1983.
 [13] A. Okabe, A. Boots, B. Sugihara, and S.N. Chui, Spatial Tesselations, 2nd ed., Wiley, 2000.
 [14] K.F. Roth, On a problem of Heilbronn, Proc. London Mathematical Society, 26 (1951), 198–204.
 [15] M.I. Shamos and D. Hoey, Closestpoint problems, Proc. 16th Ann. IEEE Symp. on Foundations of Computer Science, Berkeley, CA, 151—162, 1975.
 [16] M. Sharir and P.K. Agarwal, DavenportSchinzel Sequences and Their Geometric Application, Cambridge University Press, 1995.
 [17] K. Vyatkina and G. Barequet, On 2site Voronoi diagrams under arithmetic combinations of pointtopoint distances, Proc. 7th Int. Symp. on Voronoi Diagrams, Québec City, Québec, Canada, 33–41, 2010.