On K Sets in Arrangements of Curves And Surfaces
On K Sets in Arrangements of Curves And Surfaces
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Since, as shown in [13], the boundaries of any pair of sets Ki intersect at most twice (assimiing general position), the claim follows immediately from Theorem 1. 1. □ 3. 3 Partial stabbing Let C = {Ci, . . . , C„} be a collection of n pairwise disjoint compact convex sets in the plane. For each line £, let a(i) denote the sequence of those sets d that intersect I, ordered in their order along £ (we will identify a(£) with its reverse sequence). A 11 line ^ is a j -stabber oi C if \(t(£)\ = j; ...an n-stabber is also called a common transversal of C. Edelsbrunner and Sharir [9] have shown that the maximum number of distinct sequences cr(^), over all n-stabbers i of C, is 2n — 2. We can extend this result to show Theorem 3. 7 The maximum number of distinct sequences cr{£), over all j -stabbers C of C, for j = n, n — 1, . . . , n — k + 1, is 0{nk). This is tight in the worst case.
Proof: Let ^ be a j-stabber of C Arguing as in Lemma 1 of [9], one can show that £ can be continuously moved to an extreme line £*, such that a(£) = a{t') and C* is tangent to two sets C, C' E C that lie on the same side of £*.
What to read after On K Sets in Arrangements of Curves And Surfaces?
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:
Since, as shown in [13], the boundaries of any pair of sets Ki intersect at most twice (assimiing general position), the claim follows immediately from Theorem 1. 1. □ 3. 3 Partial stabbing Let C = {Ci, . . . , C„} be a collection of n pairwise disjoint compact convex sets in the plane. For each line £, let a(i) denote the sequence of those sets d that intersect I, ordered in their order along £ (we will identify a(£) with its reverse sequence). A 11 line ^ is a j -stabber oi C if \(t(£)\ = j; ...an n-stabber is also called a common transversal of C. Edelsbrunner and Sharir [9] have shown that the maximum number of distinct sequences cr(^), over all n-stabbers i of C, is 2n — 2. We can extend this result to show Theorem 3. 7 The maximum number of distinct sequences cr{£), over all j -stabbers C of C, for j = n, n — 1, . . . , n — k + 1, is 0{nk). This is tight in the worst case.
Proof: Let ^ be a j-stabber of C Arguing as in Lemma 1 of [9], one can show that £ can be continuously moved to an extreme line £*, such that a(£) = a{t') and C* is tangent to two sets C, C' E C that lie on the same side of £*.
What to read after On K Sets in Arrangements of Curves And Surfaces?
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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