Intersection And Closest Pair Problems for a Set of Planar Objects
Intersection And Closest Pair Problems for a Set of Planar Objects
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(c) There exists precisely one L-region; all other componeuts of the compIemeDt of C are R- regions.
(d) Each R-region has a connected boundary, consbting of a single (bounded or unbounded) com- ponent of C .
Proof: (a) It follows from its definition that C is a union of Voronoi edges of Vor(S). It therefore suffices to show that for each Voronoi vertex t; of Vot(S) lying on C, there are exactly two Voro- noi edges emerging from v which belong to C. Since we have ruled out degenerate configurat...ions, we can assume that t; belongs to exactly three Voronoi cells V(i), V(j) and V(k). Moreover, since v Q. C, one of the discs B, B, B^ must belong to L, and another of these discs must belong to R. Assume first that B, Bn E L and that B, Qi R . Then, in the neighborhood of V, the contour C consists of the two edges separating V(j) from V(i) and V(k) respectively. Much the same argument applies it B, B^ ^ R and B, ^ L . This proves (a).
(b) Suppose the contrary, and let i be a contour point lying on u.
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You can find similar books in the "Read Also" column, or choose other free books by Micha Sharir to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
(c) There exists precisely one L-region; all other componeuts of the compIemeDt of C are R- regions.
(d) Each R-region has a connected boundary, consbting of a single (bounded or unbounded) com- ponent of C .
Proof: (a) It follows from its definition that C is a union of Voronoi edges of Vor(S). It therefore suffices to show that for each Voronoi vertex t; of Vot(S) lying on C, there are exactly two Voro- noi edges emerging from v which belong to C. Since we have ruled out degenerate configurat...ions, we can assume that t; belongs to exactly three Voronoi cells V(i), V(j) and V(k). Moreover, since v Q. C, one of the discs B, B, B^ must belong to L, and another of these discs must belong to R. Assume first that B, Bn E L and that B, Qi R . Then, in the neighborhood of V, the contour C consists of the two edges separating V(j) from V(i) and V(k) respectively. Much the same argument applies it B, B^ ^ R and B, ^ L . This proves (a).
(b) Suppose the contrary, and let i be a contour point lying on u.
What to read after Intersection And Closest Pair Problems for a Set of Planar Objects?
You can find similar books in the "Read Also" column, or choose other free books by Micha Sharir to read onlineMoreLess
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