Improved bounds for intersecting triangles and halving planes

If a configuration of m triangles in the plane has only n points as vertices, then there must be a set ofmax { [m/(2n - 5)] Ω(m^3 /(n^6 log^2 n))triangles having a common intersection. As a consequence the number of halving planes for a three-dimensional point set is O(n^8/3 log^2/3 n). For all m and n there exist configurations of triangles in which the largest common intersection involvesmax{ [m/(2n - 5)] O(m^2 /n^3)triangles; the upper and lower bounds match for m= O(n^2). The best previous bounds were Ω(m^3 /n^ 6 log^5 n)) for intersecting triangles, and O(n^8/3 log^5/3 n) for halving planes

Eppstein's bound on intersecting triangles revisited

Let S be a set of n points in the plane, and let T be a set of m triangles with vertices in S. Then there exists a point in the plane contained in Omega(m^3 / (n^6 log^2 n)) triangles of T. Eppstein (1993) gave a proof of this claim, but there is a problem with his proof. Here we provide a correct proof by slightly modifying Eppstein's argument.Comment: Minor revision following referee's suggestions. To appear in Journal of Combinatorial Theory, Series A. 5 pages, 1 figur

Consensus-halving via Theorems of Borsuk-Ulam and Tucker

In this paper we show how theorems of Borsuk-Ulam and Tucker can be used to construct a consensus-halving: a division of an object into two portions so that each of n people believe the portions are equally split. Moreover, the division takes at most n cuts, which is best possible. This extends prior work using methods from combinatorial topology to solve fair division problems. Several applications of consensus-halving are discussed.

Input description for Jameson's three-dimensional transonic airfoil analysis program

The input parameters are presented for a computer program which performs calculations for inviscid isentropic transonic flow over three dimensional airfoils with straight leading edges. The free stream Mach number is restricted only by the isentropic assumption. Weak shock waves are automatically located where they occur in the flow. The finite difference form of the full equation for the velocity potential is solved by the method of relaxation, after the flow exterior to the airfoil is mapped to the upper half plane

Autocalibration with the Minimum Number of Cameras with Known Pixel Shape

In 3D reconstruction, the recovery of the calibration parameters of the cameras is paramount since it provides metric information about the observed scene, e.g., measures of angles and ratios of distances. Autocalibration enables the estimation of the camera parameters without using a calibration device, but by enforcing simple constraints on the camera parameters. In the absence of information about the internal camera parameters such as the focal length and the principal point, the knowledge of the camera pixel shape is usually the only available constraint. Given a projective reconstruction of a rigid scene, we address the problem of the autocalibration of a minimal set of cameras with known pixel shape and otherwise arbitrarily varying intrinsic and extrinsic parameters. We propose an algorithm that only requires 5 cameras (the theoretical minimum), thus halving the number of cameras required by previous algorithms based on the same constraint. To this purpose, we introduce as our basic geometric tool the six-line conic variety (SLCV), consisting in the set of planes intersecting six given lines of 3D space in points of a conic. We show that the set of solutions of the Euclidean upgrading problem for three cameras with known pixel shape can be parameterized in a computationally efficient way. This parameterization is then used to solve autocalibration from five or more cameras, reducing the three-dimensional search space to a two-dimensional one. We provide experiments with real images showing the good performance of the technique.Comment: 19 pages, 14 figures, 7 tables, J. Math. Imaging Vi

A calculation of the structure and energy of the Nb/Al2O3 interface

We have modelled the (111)(Nb)/(0001)(s)Nb/Al2O3 interface using an atomistic, static lattice simulation technique. The interaction between the metal and the oxide combines the short range interaction between the metal atoms and the oxide ions, the Coulomb interaction between the oxide ions and the induced image charge of the metal, and the energy required to immerse the ionic cores in the metal jellium. The short range interaction between the Al3+ ion and the Nb atom was found to be repulsive, but the O2-/Nb interaction was found to be attractive at separations greater than 0.23 nm. As a result the lowest energy interface was found to terminate on an oxygen plane of the Al2O3; crystal, with the Nb atoms placed over the vacant sites in the Al lattice. The interfacial energy of this interface was calculated to be -3.6 J/m(2). As in previous work the results agree well with LDF calculations. The calculated structure is also in good agreement with the interpretation of the HREM images of Nb films grown on the (0001) face of Al2O3 using Molecular Beam Epitaxy. Copyright (C) 1996 Acta Metallurgica Inc