A unified Pythagorean hodograph approach to the medial axis transform and offset approximation

AbstractAlgorithms based on Pythagorean hodographs (PH) in the Euclidean plane and in Minkowski space share common goals, the main one being rationality of offsets of planar domains. However, only separate interpolation techniques based on these curves can be found in the literature. It was recently revealed that rational PH curves in the Euclidean plane and in Minkowski space are very closely related. In this paper, we continue the discussion of the interplay between spatial MPH curves and their associated planar PH curves from the point of view of Hermite interpolation. On the basis of this approach we design a new, simple interpolation algorithm. The main advantage of the unifying method presented lies in the fact that it uses, after only some simple additional computations, an arbitrary algorithm for interpolation using planar PH curves also for interpolation using spatial MPH curves. We present the functionality of our method for G1 Hermite data; however, one could also obtain higher order algorithms

Measuring Planck beams with planets

Aims. Accurate measurement of the cosmic microwave background (CMB) anisotropy requires precise knowledge of the instrument beam. We explore how well the Planck beams will be determined from observations of planets, developing techniques that are also appropriate for other experiments. Methods. We simulate planet observations with a Planck-like scanning strategy, telescope beams, noise, and detector properties. Then we employ both parametric and non-parametric techniques, reconstructing beams directly from the time-ordered data. With a faithful parameterization of the beam shape, we can constrain certain detector properties, such as the time constants of the detectors, to high precision. Alternatively, we decompose the beam using an orthogonal basis. For both techniques, we characterize the errors in the beam reconstruction with Monte Carlo realizations. For a simplified scanning strategy, we study the impact on estimation of the CMB power spectrum. Finally, we explore the consequences for measuring cosmological parameters, focusing on the spectral index of primordial scalar perturbations, n_s. Results. The quality of the power spectrum measurement will be significantly influenced by the optical modeling of the telescope. In our most conservative case, using no information about the optics except the measurement of planets, we find that a single transit of Jupiter across the focal plane will measure the beam window functions to better than 0.3% for the channels at 100–217 GHz that are the most sensitive to the CMB. Constraining the beam with optical modeling can lead to much higher quality reconstruction. Conclusions. Depending on the optical modeling, the beam errors may be a significant contribution to the measurement systematics for n_s

Singular forces and point-like colloids in lattice Boltzmann hydrodynamics

We present a second-order accurate method to include arbitrary distributions of force densities in the lattice Boltzmann formulation of hydrodynamics. Our method may be used to represent singular force densities arising either from momentum-conserving internal forces or from external forces which do not conserve momentum. We validate our method with several examples involving point forces and find excellent agreement with analytical results. A minimal model for dilute sedimenting particles is presented using the method which promises a substantial gain in computational efficiency.Comment: 22 pages, 9 figures. Submitted to Phys. Rev.

Measuring nonlocal Lagrangian peak bias

We investigate nonlocal Lagrangian bias contributions involving gradients of the linear density field, for which we have predictions from the excursion set peak formalism. We begin by writing down a bias expansion which includes all the bias terms, including the nonlocal ones. Having checked that the model furnishes a reasonable fit to the halo mass function, we develop a 1-point cross-correlation technique to measure bias factors associated with 2-distributed quantities. We validate the method with numerical realizations of peaks of Gaussian random fields before we apply it to N-body simulations. We focus on the lowest (quadratic) order nonlocal contributions. We can reproduce our measurement of \chi_{10} if we allow for an offset between the Lagrangian halo center-of-mass and the peak position. The sign and magnitude of \chi_{10} is consistent with Lagrangian haloes sitting near linear density maxima. The resulting contribution to the halo bias can safely be ignored for M = 10^13 Msun/h, but could become relevant at larger halo masses. For the second nonlocal bias \chi_{01} however, we measure a much larger magnitude than predicted by our model. We speculate that some of this discrepancy might originate from nonlocal Lagrangian contributions induced by nonspherical collapse.Comment: (v2): presentation clarified. agreement with the simulation improved. accepted for publication. 11 pages, 8 figure

Dynamical models of NGC 3115

We present new dynamical models of the S0 galaxy N3115, making use of the available published photometry and kinematics as well as of two-dimensional TIGER spectrography. We first examined the kinematics in the central 40 arcsec in the light of two integral f(E,J) models. Jeans equations were used to constrain the mass to light ratio, and the central dark mass whose existence was suggested by previous studies. The even part of the distribution function was then retrieved via the Hunter & Qian formalism. We thus confirmed that the velocity and dispersion profiles in the central region could be well fit with a two-integral model, given the presence of a central dark mass of ~10^9 Msun. However, no two integral model could fit the h_3 profile around a radius of 25 arcsec where the outer disc dominates the surface brightness distribution. Three integral analytical models were therefore built using a Quadratic Programming technique. These models showed that three integral components do indeed provide a reasonable fit to the kinematics, including the higher Gauss-Hermite moments. Again, models without a central dark mass failed to reproduce the observed kinematics in the central arcseconds. This clearly supports the presence of a nuclear black hole of at least 6.5 10^8 Msun in the centre of NGC 3115. These models were finally used to estimate the importance of the dark matter in the outer part of NGC 3115, suggested by the flat stellar rotation curve observed by Capaccioli et al. (1993).Comment: 18 pages, 22 figures, accepted for publication in MNRA