By Dr. Mark de Berg, Dr. Marc van Kreveld, Prof. Dr. Mark Overmars, Dr. Otfried Schwarzkopf (auth.)

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This structure, or in fact a variant of it, was described by Muller and Preparata [252]. There are also other data structures for storing subdivisions, such as the winged edge structure by Baumgart [28] and 41 Chapter 2 LINE SEGMENT INTERSECTION the quad edge structure by Guibas and Stolfi [171]. The difference between all these structures is small. They all have more or less the same functionality, but some save a few bytes of storage per edge. ,. . 1\ . . 1 Let S be a set of n disjoint line segments whose upper endpoints lie on the line y = 1 and whose lower endpoints lie on the line y = 0.

But eventually we must end up linking a hole to the outer boundary, as the next lemma shows. 5 Each connected component of the graph to the set of cycles incident to one face. q corresponds exactly Proof Consider a cycle C bounding a hole in a face f. Because f lies locally to the left of the leftmost vertex of C, C must be linked to another cycle that also bounds f. It follows that cycles in the same connected component of q bound the same face. To finish the proof, we show that every cycle bounding a hole in f is in the same connected component as the outer boundary of f.

At each internal node, we store the segment from the rightmost leaf in its left subtree. (Alternatively, we could store the segments only in interior nodes. This will save some storage. However, it is conceptually simpler to think about the segments in interior nodes as values to guide the search, not as data items. ) Suppose we search in

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