The Complexity of Many Faces in Arrangements of Lines And of Segments
The Complexity of Many Faces in Arrangements of Lines And of Segments
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On the other hand, if Cj and 62 belong to a common contour cycle 7 of some E^, then we distinguish two cases. Let '/g be the piece of 7 connecting the last point of e^ with the first point of e2- (ii) If 7q contains no blue boundary piece, then we charge ^2 to the (first) reflex red vertex that lies on 7o - such a vertex must exist since, other- . Wise, we could not be led back to the same blue edge (see Figure 5. 4(b)).
(iii) Otherwise, we charge e^ to the arc of ^ that is induced by the first... blue boundary piece on 70 (see Figure 5. 4(c)).
It is easy to see that each reflex vertex is charged at most once. The definition of C, guarantees that in each case where we charge ^ there is in fact an arc that takes the charge. We now argue that the mechanism we use can charge an arc of Q at most twice. In case (i) the arc takes the charge for an edge, e^' that is the first edge of the corresponding blue boundary piece. Since every blue boundary piece has only one first edge, this case can occur only once.
What to read after The Complexity of Many Faces in Arrangements of Lines And of Segments?
You can find similar books in the "Read Also" column, or choose other free books by Herbert Edelsbrunner to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
On the other hand, if Cj and 62 belong to a common contour cycle 7 of some E^, then we distinguish two cases. Let '/g be the piece of 7 connecting the last point of e^ with the first point of e2- (ii) If 7q contains no blue boundary piece, then we charge ^2 to the (first) reflex red vertex that lies on 7o - such a vertex must exist since, other- . Wise, we could not be led back to the same blue edge (see Figure 5. 4(b)).
(iii) Otherwise, we charge e^ to the arc of ^ that is induced by the first... blue boundary piece on 70 (see Figure 5. 4(c)).
It is easy to see that each reflex vertex is charged at most once. The definition of C, guarantees that in each case where we charge ^ there is in fact an arc that takes the charge. We now argue that the mechanism we use can charge an arc of Q at most twice. In case (i) the arc takes the charge for an edge, e^' that is the first edge of the corresponding blue boundary piece. Since every blue boundary piece has only one first edge, this case can occur only once.
What to read after The Complexity of Many Faces in Arrangements of Lines And of Segments?
You can find similar books in the "Read Also" column, or choose other free books by Herbert Edelsbrunner to read onlineMoreLess
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