Computing a Compact Spline Representation of the Medial Axis Transform of a 2D Shape

We present a full pipeline for computing the medial axis transform of an arbitrary 2D shape. The instability of the medial axis transform is overcome by a pruning algorithm guided by a user-defined Hausdorff distance threshold. The stable medial axis transform is then approximated by spline curves in 3D to produce a smooth and compact representation. These spline curves are computed by minimizing the approximation error between the input shape and the shape represented by the medial axis transform. Our results on various 2D shapes suggest that our method is practical and effective, and yields faithful and compact representations of medial axis transforms of 2D shapes.Comment: GMP14 (Geometric Modeling and Processing

Deconvolution, differentiation and Fourier transformation algorithms for noise-containing data based on splines and global approximation

One of the main problems in the analysis of measured spectra is how to reduce the influence of noise in data processing. We show a deconvolution, a differentiation and a Fourier Transform algorithm that can be run on a small computer (64 K RAM) and suffer less from noise than commonly used routines. This objective is achieved by implementing spline based functions in mathematical operations to obtain global approximation properties in our routines. The convenient behaviour and the pleasant mathematical character of splines makes it possible to perform these mathematical operations on large data input in a limited computing time on a small computer system. Comparison is made with widely used routines

Smooth quasi-developable surfaces bounded by smooth curves

Computing a quasi-developable strip surface bounded by design curves finds wide industrial applications. Existing methods compute discrete surfaces composed of developable lines connecting sampling points on input curves which are not adequate for generating smooth quasi-developable surfaces. We propose the first method which is capable of exploring the full solution space of continuous input curves to compute a smooth quasi-developable ruled surface with as large developability as possible. The resulting surface is exactly bounded by the input smooth curves and is guaranteed to have no self-intersections. The main contribution is a variational approach to compute a continuous mapping of parameters of input curves by minimizing a function evaluating surface developability. Moreover, we also present an algorithm to represent a resulting surface as a B-spline surface when input curves are B-spline curves.Comment: 18 page

Application of B-splines to determining eigen-spectrum of Feshbach molecules

The B-spline basis set method is applied to determining the rovibrational eigen-spectrum of diatomic molecules. A particular attention is paid to a challenging numerical task of an accurate and efficient description of the vibrational levels near the dissociation limit (halo-state and Feshbach molecules). Advantages of using B-splines are highlighted by comparing the performance of the method with that of the commonly-used discrete variable representation (DVR) approach. Several model cases, including the Morse potential and realistic potentials with 1/R^3 and 1/R^6 long-range dependence of the internuclear separation are studied. We find that the B-spline method is superior to the DVR approach and it is robust enough to properly describe the Feshbach molecules. The developed numerical method is applied to studying the universal relation of the energy of the last bound state to the scattering length. We numerically illustrate the validity of the quantum-defect-theoretic formulation of such a relation for a 1/R^6 potential.Comment: submitted to can j phys: Walter Johnson symposu

Local RBF approximation for scattered data fitting with bivariate splines

In this paper we continue our earlier research [4] aimed at developing effcient methods of local approximation suitable for the first stage of a spline based two-stage scattered data fitting algorithm. As an improvement to the pure polynomial local approximation method used in [5], a hybrid polynomial/radial basis scheme was considered in [4], where the local knot locations for the RBF terms were selected using a greedy knot insertion algorithm. In this paper standard radial local approximations based on interpolation or least squares are considered and a faster procedure is used for knot selection, signicantly reducing the computational cost of the method. Error analysis of the method and numerical results illustrating its performance are given

A new code for orbit analysis and Schwarzschild modelling of triaxial stellar systems

We review the methods used to study the orbital structure and chaotic properties of various galactic models and to construct self-consistent equilibrium solutions by Schwarzschild's orbit superposition technique. These methods are implemented in a new publicly available software tool, SMILE, which is intended to be a convenient and interactive instrument for studying a variety of 2D and 3D models, including arbitrary potentials represented by a basis-set expansion, a spherical-harmonic expansion with coefficients being smooth functions of radius (splines), or a set of fixed point masses. We also propose two new variants of Schwarzschild modelling, in which the density of each orbit is represented by the coefficients of the basis-set or spline spherical-harmonic expansion, and the orbit weights are assigned in such a way as to reproduce the coefficients of the underlying density model. We explore the accuracy of these general-purpose potential expansions and show that they may be efficiently used to approximate a wide range of analytic density models and serve as smooth representations of discrete particle sets (e.g. snapshots from an N-body simulation), for instance, for the purpose of orbit analysis of the snapshot. For the variants of Schwarzschild modelling, we use two test cases - a triaxial Dehnen model containing a central black hole, and a model re-created from an N-body snapshot obtained by a cold collapse. These tests demonstrate that all modelling approaches are capable of creating equilibrium models.Comment: MNRAS, 24 pages, 18 figures. Software is available at http://td.lpi.ru/~eugvas/smile

Fast B-spline Curve Fitting by L-BFGS

We propose a novel method for fitting planar B-spline curves to unorganized data points. In traditional methods, optimization of control points and foot points are performed in two very time-consuming steps in each iteration: 1) control points are updated by setting up and solving a linear system of equations; and 2) foot points are computed by projecting each data point onto a B-spline curve. Our method uses the L-BFGS optimization method to optimize control points and foot points simultaneously and therefore it does not need to perform either matrix computation or foot point projection in every iteration. As a result, our method is much faster than existing methods