32,621 research outputs found
Triangles in arrangements of lines
AbstractA set of n lines in the projective plane divides the plane into a certain number of polygonal cells. We show that if we insist that all of these cells be triangles, then there are at most 13n(n − 1) + 4 − 27n of them. We also observe that if no point of the plane belongs to more than two of the lines and n is at least 4, some of the cells must be either quadrangles or pentagons. We further show that for n ≧ 4, there is a set of n lines which divides the plane into at least 13n(n − 3) + 4 triangles
Congruent triangles in arrangements of lines
We study the maximum number of congruent triangles in finite arrangements of I lines in the Euclidean plane. Denote this number by f (l). 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 l 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
On simple arrangements of lines and pseudo-lines in P^2 and R^2 with the maximum number of triangles
We give some new advances in the research of the maximum number of triangles
that we may obtain in a simple arrangements of n lines or pseudo-lines.Comment: 12 pages, 9 figure
Triangle areas in line arrangements
A widely investigated subject in combinatorial geometry, originated from
Erd\H{o}s, is the following. Given a point set of cardinality in the
plane, how can we describe the distribution of the determined distances? This
has been generalized in many directions. In this paper we propose the following
variants. Consider planar arrangements of lines. Determine the maximum
number of triangles of unit area, maximum area or minimum area, determined by
these lines. Determine the minimum size of a subset of these lines so that
all triples determine distinct area triangles.
We prove that the order of magnitude for the maximum occurrence of unit areas
lies between and . This result is strongly connected
to both additive combinatorial results and Szemer\'edi--Trotter type incidence
theorems. Next we show a tight bound for the maximum number of minimum area
triangles. Finally we present lower and upper bounds for the maximum area and
distinct area problems by combining algebraic, geometric and combinatorial
techniques.Comment: Title is shortened. Some typos and small errors were correcte
Flag arrangements and triangulations of products of simplices
We investigate the line arrangement that results from intersecting d complete
flags in C^n. We give a combinatorial description of the matroid T_{n,d} that
keeps track of the linear dependence relations among these lines. We prove that
the bases of the matroid T_{n,3} characterize the triangles with holes which
can be tiled with unit rhombi. More generally, we provide evidence for a
conjectural connection between the matroid T_{n,d}, the triangulations of the
product of simplices Delta_{n-1} x \Delta_{d-1}, and the arrangements of d
tropical hyperplanes in tropical (n-1)-space. Our work provides a simple and
effective criterion to ensure the vanishing of many Schubert structure
constants in the flag manifold, and a new perspective on Billey and Vakil's
method for computing the non-vanishing ones.Comment: 39 pages, 12 figures, best viewed in colo
BoolSurf: Boolean Operations on Surfaces
We port Boolean set operations between 2D shapes to surfaces of any genus, with any number of open boundaries. We combine shapes bounded by sets of freely intersecting loops, consisting of geodesic lines and cubic Bézier splines lying on a surface. We compute the arrangement of shapes directly on the surface and assign integer labels to the cells of such arrangement. Differently from the Euclidean case, some arrangements on a manifold may be inconsistent. We detect inconsistent arrangements and help the user to resolve them. Also, we extend to the manifold setting recent work on Boundary-Sampled Halfspaces, thus supporting operations more general than standard Booleans, which are well defined on inconsistent arrangements, too. Our implementation discretizes the input shapes into polylines at an arbitrary resolution, independent of the level of resolution of the underlying mesh. We resolve the arrangement inside each triangle of the mesh independently and combine the results to reconstruct both the boundaries and the interior of each cell in the arrangement. We reconstruct the control points of curves bounding cells, in order to free the result from discretization and provide an output in vector format. We support interactive usage, editing shapes consisting up to 100k line segments on meshes of up to 1M triangles
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