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?
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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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