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 $P$ of cardinality $n$ 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 $n$ 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 $n$ 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 $\Omega(n^2)$ and $O(n^{9/4})$. 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

On the number of tetrahedra with minimum, unit, and distinct volumes in three-space

We formulate and give partial answers to several combinatorial problems on volumes of simplices determined by $n$ points in 3-space, and in general in $d$ dimensions. (i) The number of tetrahedra of minimum (nonzero) volume spanned by $n$ points in \RR^3 is at most ${2/3}n^3-O(n^2)$, and there are point sets for which this number is ${3/16}n^3-O(n^2)$. We also present an $O(n^3)$ time algorithm for reporting all tetrahedra of minimum nonzero volume, and thereby extend an algorithm of Edelsbrunner, O'Rourke, and Seidel. In general, for every k,d\in \NN, $1\leq k \leq d$, the maximum number of $k$-dimensional simplices of minimum (nonzero) volume spanned by $n$ points in \RR^d is $\Theta(n^k)$. (ii) The number of unit-volume tetrahedra determined by $n$ points in \RR^3 is $O(n^{7/2})$, and there are point sets for which this number is $\Omega(n^3 \log \log{n})$. (iii) For every d\in \NN, the minimum number of distinct volumes of all full-dimensional simplices determined by $n$ points in \RR^d, not all on a hyperplane, is $\Theta(n)$.Comment: 19 pages, 3 figures, a preliminary version has appeard in proceedings of the ACM-SIAM Symposium on Discrete Algorithms, 200

Micro-Macro relations for flow through random arrays of cylinders

The transverse permeability for creeping flow through unidirectional random arrays of fibers with various structures is revisited theoretically and numerically using the finite element method (FEM). The microstructure at various porosities has a strong effect on the transport properties, like permeability, of fibrous materials. We compare different microstructures (due to four random generator algorithms) as well as the effect of boundary conditions, finite size, homogeneity and isotropy of the structure on the macroscopic permeability of the fibrous medium. Permeability data for different minimal distances collapse when their minimal value is subtracted, which yields an empirical macroscopic permeability master function of porosity. Furthermore, as main result, a microstructural model is developed based on the lubrication effect in the narrow channels between neighboring fibers. The numerical experiments suggest a unique, scaling power law relationship between the permeability obtained from fluid flow simulations and the mean value of the shortest Delaunay triangulation edges (constructed using the centers of the fibers), which is identical to the averaged second nearest neighbor fiber distances. This universal lubrication relation, as valid in a wide range of porosities, accounts for the microstructure, e.g. hexagonally ordered or disordered fibrous media. It is complemented by a closure relation that relates the effective microscopic length to the packing fraction

Simplifying intensity-modulated radiotherapy plans with fewer beam angles for the treatment of oropharyngeal carcinoma.

The first aim of the present study was to investigate the feasibility of using fewer beam angles to improve delivery efficiency for the treatment of oropharyngeal cancer (OPC) with inverse-planned intensity-modulated radiation therapy (IP-IMRT). A secondary aim was to evaluate whether the simplified IP-IMRT plans could reduce the indirect radiation dose. The treatment plans for 5 consecutive OPC patients previously treated with a forward-planned IMRT (FP-IMRT) technique were selected as benchmarks for this study. The initial treatment goal for these patients was to deliver 70 Gy to > or = 95% of the planning gross tumor volume (PTV-70) and 59.4 Gy to > or = 95% of the planning clinical tumor volume (PTV-59.4) simultaneously. Each case was re-planned using IP-IMRT with multiple beam-angle arrangements, including four complex IP-IMRT plans using 7 or more beam angles, and one simple IMRT plan using 5 beam angles. The complex IP-IMRT plans and simple IP-IMRT plans were compared to each other and to the FPIMRT plans by analyzing the dose coverage of the target volumes, the plan homogeneity, the dose-volume histograms of critical structures, and the treatment delivery parameters including delivery time and the total number of monitor units (MUs). When comparing the plans, we found no significant difference between the complex IP-IMRT, simple IP-IMRT, and FP-IMRT plans for tumor target coverage (PTV-70: p = 0.56; PTV-59.4: p = 0.20). The plan homogeneity, measured by the mean percentage isodose, did not significantly differ between the IP-IMRT and FP-IMRT plans (p = 0.08), although we observed a trend toward greater inhomogeneity of dose in the simple IP-IMRT plans. All IP-IMRT plans either met or exceeded the quality of the FP-IMRT plans in terms of dose to adjacent critical structures, including the parotids, spinal cord, and brainstem. As compared with the complex IP-IMRT plans, the simple IP-IMRT plans significantly reduced the mean treatment time (maximum probability for four pairwise comparisons: p = 0.0003). In conclusion, our study demonstrates that, as compared with complex IP-IMRT, simple IP-IMRT can significantly improve treatment delivery efficiency while maintaining similar target coverage and sparing of critical structures. However, the improved efficiency does not significantly reduce the total number of MUs nor the indirect radiation dose

Multitriangulations, pseudotriangulations and primitive sorting networks

We study the set of all pseudoline arrangements with contact points which cover a given support. We define a natural notion of flip between these arrangements and study the graph of these flips. In particular, we provide an enumeration algorithm for arrangements with a given support, based on the properties of certain greedy pseudoline arrangements and on their connection with sorting networks. Both the running time per arrangement and the working space of our algorithm are polynomial. As the motivation for this work, we provide in this paper a new interpretation of both pseudotriangulations and multitriangulations in terms of pseudoline arrangements on specific supports. This interpretation explains their common properties and leads to a natural definition of multipseudotriangulations, which generalizes both. We study elementary properties of multipseudotriangulations and compare them to iterations of pseudotriangulations.Comment: 60 pages, 40 figures; minor corrections and improvements of presentatio