# Chapter 43
2D Segment Voronoi Diagrams

Menelaos Karavelas

This chapter describes the two-dimensional segment Voronoi diagram package of CGAL. We start with a few definitions in Section . The software design of the 2D segment Voronoi diagram package is described in Section . In Section we discuss the geometric traits of the 2D segment Voronoi diagram package and in Section the segment Voronoi diagram hierarchy, a data structure suitable for fast nearest neighbor queries, is briefly described.

## 43.1 Definitions

Figure: The segment Voronoi diagram for a set of weakly (left) and strongly (right) intersecting sites.

The 2D segment Voronoi diagram package of CGAL is designed to compute the dual of the Voronoi diagram of a set of possibly intersecting segments on the plane. Although we compute the dual, we will often refer to the diagram itself, since it is easier to explain an d understand. The algorithm that has been implemented is incremental. The corresponding CGAL class is called Segment_Voronoi_diagram_2<SegmentVoronoiDiagramTraits_2,SegmentVoronoiDiagramStructure_2> and will be discussed in more detail in the sequel. The interested reader may want to refer to the paper by Karavelas [Kar04] for the general idea as well as the details of the algorithm implemented.

Definitions. Before describing the details of the implementation we make a brief introduction to the theory of segment Voronoi diagrams. The segment Voronoi diagram is defined over a set of non-intersecting sites, which can either be points or linear segments, which we assume that are given through their endpoints. The segment Voronoi diagram a subdivision of the plane into connected regions, called cells, associated with the sites. The cell of a site ti is the locus of points on the plane that are closer to ti than any other site tj, j i. The distance (x, ti) of a point x in the plane to a site ti is defined as the minimum of the Euclidean distances of x from the points in ti. Hence, if ti is a point pi, then

(x,ti)= x-ti ,

whereas if ti, is a segment, then

(x,ti)=min y ti x-y ,

where · denotes the Euclidean norm. It can easily be seen that it is a generalization of the Voronoi diagram for points.

In many applications the restriction that sites are non-intersecting is too strict. Often we want to allow segments that touch at their endpoints, or even segments that overlap or intersect properly at their interior (for example, see Fig. ). Allowing such configurations poses certain problems. More specifically, when we allow segments to touch at their endpoints we may end up with pairs of segments whose bisector is two-dimensional. If we allow pairs of segments that intersect properly at their interior, the interiors of their Voronoi cells are no longer simply connected. In both cases above the resulting Voronoi diagrams are no longer instances of abstract Voronoi diagrams (cf. [Kle89]), which has a direct consequence on the efficient computation of the corresponding Voronoi diagram. The remedy to these problems is to consider linear segments not as one object, but rather as three, namely the two endpoints and the interior. This choice guarantees that all bisectors in the Voronoi diagram are one-dimensional and that all Voronoi cells are simply connected. Moreover, we further distinguish between two cases, according to the type of intersecting pair that our input data set contains. A pair of sites is called weakly intersecting if they a single common point and this common point does not lie in the interior of any of the two sites. A pair of sites is called strongly intersecting if they intersect and they either have more than one common point or their common point lies in the interior of at least one of the two sites. As it will be seen later the two cases have different representation (and thus storage) requirements, as well as they require a somehow different treatment on how the predicates are evaluated. Having made the distinction between weakly and strongly intersecting sites, and having said that segment sites are treated as three objects, we are now ready to precisely define the Voronoi diagram we compute. Given a set of input sites, let be the set of points and (open) segments in the arrangement () of . The 2D segment Voronoi diagram package of CGAL computes the dual of the Voronoi diagram of the sites in the set .

The dual of the segment Voronoi diagram is uniquely defined once we have the segment Voronoi diagram. If the all sites are in general position, then dual graph is a graph with triangular faces away from the convex hull of the set of sites. To unify our approach and handling of the dual graph we add to the set of (finite) sites a fictitious site at infinity, which we call the site at infinity. We can then connect all vertices of the outer face of the dual graph to the site at infinity which gives us a graph with the property that all of its faces are now triangular. However, the dual graph is not a triangulation for two main reasons: we cannot always embed it on the plane with straight line segments that yield a triangulation and, moreover, we may have two faces of the graph that have two edges in common, which is not allowed in a triangulation.

We would like to finish our brief introduction to the theory of segment Voronoi diagrams by discussing the concept of general position. We say that a set of sites is in general position if no two triplets of sites have the same tritangent Voronoi circle. This statement is rather technical and it is best understood in the context of points. The equivalent statement for points is that we have no two triplets of points that define the same circumcircle, or equivalently that no four points are co-circular. The statement about general position made above is a direct generalization of the (much simpler to understand) statement about points. On the contrary, when we have sites in degenerate position, the dual graph has faces with more than three edges on their boundary. We can get a triangulated version of the graph by simply triangulating the corresponding faces in an arbitrary way. In fact the algorithm that has been implemented in CGAL has the property that it always returns a valid triangulated version of the dual graph of the segment Voronoi diagram. By valid we mean that it contains the actual dual graph (i.e., the actual dual of the segment Voronoi diagram) and whenever there are faces with more than three faces then they are triangulated. The way that they are triangulated depends on the order of insertion of the sites in the diagram.

One final remark has to be made with respect to the difference between the set of input sites and the set of output sites. The set of input sites consists of the closed sites that the user inserts in the diagram. Since segment sites are treated as three objects, internally our algorithm sees only points and open segments. As a result, from the point of view of the algorithm, the input sites have no real meaning. What has real meaning is the set of sites that correspond to cells of the Voronoi diagrams and this is the set of output sites.

Degenerate dimensions. The dimension of the segment Voronoi diagram is in general 2. The exceptions to this rule are as follows:

- The dimension is -1 if the segment Voronoi diagram contains no sites.
- The dimension is 0 if the segment Voronoi diagram contains exactly one (output) site.
- The dimension is 1 is the segment Voronoi diagram contains exactly two (output) sites.

## 43.2 Software Design

The 2D segment Voronoi diagram class Segment_Voronoi_diagram_2<SegmentVoronoiDiagramTraits_2,SegmentVoronoiDiagramDataStructure_2> follows the design of the triangulation package of CGAL. It is parametrized by two arguments:

- the geometric traits class. It provides the basic geometric objects involved in the algorithm, such as sites, points etc. It also provides the geometric predicates for the computation of the segment Voronoi diagram, as well as some basic constructions that can be used, for example, to visualize the diagram. The geometric traits for the segment Voronoi diagram will be discussed in more detail in the next section.
- the segment Voronoi diagram data structure. This is essentially the same as the Apollonius graph data structure (discussed in Chapter ), augmented with some additional operations that are specific to segment Voronoi diagrams. The corresponding concept is that of SegmentVoronoiDiagramDataStructure_2, which in fact is a refinement of the ApolloniusGraphDataStructure_2 concept. The class Triangulation_data_structure_2<Vb,Fb> is a model of the concept SegmentVoronoiDiagramDataStructure_2. A default value for the corresponding template parameter is provided, so the user does not need to specify it.

Strongly intersecting sites and their representation. As we have mentioned above, the segment Voronoi diagram package of CGAL is designed to support the computation of the segment Voronoi diagram even when the input segment sites are intersecting. This choice poses certain issues for the design of the software package. The major concern is the representation of the subsegments that appear in the arrangement of the these sites, because the sites in the arrangement are the ones over which the diagram is actually defined. A direct consequence of the choice of representation is the algebraic degree of the predicates involved in the computation of the segment Voronoi diagram, as well as the storage requirements for the subsegments and points on intersection in the arrangement.

The case of weakly intersecting sites does not require any special treatment. We can simply represent points by their coordinates and segments by their endpoints. In the case of strongly intersecting sites, the obvious choice to use the afore-mentioned representation has severe disadvantages. Consider two strongly intersecting segments ti and tj, whose endpoints have homogeneous coordinates of size b. Their intersection point will have homogeneous coordinates of bit size 6b+O(1). This effect can be cascaded, which implies that after inserting k (input) segments we can arrive at having points of intersection whose bit sizes are exponential with respect to k, i.e., their homogeneous coordinates will have bit size (2kb). Not only the points of intersection, but also the adjacent subsegments will be represented by quantities of arbitrarily high bit size, and as a result we would not be able to give a bound on the bit sizes of the coordinates of the points of intersection. As a result, we would not be able to give a bound on the memory needed to store these coordinates. An equally important consequence is that we would also not be able to give a bound on the algebraic degree of the algebraic expressions involved in the evaluation of the predicates.

Such a behavior is obviously undesirable. For robustness, efficiency, and scalability purposes, it is critical that the bit size of the algebraic expressions in the predicates does not depend on the input size. For this reason, as well as for others to be discussed below, we decided to represent sites in a implicit manner, which somehow encodes the history of their construction. In particular, we exploit the fact that points of intersection always lie on two input segments, and that segments that are not part of the input are always supported by input segments.

For example, let us consider the configuration in Fig. . We assume that the segments ti=piqi, i=1,2,3, are inserted in that order. Upon the insertion of t2, our algorithm will split the segment t1 into the subsegments p1s1 and s1q1, then add s1, and finally insert the subsegments p2s1 and s1q2. How do we represent the five new sites? s1 will be represented by its two defining segments t1 and t2. The segment p1s1 will be represented by two segments, a point, and a boolean. The first segment is t1, which is always the segment with the same support as the newly created segment. The second segment is t2 and the point is p1. The boolean indicates whether the first endpoint of p1s1 is an input point; in this case the boolean is equal to true. The segment s1q1 will also be represented by two segments, a point, and a boolean, namely, t1 (the supporting segment of s1q1), t2 and false (it is the second endpoint of s1q1 that is an input point). Subsegments p2s2 and s2q2 are represented analogously. Consider now what happens when we insert t3. The point s2 will again be represented by two segments, but not s1q1 and t3. In fact, it will be represented by t1 (the supporting segment of s1q1) and t3. s2q1 will be represented by two segments, a point, and a boolean (t1, t3 and false), and similarly for p3s2 and s2q3. On the other hand, both endpoints of s1s2 are non-input points. In such a case we represent the segment by three input segments. More precisely, s1s2 is represented by the segments t1 (the supporting segment of s1q1), t2 (it defines s1 along with t1) and t3 (it defines s2 along with t1).

Figure: Site representation. The point s1 is represented by the four points p1, q1, p2 and q2. The segment p1s1 is represented by the points p1, q1, p2, q2 and a boolean which is set to true to indicate that the first endpoint in not a point of intersection. The segment s1s2 is represented by the six points: p1, q1, p2, q2, p3 and q3. The remaining (non-input) points and segments in the figure are represented similarly.

The five different presentations, two for points (coordinates; two input segments) and three for segments (two input points; two input segments, an input point and a boolean; three input segments), form a closed set of representations and thus represent any point of intersection or subsegment regardless of the number of input segments. Moreover, every point (input or intersection) has homogeneous coordinates of bit size at most 3b+O(1). The supporting lines of the segments (they are needed in some of the predicates) have coefficients which are always of bit size 2b+O(1). As a result, the bit size of the expressions involved in our predicates will always be O(b), independently of the size of the input. The SegmentVoronoiDiagramSite_2 concept encapsulates the ideas presented above. A site is represented in this concept by up to four points and a boolean, or up to six points, depending on its type. The class Segment_Voronoi_diagram_site_2<K> implements this concept.

Even this representation, however, has some degree of redundancy. The endpoint of a segment appears in both the representation of the (open) segment site as well as the representation of the point site itself. The situation becomes even worse in the presence of strongly intersecting sites: a point may appear in the representation of multiple subsegments and/or points of intersection. To avoid this redundancy, input points are stored in a container, and the various types of sites (input points and segments, points of intersection, subsegments with one or two points of intersection as endpoints) only store handles to the points in the container. This is achieved by the Segment_Voronoi_diagram_storage_site_2<Gt> class which is a model of the corresponding concept: SegmentVoronoiDiagramStorageSite_2. This concept enforces a site to be represented by up to 6 handles (which are very lightweight objects) instead of 6 points, which are, compared to handles of course, very heavy objects.

Optimizing memory allocation. There are applications where we know beforehand that the input consists of only weakly intersecting sites. In these cases the site representation described above poses a significant overhead in the memory requirements of our implementation: instead of representing sites with up to two points (or ultimately with to two handles), we require sites to store six points (respectively, six handles). To avoid this overhead we have introduced two series of traits classes:

- One that supports the full-fledged sites, and is suitable when the input consists of strongly intersecting sites. This series consists of the Segment_Voronoi_diagram_traits_2<K,MTag> and Segment_Voronoi_diagram_filtered_traits_2<CK,CM,EK,EM,FK,FM> classes.
- One that is customized for input that contain only weakly intersecting sites. This series consists of the Segment_Voronoi_diagram_traits_without_intersections_2<K,MTag> and Segment_Voronoi_diagram_filtered_traits_without_intersections_2<CK,CM,EK,EM,FK,FM> classes.

The advantages of having different traits classes are as follows:

- When the user chooses to use one of the traits classes in the second series we only store two handles per site. This implies a reduction by a factor of three in the memory allocated per site with respect to the first series of traits classes.
- In the case of the first series of traits classes, we can better exploit the knowledge that have strongly intersecting sites, in order to further apply geometric filters (see below) during the evaluation of the predicates. On the contrary, if the second series of traits classes is used, we can avoid geometric filtering tests that have meaning only in the case of strongly intersecting sites.

## 43.3 The Geometric Traits

The predicates required for the computation of the segment Voronoi diagram are rather complicated. It is not the purpose of this document to discuss them in detail. The interested reader may refer to Burnikel's thesis [Bur96], where it is shown that in the case of weakly intersecting sites represented in homogeneous coordinates of bit size b, the maximum bit size of the algebraic expressions involved in the predicates is 40b+O(1). Given our site representation given above we can guarantee that even in the case of strongly intersecting sites, the algebraic degree of the predicates remains O(b), independently of the size of the input. What we want to focus in the remainder of this section are the different kinds of filtering techniques that we have employed in our implementation.

Geometric filtering. Our representation of sites is coupled very naturally, with what we call geometric filtering. The technique amounts to performing simple geometric tests exploiting the representation of our data, as well as the geometric structure inherent in our problem, in order to evaluate predicates in seemingly degenerate configurations. Geometric filtering can be seen as a preprocessing step before performing arithmetic filtering. Roughly speaking, by arithmetic filtering we mean that we first try to evaluate the predicates using a fixed-precision floating-point number type (such as `double`), and at the same time keep error bounds on the numerical errors of the computations we perform. If the numerical errors are too big and do not permit us to evaluate the predicate, we switch to an exact number type, and repeat the evaluation of the predicate. Geometric filtering can help by eliminating situations in which the arithmetic filter will fail, thus decreasing the number of times we need to evaluate a predicate using exact arithmetic.

To illustrate the application and effectiveness of this approach, let us consider a very simple example usage. Suppose we want to determine if two non-input points are identical (we assume here that the input sites are represented by doubles). In order to do that we need to compute their coordinates and compare them. If the two points are identical, the answer to our question using double arithmetic may be wrong (due to numerical errors), in which case we will have to reside to the more expensive exact computation. Instead, before testing the coordinates for equality, we can use the representation of the points to potentially answer the question. More specifically, and this is the geometric filtering part of the computation, we can first test if the defining segments of the two points are the same. If they are not, then we proceed to comparing their coordinates as usual. Testing the defining segments for equality does not involve any arithmetic operations on the input, but rather only comparisons on doubles. By performing this very simple test we avoid a numerically difficult computation, which could be performed thousands of times during the computation of a Voronoi diagram.

Geometric filtering has been implemented in all our models of the SegmentVoronoiDiagramTraits_2 concept. These models are the classes: Segment_Voronoi_diagram_traits_2<K,MTag>, Segment_Voronoi_diagram_traits_without_intersections_2<K,MTag>, Segment_Voronoi_diagram_filtered_traits_2<CK,CM,EK,EM,FK,FM> and Segment_Voronoi_diagram_filtered_traits_without_intersections_2<CK,CM,EK,EM,FK,FM>.

Arithmetic filtering. As mentioned above, performing computations with exact arithmetic can be very costly. For this reason we have devoted considerable effort in implementing different kinds of arithmetic filtering mechanisms. Presently, there two ways of performing arithmetic filtering for the predicates involved in the computation of segment Voronoi diagrams:

1. The user can define his/her kernel using as number type, a number type of the form CGAL::Filtered_exact<CT,ET>. Then this kernel can be entered as the first template parameter in the Segment_Voronoi_diagram_2<K,MTag> or Segment_Voronoi_diagram_with_intersections_2<K,MTag> class.
2. The user can define up to three different kernels CK, FK and EK (default values are provided for most parameters). The first kernel CK is used only for constructions. The second kernel FK is the filtering kernel: the traits class will attempt to compute the predicates using this kernel. If the filtering kernel fails to successfully compute a predicate, the exact kernel EK will be used. These three kernels are then used in the Segment_Voronoi_diagram_filtered_traits_2<CK,CM,EK,EM,FK,FM> and Segment_Voronoi_diagram_filtered_traits_without_intersections_2<CK,CM,EK,EM,FK,FM> classes, which have been implemented using the Filtered_predicate<EP,FP> mechanism.

Our experience so far has shown that for all reasonable and valid values of the template parameters, the second method for arithmetic filtering is more efficient among the two.

Let's consider once more the classes Segment_Voronoi_diagram_2<K,MTag> and Segment_Voronoi_diagram_with_intersections_2<K,MTag>. The template parameter MTag provides another degree of freedom to the user, who can indicate the type of arithmetic operations to be used in the evaluation of the predicates. More specifically, in both classes, MTag can be CGAL::Sqrt_field_tag, in which case the predicates will be evaluated using all four basic arithmetic operations plus square roots; this requires, of course, that the number type used in the kernel K supports these operations exactly. The second choices are CGAL::Field_tag for the Segment_Voronoi_diagram_2<K,MTag> class, and CGAL::Ring_tag for the Segment_Voronoi_diagram_with_intersections_2<K,MTag> class. In the first case we indicate that we want the predicates to be computed using only the four basic arithmetic operations, whereas in the second case we evaluate the predicates using only ring operations. Again, for the predicates to be evaluated correctly, the number type used in the kernel K must support the corresponding operations exactly.

The semantics for the template parameters CM, FM and EM in the Segment_Voronoi_diagram_filtered_traits_2<CK,CM,EK,EM,FK,FM> and Segment_Voronoi_diagram_filtered_traits_without_intersections_2<CK,CM,EK,EM,FK,FM> classes are analogous. With each of these template parameters we can control the type of arithmetic operations that are going to be used in calculations involving each of the corresponding kernels CK, FK and EK. When the Segment_Voronoi_diagram_filtered_traits_2<CK,CM,EK,EM,FK,FM> is used the possible values for CM, FM and EM are CGAL::Sqrt_field_tag and CGAL::Field_tag, whereas if the Segment_Voronoi_diagram_filtered_traits_without_intersections_2<CK,CM,EK,EM,FK,FM> class is used, the possible values are CGAL::Sqrt_field_tag and CGAL::Ring_tag. The semantics are the same as in the case of the Segment_Voronoi_diagram_2<K,MTag> and Segment_Voronoi_diagram_with_intersections_2<K,MTag> classes.

## 43.4 The segment Voronoi diagram hierarchy

The Segment_Voronoi_diagram_hierarchy_2<SegmentVoronoiDiagramTraits_2, SSTag, SegmentVoronoiDiagramDataStructure_2> class is the analogue of the Triangulation_hierarchy_2 or the Apollonius_graph_hierarchy_2 classes, applied to the segment Voronoi diagram. It consists of a hierarchy of segment Voronoi diagrams constructed in a manner analogous to the Delaunay hierarchy by Devillers [Dev02]. Unlike the triangulation hierarchy or the Apollonius graph hierarchy, the situation here is more complicated because of two factors: firstly, segments are treated as three objects instead of one (the two endpoints and the interior of the segments), and secondly, the presence of strongly intersecting sites complicates significantly the way the hierarchy is constructed. The interested reader may refer to the paper by Karavelas [Kar04] for the details of the construction of the hierarchy. Another alternative is to have a hybrid hierarchy that consists of the segment Voronoi diagram at the bottom-most level and point Voronoi diagrams at all other levels. This choice seems to work very well in practice , primarily because it avoids the overhead of maintaining a Voronoi diagram for segments at the upper levels of the hierarchy. However, it seems much less likely to be possible to give any theoretical guarantees for its performance, in contrast to the hierarchy with segment Voronoi diagrams at all levels (cf. [Kar04]). The user can choose between the two types of hierarchies by means of the template parameter SSTag. If SSTag is set to false (which is also the default value), the upper levels of the hierarchy consist of point Voronoi diagrams. If SSTag is set to true, we have segment Voronoi diagrams at all levels of the hierarchy.

The class Segment_Voronoi_diagram_hierarchy_2<SegmentVoronoiDiagramTraits_2, SSTag, SegmentVoronoiDiagramDataStructure_2> has exactly the same interface and functionality as the Segment_Voronoi_diagram_2<SegmentVoronoiDiagramTraits_2,SegmentVoronoiDiagramDataStructure_2> class. Using the segment Voronoi diagram hierarchy involves an additional cost in space and time for maintaining the hierarchy. Our experiments have shown that it usually pays off to use the hierarchy for inputs consisting of more than about 1,000 sites.

## 43.5 Examples

### 43.5.1 First example

The following example shows to use the segment Voronoi diagram traits in conjunction with the Filtered_exact<CT,ET> mechanism. In addition it shows how to use a few of the iterators provided by the Segment_Voronoi_diagram_2 class in order to count a few site-related quantities.

    // file: svd-count-sites.C
    #include <CGAL/basic.h>

    // standard includes
    #include <iostream>
    #include <fstream>
    #include <cassert>

    // define the number type
    #  include <CGAL/Quotient.h>
    #  include <CGAL/MP_Float.h>
    typedef CGAL::Quotient<CGAL::MP_Float>  EFT;

    // define the kernel
    #include <CGAL/Simple_cartesian.h>
    #include <CGAL/Filtered_exact.h>

    typedef CGAL::Filtered_exact<double,EFT>  FT;
    typedef CGAL::Simple_cartesian<FT>        Kernel;

    // typedefs for the traits and the algorithm
    #include <CGAL/Segment_Voronoi_diagram_traits_2.h>
    #include <CGAL/Segment_Voronoi_diagram_2.h>

    typedef CGAL::Segment_Voronoi_diagram_traits_2<Kernel>  Gt;
    typedef CGAL::Segment_Voronoi_diagram_2<Gt>             SVD2;

    using namespace std;

    int main() {
      ifstream ifs("data/sitesx.cin");
      assert( ifs );

      SVD2          svd;
      SVD2::Site_2  site;

      while ( ifs >> site ) { svd.insert( site ); }

      ifs.close();

      assert( svd.is_valid(true, 1) );
      cout << endl << endl;

      // print the number of input and output sites
      cout << "# of input sites : " << svd.number_of_input_sites() << endl;
      cout << "# of output sites: " << svd.number_of_output_sites() << endl;

      unsigned int n_ipt(0), n_iseg(0), n_opt(0), n_oseg(0), n_ptx(0);

      // count the number of input points and input segments
      SVD2::Input_sites_iterator iit;
      for (iit = svd.input_sites_begin(); iit != svd.input_sites_end(); ++iit)
        {
          if ( iit->is_point() ) { n_ipt++; } else { n_iseg++; }
        }

      // count the number of output points and output segments, as well
      // as the number of points that are points of intersection of pairs
      // of strongly intersecting sites
      SVD2::Output_sites_iterator oit;
      for (oit = svd.output_sites_begin(); oit != svd.output_sites_end(); ++oit)
        {
          if ( oit->is_segment() ) { n_oseg++; } else {
            n_opt++;
            if ( !oit->is_input() ) { n_ptx++; }
          }
        }

      cout << endl << "# of input segments:  " << n_iseg << endl;
      cout << "# of input points:    " << n_ipt << endl << endl;
      cout << "# of output segments: " << n_oseg << endl;
      cout << "# of output points:   " << n_opt << endl << endl;
      cout << "# of intersection points: " << n_ptx << endl;

      return 0;
    }

### 43.5.2 Second example

The following example shows how to use the segment Voronoi diagram hierarchy along with the filtered traits class that supports intersecting sites.

    // file: svd-filtered-traits.C
    #include <CGAL/basic.h>

    // standard includes
    #include <iostream>
    #include <fstream>
    #include <cassert>

    // example that uses the filtered traits and
    // the segment Voronoi diagram hierarchy

    // choose the kernel
    #include <CGAL/Simple_cartesian.h>

    struct Rep : public CGAL::Simple_cartesian<double> {};

    // typedefs for the traits and the algorithm
    #include <CGAL/Segment_Voronoi_diagram_hierarchy_2.h>
    #include <CGAL/Segment_Voronoi_diagram_filtered_traits_2.h>

    struct Gt
      : public CGAL::Segment_Voronoi_diagram_filtered_traits_2<Rep> {};

    typedef CGAL::Segment_Voronoi_diagram_hierarchy_2<Gt>  SVD2;

    int main()
    {
      std::ifstream ifs("data/sites.cin");
      assert( ifs );

      SVD2          svd;
      SVD2::Site_2  site;

      // read the sites and insert them in the segment Voronoi diagram
      while ( ifs >> site ) {
        svd.insert(site);
      }

      // validate the segment Voronoi diagram
      assert( svd.is_valid(true, 1) );

      return 0;
    }

### 43.5.3 Third example

The following example demonstrates how to recover the defining sites for the edges of the Voronoi diagram.

    // file: svd-voronoi-vertices.C
    #include <CGAL/basic.h>

    // standard includes
    #include <iostream>
    #include <fstream>
    #include <cassert>
    #include <string>

    // define the kernel
    #include <CGAL/Simple_cartesian.h>
    #include <CGAL/Filtered_kernel.h>

    typedef CGAL::Simple_cartesian<double>    CK;
    typedef CGAL::Filtered_kernel<CK>         Kernel;

    // typedefs for the traits and the algorithm
    #include <CGAL/Segment_Voronoi_diagram_traits_2.h>
    #include <CGAL/Segment_Voronoi_diagram_2.h>

    typedef CGAL::Segment_Voronoi_diagram_traits_2<Kernel>  Gt;
    typedef CGAL::Segment_Voronoi_diagram_2<Gt>             SVD2;

    using namespace std;

    int main()
    {
      ifstream ifs("data/sites2.cin");
      assert( ifs );

      SVD2          svd;
      SVD2::Site_2  site;

      // read the sites from the stream and insert them in the diagram
      while ( ifs >> site ) { svd.insert( site ); }

      ifs.close();

      // validate the diagram
      assert( svd.is_valid(true, 1) );
      cout << endl << endl;

      /*
      // now walk through the non-infinite edges of the dual (which are
      // dual to the edges in the Voronoi diagram) and print the sites
      // defining each Voronoi edge.
      //
      // Each oriented Voronoi edge (horizontal segment in the figure
      // below) is defined by four sites A, B, C and D.
      //
      //     \                     /
      //      \         B         /
      //       \                 /
      //     C  -----------------  D
      //       /                 \
      //      /         A         \
      //     /                     \
      //
      // The sites A and B define the (oriented) bisector on which the
      // edge lies whereas the sites C and D, along with A and B define
      // the two endpoints of the edge. These endpoints are the Voronoi
      // vertices of the triples A, B, C and B, A, D.
      // If one of these vertices is the vertex at infinity the string
      // "infinite vertex" is printed; the corresponding Voronoi edge is
      // actually a stright-line or parabolic ray.
      // The sites below are printed in the order A, B, C, D.
      */

      string inf_vertex("infinite vertex");
      char vid[] = {'A', 'B', 'C', 'D'};

      SVD2::Finite_edges_iterator eit = svd.finite_edges_begin();
      for (int k = 1; eit != svd.finite_edges_end(); ++eit, ++k) {
        SVD2::Edge e = *eit;
        // get the vertices defining the Voronoi edge
        SVD2::Vertex_handle v[] = { e.first->vertex( svd.ccw(e.second) ),
                                    e.first->vertex( svd.cw(e.second) ),
                                    e.first->vertex( e.second ),
                                    e.first->mirror_vertex(e.second) };

        cout << "--- Edge " << k << " ---" << endl;
        for (int i = 0; i < 4; i++) {
          // check if the vertex is the vertex at infinity; if yes, print
          // the corresponding string, otherwise print the site
          if ( svd.is_infinite(v[i]) ) {
            cout << vid[i] << ": " << inf_vertex << endl;
          } else {
            cout << vid[i] << ": " << v[i]->site() << endl;
          }
        }
        cout << endl;
      }

      return 0;
    }