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.
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
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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