Straight line arrangements in the real projective plane

Forge, David; Alfonsín, Jorge Luis Ramírez

doi:10.1007/pl00009373

Cited by 6 publications

(8 citation statements)

References 3 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…On the one hand, there is an algorithm for finding all triangular pseudoline arrangements in [8] and there are various constructions of Harborth [16], Roudneff [33,34] and Forge and Ramirez Alfonsin [13] for finding infinite classes of such triangular pseudoline arrangements from given ones. On the other hand, there are recent contributions by Bonnington et al [9], Korzhik and Voss [20], and Bracho and Strausz [10] aiming at non-isomorphic complete triangulations of surfaces.…”

Section: Motivation and Historical Remarksmentioning

confidence: 99%

See 1 more Smart Citation

Oriented matroids and complete-graph embeddings on surfaces

Bokowski

Pisanski

2007

Journal of Combinatorial Theory, Series A

View full text Add to dashboard Cite

We provide a link between topological graph theory and pseudoline arrangements from the theory of oriented matroids. We investigate and generalize a function f that assigns to each simple pseudoline arrangement with an even number of elements a pair of complete-graph embeddings on a surface. Each element of the pair keeps the information of the oriented matroid we started with. We call a simple pseudoline arrangement triangular, when the cells in the cell decomposition of the projective plane are 2-colorable and when one color class of cells consists of triangles only. Precisely for triangular pseudoline arrangements, one element of the image pair of f is a triangular complete-graph embedding on a surface. We obtain all triangular complete-graph embeddings on surfaces this way, when we extend the definition of triangular complete pseudoline arrangements in a natural way to that of triangular curve arrangements on surfaces in which each pair of curves has a point in common where they cross. Thus Ringel's results on the triangular complete-graph embeddings can be interpreted as results on curve arrangements on surfaces. Furthermore, we establish the relationship between 2-colorable curve arrangements and Petrie dual maps. A data structure, called intersection pattern is provided for the study of curve arrangements on surfaces. Finally we show that an orientable surface of genus g admits a complete curve arrangement with at most 2g + 1 curves in contrast to the non-orientable surface where the number of curves is not bounded.

show abstract

Section: Motivation and Historical Remarksmentioning

confidence: 99%

“…[Note that (S 13 ) is an Ringel's notation for the complete graph on 13 vertices, nowadays denoted by (K 13 ).] It leads to a triangular complete curve arrangement with 13 curves.…”

Section: Triangular Curve Arrangementsmentioning

confidence: 99%

Oriented matroids and complete-graph embeddings on surfaces

Bokowski

Pisanski

2007

Journal of Combinatorial Theory, Series A

View full text Add to dashboard Cite

show abstract

“…Füredi and Palásti [9] construct an arrangement proving K( ) ≥ ( − 3)/3. See also the article of Forge and Ramírez Alfonsín [7].…”

Section: Preparationmentioning

confidence: 99%

Congruent triangles in arrangements of lines

Zamfirescu

2017

Ars Math. Contemp.

View full text Add to dashboard Cite

We study the maximum number of congruent triangles in finite arrangements of lines in the Euclidean plane. Denote this number by f ( ). We show that f (5) = 5 and that the construction realizing this maximum is unique, f (6) = 8, and f (7) = 14. We also discuss for which integers c there exist arrangements on lines with exactly c congruent triangles. In parallel, we treat the case when the triangles are faces of the plane graph associated to the arrangement (i.e. the interior of the triangle has empty intersection with every line in the arrangement). Lastly, we formulate four conjectures.

show abstract

“…The converse has recently been shown as well; one of the mentioned constructions can be specialized in order to preserve realizability, in other words: there exists an infinite family of stretchable p 3 -maximal arrangements (see [11]). …”

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

View full text Add to dashboard Cite

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.

show abstract

Straight line arrangements in the real projective plane

Abstract: Abstract. Let In this paper we answer this question affirmatively.

Cited by 6 publications

References 3 publications

Oriented matroids and complete-graph embeddings on surfaces

Oriented matroids and complete-graph embeddings on surfaces

Congruent triangles in arrangements of lines

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

Contact Info

Product

Resources

About