Control point based exact description of curves and surfaces in extended Chebyshev spaces

Extended Chebyshev spaces that also comprise the constants represent large families of functions that can be used in real-life modeling or engineering applications that also involve important (e.g. transcendental) integral or rational curves and surfaces. Concerning computer aided geometric design, the unique normalized B-bases of such vector spaces ensure optimal shape preserving properties, important evaluation or subdivision algorithms and useful shape parameters. Therefore, we propose global explicit formulas for the entries of those transformation matrices that map these normalized B-bases to the traditional (or ordinary) bases of the underlying vector spaces. Then, we also describe general and ready to use control point configurations for the exact representation of those traditional integral parametric curves and (hybrid) surfaces that are specified by coordinate functions given as (products of separable) linear combinations of ordinary basis functions. The obtained results are also extended to the control point and weight based exact description of the rational counterpart of these integral parametric curves and surfaces. The universal applicability of our methods is presented through polynomial, trigonometric, hyperbolic or mixed extended Chebyshev vector spaces.Comment: 24 pages, 7 figures, 2 appendices, 2 listings (some new materials have been added

Control point based exact description of higher dimensional trigonometric and hyperbolic curves and multivariate surfaces

Using the normalized B-bases of vector spaces of trigonometric and hyperbolic polynomials of finite order, we specify control point configurations for the exact description of higher dimensional (rational) curves and (hybrid) multivariate surfaces determined by coordinate functions that are exclusively given either by traditional trigonometric or hyperbolic polynomials in each of their variables. The usefulness and applicability of theoretical results and proposed algorithms are illustrated by many examples that also comprise the control point based exact description of several famous curves (like epi- and hypocycloids, foliums, torus knots, Bernoulli's lemniscate, hyperbolas), surfaces (such as pure trigonometric or hybrid surfaces of revolution like tori and hyperboloids, respectively) and 3-dimensional volumes. The core of the proposed modeling methods relies on basis transformation matrices with entries that can be efficiently obtained by order elevation. Providing subdivision formulae for curves described by convex combinations of these normalized B-basis functions and control points, we also ensure the possible incorporation of all proposed techniques into today's CAD systems.Comment: 25 pages, 16 figure

An OpenGL and C++ based function library for curve and surface modeling in a large class of extended Chebyshev spaces

We propose a platform-independent multi-threaded function library that provides data structures to generate, differentiate and render both the ordinary basis and the normalized B-basis of a user-specified extended Chebyshev (EC) space that comprises the constants and can be identified with the solution space of a constant-coefficient homogeneous linear differential equation defined on a sufficiently small interval. Using the obtained normalized B-bases, our library can also generate, (partially) differentiate, modify and visualize a large family of so-called B-curves and tensor product B-surfaces. Moreover, the library also implements methods that can be used to perform dimension elevation, to subdivide B-curves and B-surfaces by means of de Casteljau-like B-algorithms, and to generate basis transformations for the B-representation of arbitrary integral curves and surfaces that are described in traditional parametric form by means of the ordinary bases of the underlying EC spaces. Independently of the algebraic, exponential, trigonometric or mixed type of the applied EC space, the proposed library is numerically stable and efficient up to a reasonable dimension number and may be useful for academics and engineers in the fields of Approximation Theory, Computer Aided Geometric Design, Computer Graphics, Isogeometric and Numerical Analysis.Comment: 29 pages, 20 figures, 2 tables, additional references have been included, some cross-references have been update

Nielson-type transfinite triangular interpolants by means of quadratic energy functional optimizations

We generalize the transfinite triangular interpolant of (Nielson, 1987) in order to generate visually smooth (not necessarily polynomial) local interpolating quasi-optimal triangular spline surfaces. Given as input a triangular mesh stored in a half-edge data structure, at first we produce a local interpolating network of curves by optimizing quadratic energy functionals described along the arcs as weighted combinations of squared length variations of first and higher order derivatives, then by optimizing weighted combinations of first and higher order quadratic thin-plate-spline-like energies we locally interpolate each curvilinear face of the previous curve network with triangular patches that are usually only $C^0$ continuous along their common boundaries. In a following step, these local interpolating optimal triangular surface patches are used to construct quasi-optimal continuous vector fields of averaged unit normals along the joints, and finally we extend the $G^1$ continuous transfinite triangular interpolation scheme of (Nielson, 1987) by imposing further optimality constraints concerning the isoparametric lines of those groups of three side-vertex interpolants that have to be convexly blended in order to generate the final visually smooth local interpolating quasi-optimal triangular spline surface. While we describe the problem in a general context, we present examples in special polynomial, trigonometric, hyperbolic and algebraic-trigonometric vector spaces of functions that may be useful both in computer-aided geometric design and in computer graphics.Comment: 17 + 7 = 24 pages (main body text + 2 appendices listing useful univariate and double integrals), 11 figure

Hyperbolic Cross Approximation

Hyperbolic cross approximation is a special type of multivariate approximation. Recently, driven by applications in engineering, biology, medicine and other areas of science new challenging problems have appeared. The common feature of these problems is high dimensions. We present here a survey on classical methods developed in multivariate approximation theory, which are known to work very well for moderate dimensions and which have potential for applications in really high dimensions. The theory of hyperbolic cross approximation and related theory of functions with mixed smoothness are under detailed study for more than 50 years. It is now well understood that this theory is important both for theoretical study and for practical applications. It is also understood that both theoretical analysis and construction of practical algorithms are very difficult problems. This explains why many fundamental problems in this area are still unsolved. Only a few survey papers and monographs on the topic are published. This and recently discovered deep connections between the hyperbolic cross approximation (and related sparse grids) and other areas of mathematics such as probability, discrepancy, and numerical integration motivated us to write this survey. We try to put emphases on the development of ideas and methods rather than list all the known results in the area. We formulate many problems, which, to our knowledge, are open problems. We also include some very recent results on the topic, which sometimes highlight new interesting directions of research. We hope that this survey will stimulate further active research in this fascinating and challenging area of approximation theory and numerical analysis.Comment: 185 pages, 24 figure

Critical length: an alternative approach

We provide a numerical method to determine the critical lengths of linear differential operators with constant real coefficients. The need for such a procedure arises when the orders increase. The interest of this article is clearly on the practical side since knowing the critical lengths permits an optimal use of the associated kernels. The efficiency of the procedure is due to its being based on crucial features of Extended Chebyshev spaces on closed bounded intervals

Multivariate approximation by translates of the Korobov function on Smolyak grids

For a set \mathbb{W} \subset L_p(\bT^d), $1 < p < \infty$, of multivariate periodic functions on the torus \bT^d and a given function \varphi \in L_p(\bT^d), we study the approximation in the L_p(\bT^d)-norm of functions $f \in \mathbb{W}$ by arbitrary linear combinations of $n$ translates of $\varphi$. For \mathbb{W} = U^r_p(\bT^d) and $\varphi = \kappa_{r,d}$, we prove upper bounds of the worst case error of this approximation where U^r_p(\bT^d) is the unit ball in the Korobov space K^r_p(\bT^d) and $\kappa_{r,d}$ is the associated Korobov function. To obtain the upper bounds, we construct approximation methods based on sparse Smolyak grids. The case $p=2, \ r > 1/2$, is especially important since K^r_2(\bT^d) is a reproducing kernel Hilbert space, whose reproducing kernel is a translation kernel determined by $\kappa_{r,d}$. We also provide lower bounds of the optimal approximation on the best choice of $\varphi$

Dynamic evaluation of exponential polynomial curves and surfaces via basis transformation

It is shown in "SIAM J. Sci. Comput. 39 (2017):B424-B441" that free-form curves used in computer aided geometric design can usually be represented as the solutions of linear differential systems and points and derivatives on the curves can be evaluated dynamically by solving the differential systems numerically. In this paper we present an even more robust and efficient algorithm for dynamic evaluation of exponential polynomial curves and surfaces. Based on properties that spaces spanned by general exponential polynomials are translation invariant and polynomial spaces are invariant with respect to a linear transform of the parameter, the transformation matrices between bases with or without translated or linearly transformed parameters are explicitly computed. Points on curves or surfaces with equal or changing parameter steps can then be evaluated dynamically from a start point using a pre-computed matrix. Like former dynamic evaluation algorithms, the newly proposed approach needs only arithmetic operations for evaluating exponential polynomial curves and surfaces. Unlike conventional numerical methods that solve a linear differential system, the new method can give robust and accurate evaluation results for any chosen parameter steps. Basis transformation technique also enables dynamic evaluation of polynomial curves with changing parameter steps using a constant matrix, which reduces time costs significantly than computing each point individually by classical algorithms.Comment: 18 pages, 4 figure

Sparse approximation by greedy algorithms

It is a survey on recent results in constructive sparse approximation. Three directions are discussed here: (1) Lebesgue-type inequalities for greedy algorithms with respect to a special class of dictionaries, (2) constructive sparse approximation with respect to the trigonometric system, (3) sparse approximation with respect to dictionaries with tensor product structure. In all three cases constructive ways are provided for sparse approximation. The technique used is based on fundamental results from the theory of greedy approximation. In particular, results in the direction (1) are based on deep methods developed recently in compressed sensing. We present some of these results with detailed proofs.Comment: arXiv admin note: substantial text overlap with arXiv:1303.6811, arXiv:1303.359

Effect of Position-dependent Mass on Dynamical Breaking of Type B and Type X_2 N-fold Supersymmetry

We investigate effect of position-dependent mass profiles on dynamical breaking of N-fold supersymmetry in several type B and type X_2 models. We find that N-fold supersymmetry in rational potentials in the constant-mass background are steady against the variation of mass profiles. On the other hand, some physically relevant mass profiles can change the pattern of dynamical N-fold supersymmetry breaking in trigonometric, hyperbolic, and exponential potentials of both type B and type X_2. The latter results open the possibility of detecting experimentally phase transition of N-fold as well as ordinary supersymmetry at a realistic energy scale.Comment: 24 pages, no figures, to appear in Journal of Physics