Arrangements of lines with a minimum number of triangles are simple

Roudneff, Jean-Pierre

doi:10.1007/bf02187900

Cited by 9 publications

(9 citation statements)

References 2 publications

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“…The second one is that of general position. An arrangement of lines A is said to be simple if no point of the plane belongs to more than two lines of A, i.e., if no three lines are concurrent [Ro88]. An arrangement of lines A is said to be in general position if (i) it is simple and (ii) no two lines of A are parallel.…”

Section: Sail Arrangementsmentioning

confidence: 99%

On Envelopes of Arrangements of Lines

Eu¹,

Guévremont

Toussaint

1996

Journal of Algorithms

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Section: Sail Arrangementsmentioning

confidence: 99%

On Envelopes of Arrangements of Lines

Eu¹,

Guévremont

Toussaint

1996

Journal of Algorithms

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“…A similar effect in the projective setting was conjectured by Grünbaum and proved by Roudneff [Rou2]. A nonsimple projective arrangement with p 3 = n is nonstrechable.…”

Section: Corollary 3 If P 3 (B) < N − 2 For An Arrangement B Of N Psmentioning

confidence: 57%

Triangles in Euclidean Arrangements

Felsner¹,

Kriegel²

1999

Discrete Comput Geom

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The number of triangles in arrangements of lines and pseudolines has been the object of some research. Most results, however, concern arrangements in the projective plane. In this article we add results for the number of triangles in Euclidean arrangements of pseudolines. Though the change in the embedding space from projective to Euclidean may seem small there are interesting changes both in the results and in the techniques required for the proofs.In 1926 Levi proved that a nontrivial arrangement-simple or not-of n pseudolines in the projective plane contains at least n triangles. To show the corresponding result for the Euclidean plane, namely, that a simple arrangement of n pseudolines contains at least n − 2 triangles, we had to find a completely different proof. On the other hand a nonsimple arrangement of n pseudolines in the Euclidean plane can have as few as 2n/3 triangles and this bound is best possible. We also discuss the maximal possible number of triangles and some extensions.

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“…The general upper bound p 3 ≤ n(n − 1)/3 is true for n ≥ 9, see [21]. Moreover, for simple arrangements, equality holds if and only if the triangles alternate along each pseudoline, which requires n to be even.…”

Section: Inductive Constructions For Generating P 3 -Maximal Pseudolimentioning

confidence: 99%

Cell Decompositions of the Projective Plane with Petrie Polygons of Constant Length

Bokowski¹,

Roudneff²,

Strempel³

1997

Discrete Comput Geom

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We study dual pairs of combinatorial face-to-face cell decompositions (D P 2 , D * P 2 ) of the real projective plane P 2 such that their canonically induced cell decompositions (D S 2 , D * S 2 ) on the 2-sphere S 2 form dual pairs of combinatorical types of convex polyhedra, and such that these dual pairs share two natural properties with those induced by dual pairs of Platonic solids: (1) Every Petrie polygon is a finite simple closed polygon of length 2(n − 1) for some fixed n.(2) Every pair of Petrie polygons has precisely two common edges. Such pairs of face-to-face cell decompositions of the projective plane are in one-to-one correspondence with n-element pseudoline arrangements. We study in particular those dual pairs of cell decompositions (D P 2 , D * P 2 ) in which D P 2 or D * P 2 has only 3-valent vertices, i.e., via the above one-to-one correspondence: p 3 -maximal pseudoline arrangements. A p 3 -maximal pseudoline arrangement with n elements in turn determines a neighborly 2-manifold with Euler characteristic χ = n(7−n)/6, and vice versa, this neighborly 2-manifold uniquely determines its generating p 3 -maximal pseudoline arrangement. We provide new inductive constructions for finding infinite example classes of p 3 -maximal pseudoline arrangements from small existing ones, we describe an algorithm for generating them, we provide a complete list of existence up to n = 40, and we discuss their properties.

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Arrangements of lines with a minimum number of triangles are simple

Cited by 9 publications

References 2 publications

On Envelopes of Arrangements of Lines

On Envelopes of Arrangements of Lines

Triangles in Euclidean Arrangements

Cell Decompositions of the Projective Plane with Petrie Polygons of Constant Length

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