Graph Theory

Hypergraph Seminar by C. Berge, D. Ray-Chaudhuri

By C. Berge, D. Ray-Chaudhuri

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There is, however, a subtlety regarding this combined approach since, as we will describe next, incremental construction of arrangements requires answering point-location queries. We discuss this issue after presenting various point-location strategies. The Insertion Process Fig. 5. The insertion process of a new curve (dotted) into an existing arrangement of planar curves. The zone of the new curve is lightly shaded and all the newly introduced vertices are marked k) log n), while the incremental construction may take O(n2 ) time.

The traits class exploits the functionality of the parameterized SegmentTraits type to handle the segments that comprise the polyline curves. , Arr segment traits 2 or Arr segment cached traits 2 ), and the number type used by the instantiated segment traits should satisfy the same conditions listed above. We next review the arrangement traits classes that handle non-linear curves. The first two traits classes are included in the public distribution of Cgal, while the others (Sect.

The first step consists of choosing a representation for the input data: - We will consider here implicit algebraic curves, and focus more specifically circles since this corresponds to the current Cgal release. Such a curve is represented by a bivariate polynomial of total degree two. A circular arc is represented by a supporting circle and its two delimiting endpoints, knowing that the arc is oriented counterclockwise. - Two kinds of points—intersection points and endpoints—can be considered in the same way, thus allowing us to have a unique representation.

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