Approximate Fitting of a Circular Arc When Two Points Are Known

The task of approximating points with circular arcs is performed in many applications, such as polyline compression, noise filtering, and feature recognition. However, the development of algorithms that perform a significant amount of circular arcs fitting requires an efficient way of fitting circular arcs with complexity O(1). The elegant solution to this task based on an eigenvector problem for a square nonsymmetrical matrix is described in [1]. For the compression algorithm described in [2], it is necessary to solve this task when two points on the arc are known. This paper describes a different approach to efficiently fitting the arcs and solves the task when one or two points are known.Comment: 15 pages, 4 figures, extended abstract published at the conferenc

Polynomial Meshes: Computation and Approximation

We present the software package WAM, written in Matlab, that generates Weakly Admissible Meshes and Discrete Extremal Sets of Fekete and Leja type, for 2d and 3d polynomial least squares and interpolation on compact sets with various geometries. Possible applications range from data fitting to high-order methods for PDEs

Is the Number of Giant Arcs in LCDM Consistent With Observations?

We use high-resolution N-body simulations to study the galaxy-cluster cross-sections and the abundance of giant arcs in the $\Lambda$CDM model. Clusters are selected from the simulations using the friends-of-friends method, and their cross-sections for forming giant arcs are analyzed. The background sources are assumed to follow a uniform ellipticity distribution from 0 to 0.5 and to have an area identical to a circular source with diameter 1\arcsec. We find that the optical depth scales as the source redshift approximately as \tau_{1''} = 2.25 \times 10^{-6}/[1+(\zs/3.14)^{-3.42}] (0.6<\zs<7). The amplitude is about 50% higher for an effective source diameter of 0.5\arcsec. The optimal lens redshift for giant arcs with the length-to-width ratio ($L/W$) larger than 10 increases from 0.3 for \zs=1, to 0.5 for \zs=2, and to 0.7-0.8 for \zs>3. The optical depth is sensitive to the source redshift, in qualitative agreement with Wambsganss et al. (2004). However, our overall optical depth appears to be only $\sim$ 10% to 70% of those from previous studies. The differences can be mostly explained by different power spectrum normalizations ($\sigma_8$) used and different ways of determining the $L/W$ ratio. Finite source size and ellipticity have modest effects on the optical depth. We also found that the number of highly magnified (with magnification $|\mu|>10$) and ``undistorted'' images (with $L/W<3$) is comparable to the number of giant arcs with $|\mu|>10$ and $L/W>10$. We conclude that our predicted rate of giant arcs may be lower than the observed rate, although the precise `discrepancy' is still unclear due to uncertainties both in theory and observations.Comment: Revised version after the referee's reports (32 pages,13figures). The paper has been significantly revised with many additions. The new version includes more detailed comparisons with previous studies, including the effects of source size and ellipticity. New discussions about the redshift distribution of lensing clusters and the width of giant arcs have been adde

Shock fitting applied to the prediction of high-speed rotor noise

A shock fitting method applied to the transonic small disturbance (TSD) potential equation is described. This method is then applied to a simple, two dimensional (2-D) rotating disturbance which is analogous to a shock radiating from the tip of a rotor blade in high speed hover. A comparison is made between the results of this method and the more standard shock capturing method. This comparison makes it clear that the effect of the results on the acoustic signature of the 2-D model is significant, and similar results can be expected when the method is extended to the three dimensional (3-D) case

The extended tails of Palomar 5: A ten degree arc of globular cluster tidal debris

Using wide-field photometric data from the Sloan Digital Sky Survey (SDSS) we recently showed that the Galactic globular cluster Palomar 5 is in the process of being tidally disrupted. Its tidal tails were initially detected in a 2.5 degree wide band along the celestial equator. A new analysis of SDSS data for a larger field now reveals that the tails of Pal 5 have a much larger spatial extent and can be traced over an arc of 10 deg across the sky, corresponding to a projected length of 4 kpc at the distance of the cluster. The number of former cluster stars found in the tails adds up to about 1.2 times the number of stars in the cluster. The radial profile of stellar surface density in the tails follows approximately a power law r^gamma with -1.5 < gamma < -1.2. The stream of debris from Pal 5 is significantly curved, which demonstrates its acceleration by the Galactic potential. The cluster is presently near the apocenter but has repeatedly undergone disk crossings in the inner part of the Galaxy leading to strong tidal shocks. Our results suggest that the observed debris originates mostly from mass loss within the last 2 Gyrs. The cluster is likely to be destroyed after the next disk crossing, which will happen in about 100 Myr. (abridged)Comment: 44 pages, including 14 figures (Figs.1,3 & 14 with decreased resolution), accepted for publication in the Astronomical Journa