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

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The graph corresponding to the subdivision is also shown in the figure. The hole cycles are shown as single circles, and the outer boundary cycles are shown as double circles. Observe that C3 and C6 are in the same connected component as C2. 6 A subdivision and the corresponding graph q @ boundary is C2. If there is only one hole in a face J, then the graph (j links the boundary eyde of the hole to the outer boundary of J. 6. This hole, whieh lies in the same face J, may be linked to the outer boundary of J, or it may be linked to yet another hole.

To reach the boundary of a face we just need to store one pointer in the face record to an arbitrary half-edge bounding the face. Starting from that half-edge, we can step from each half-edge to the next and walk around the face. What we just said does not quite hold for the boundaries of holes in a face: if they are traversed in counterclockwise order then the face lies to the right. It will be convenient to orient half-edges such that their face always lies to the same side, so we change the direction of traversal for the boundary of a hole to clockwise.

Furthermore, we must set the IncidentFaceO fields of the half-edges on the boundary of J so that they point to the face record of J. Finally, each of the new faces must be labeled with the names of the faces in the old subdivisions that contain it. How many face records will there be? WeIl, except for the unbounded face, every face has a unique outer boundary, so the number of face records we have to create is equal to the number of outer boundaries plus one. From the part of the doubly-connected edge list we have constructed so far we can easily extract all boundary cycles.

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