By Dr. Mark de Berg, Dr. Marc van Kreveld, Prof. Dr. Mark Overmars, Dr. Otfried Cheong Schwarzkopf (auth.)
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Extra info for Computational Geometry: Algorithms and Applications
How can we construct (j? Recall that in the plane sweep algorithm for line segment intersection we always looked for the segments immediately to the left of an event point. ) Hence, the information we need to construct (j is determined during the plane sweep. So, to construct (j, we first make anode for every cyc1e. To find the arcs of (j, we consider the leftmost vertex v of every cyc1e bounding a hole. If e is the half-edge immediately left of v, then we add an arc between the two nodes in (j representing the cyc1e containing and the hole cyc1e of which v is the leftmost vertex.
After the upper end point is handled we continue to the next event point. When the event point is an intersection, the two segments that intersect change their order. Each of them gets (at most) one new neighbor against which it is tested for intersection. Again, only intersections below the sweep line are still interesting. Suppose that four segments S j, Sb SI, and Sm appear in this order on the sweep line when the intersection point of Sk and SI is reached. Then Sk and SI switch position and we must test SI and S j for intersection below the sweep line, and also Sk and Sm.
We implement the event queue as follows. Define an order -< on the event points that represents the order in which they will be handled. Hence, if p and q are two event points then we have p -< q if and only if Py > qy holds or Py = qy and Px < qx holds. We store the event points in a balanced binary search tree, ordered according to -<. With each event point p in Q we will store the segments starting at p, that is, the segments whose upper endpoint is p. This information will be needed to handle the event.