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  3.2 The data structure for managing the parabolic front

The binary tree as a data structure for keeping the parabolic front is not chosen just because of quick updating, but also because the topological structure of the parabolic front can be represented the most naturally by the binary tree (Figure 8).

Figure 8: The representation of a topological structure of the parabolic front

The arcs on the parabolic front are represented by the tree leaves while joints between arcs are stored by the interior tree nodes. The most left arc on the parabolic front is represented by the most left leaf in the tree, the second left most arc is represented by the next left most leaf and so on.

Of course, the most interesting operation is inserting and deleting the parabolic arcs from the data structure. As we mentioned in section 2.1, when a point event happens, a new arc appears also on the parabolic front by dividing the existed arc upon the event point. In the data structure representing the parabolic front, this is manifested as replacing the leaf with a sub-tree. As shown in Figure 9, the sub-tree consists of three nodes; the middle one represents the new arc and the other two the divided parts of the old arc.

Figure 9: Inserting a new arc into the tree representing the parabolic front

As we said in section 2.2, an arc shrinks to a point and disappears from the parabolic front at the circle event. In the tree representing the parabolic front, this reflects as deleting a leaf from the tree. However, with the leaf we must delete also the upper internal node and the second internal node becomes new joint point of two neighboring arcs. The circle event determines which arc has to be deleted. Both neighbors can be found as left and right leaf of disappearing leaf. The first node is trivial to find, because it is the father of the disappearing leaf. But finding the second node is not easy. We have to know on which side of the point, defining the disappearing arc the second node is. The second node is determined as an intersection node of two searching directions (see Figure 10). The first starts from the disappearing arc and the second from the left (if we divide arc on the left side) or the right neighbor (if the arc on the right side is being divided). The intersection node represents joint of two neighboring arcs.

Figure 10: Two possibilities of deleting an arc from the tree of the parabolic front

To complete the explanation of this structure, a few words is needed to explain the attributes stored in the nodes and the leaves. We are interested especially in pointers between the tree of the parabolic front and other structures. We are also interested in how the parabola is stored in the tree. Let us remember that there is no need to store a real parabola; it is enough to store just a point, which defines the parabola. This can be seen from the following equation [BERG97]:

where represents the scanning line at the moment of observation and represents the point from P, which defines the arc .

Internal nodes of the tree the pointers pointing to the edge in the data structure of Voronoi diagram are stored. In this way, each edge belonging to the Voronoi point is directly accessible, when the circle event is serviced and so there is no need to perform any search operations in the data structure of Voronoi diagram. At the leaves of the tree, a pointer to the circle event is stored, if the arc defines a circle event. If not, pointer is set to NULL. By maintaining this pointer, we do not have to perform any search after encountering false events.