From approximating to interpolatory non-stationary subdivision schemes with the same generation properties

In this paper we describe a general, computationally feasible strategy to deduce a family of interpolatory non-stationary subdivision schemes from a symmetric non-stationary, non-interpolatory one satisfying quite mild assumptions. To achieve this result we extend our previous work [C.Conti, L.Gemignani, L.Romani, Linear Algebra Appl. 431 (2009), no. 10, 1971-1987] to full generality by removing additional assumptions on the input symbols. For the so obtained interpolatory schemes we prove that they are capable of reproducing the same exponential polynomial space as the one generated by the original approximating scheme. Moreover, we specialize the computational methods for the case of symbols obtained by shifted non-stationary affine combinations of exponential B-splines, that are at the basis of most non-stationary subdivision schemes. In this case we find that the associated family of interpolatory symbols can be determined to satisfy a suitable set of generalized interpolating conditions at the set of the zeros (with reversed signs) of the input symbol. Finally, we discuss some computational examples by showing that the proposed approach can yield novel smooth non-stationary interpolatory subdivision schemes possessing very interesting reproduction properties

A tension approach to controlling the shape of cubic spline surfaces on FVS triangulations

We propose a parametric tensioned version of the FVS macro-element to control the shape of the composite surface and remove artificial oscillations, bumps and other undesired behaviour. In particular, this approach is applied to C1 cubic spline surfaces over a four-directional mesh produced by two-stage scattered data fitting methods

Polynomial cubic splines with tension properties

In this paper we present a new class of spline functions with tension properties. These splines are composed by polynomial cubic pieces and therefore are conformal to the standard, NURBS based CAD/CAM systems

Multidimensional Trajectories Generation with Vibration Suppression Capabilities: the Role of Exponential B-splines

In this paper, exponential B-spline trajectories are presented and discussed. They are generated by means of a chain of filters characterized by a truncated exponential impulse response. If properly tuned, the filters applied to a vibrating plant are able to cancel the oscillations and in this sense the resulting splines are optimized with respect to the problem of vibrations suppression. Different types of exponential B-spline are illustrated, with one or more exponential filters in the chain, and the procedure for the interpolation of a given set of desired via-points, with a proper choice of the control points, is shown. As a matter of fact, exponential B-splines, generated by means of dynamic filters, combine the vibration suppression capability of input shapers and smoothing filters with the possibility of exactly interpolating some via-points. The advantages of these curves are experimental proved by considering the motion of a spherical pendulum connected to the flange of an industrial robot

Isogeometric Analysis in advection-diffusion problems: tension splines approximation

We present a novel approach, within the new paradigm of isogeometric analysis introduced by Hughes et al., to deal with advection dominated advection-diffusion problems. The key ingredient is the use of Galerkin approximating spaces of functions with high smoothness, as in IgA based on classical B-splines, but particularly well suited to describe sharp layers involving very strong gradients

Shape-preserving algorithms for curve and surface design

PhD ThesisThis thesis investigates, develops and implements algorithms for shape- preserving curve and surface design that aim to reflect the shape characteristics of the underlying geometry by achieving a visually pleasing interpolant to a set of data points in one or two dimensions. All considered algorithms are local and useful in computer graphics applications. The thesis begins with an introduction to existing methods which attempt to solve the shape-preserving 1 curve interpolation problem using C cubic and quadratic splines. Next, a new generalized slope estimation method involving a parameter t, which is used to control the size of the estimated slope and, in turn, produces a more visually pleasing shape of the resulting curve, is proposed. Based on this slope generation formula, new automatic and interactive algorithms for constructing 1 shape-preserving curves from C quadratic and cubic splines are developed and demonstrated on a number of data sets. The results of these numerical experiments are also presented. Finally, a method suggested by Roulier which 1 generates C surfaces interpolating arbitrary sets of convex data on rectangular grids is considered in detail and modified to achieve more visually pleasing surfaces. Some numerical examples are given to demonstrate the performance of the method.Ministry of Education, Government of Pakista

From discretization to regularization of composite discontinuous functions

Discontinuities between distinct regions, described by different equation sets, cause difficulties for PDE/ODE solvers. We present a new algorithm that eliminates integrator discontinuities through regularizing discontinuities. First, the algorithm determines the optimum switch point between two functions spanning adjacent or overlapping domains. The optimum switch point is determined by searching for a “jump point” that minimizes a discontinuity between adjacent/overlapping functions. Then, discontinuity is resolved using an interpolating polynomial that joins the two discontinuous functions. This approach eliminates the need for conventional integrators to either discretize and then link discontinuities through generating interpolating polynomials based on state variables or to reinitialize state variables when discontinuities are detected in an ODE/DAE system. In contrast to conventional approaches that handle discontinuities at the state variable level only, the new approach tackles discontinuity at both state variable and the constitutive equations level. Thus, this approach eliminates errors associated with interpolating polynomials generated at a state variable level for discontinuities occurring in the constitutive equations. Computer memory space requirements for this approach exponentially increase with the dimension of the discontinuous function hence there will be limitations for functions with relatively high dimensions. Memory availability continues to increase with price decreasing so this is not expected to be a major limitation

Comparison and Evaluation of Didactic Methods in Numerical Analysis for the Teaching of Cubic Spline Interpolation

In mathematical education it is crucial to have a good teaching plan and to execute it correctly. In particular, this is true in the field of numerical analysis. Every teacher has a different style of teaching. This thesis studies how the basic material of a particular topic in numerical analysis was developed in four different textbooks. We compare and evaluate this process in order to achieve a good teaching strategy. The topic we chose for this research is cubic spline interpolation. Although this topic is a basic one in numerical analysis it may be complicated for students to understand. The aim of the thesis is to analyze the effectiveness of different approaches of teaching cubic spline interpolation and then use this insight to write our own chapter. We intend to channel every-day thinking into a more technical/practical presentation of a topic in numerical analysis. The didactic methodology that we use here can be extended to cover other topics in numerical analysis.Methods of teaching mathematics are different for several reasons, for example, the presentation style of teacher of a particular topic. In several books we can observe a different approach of presentation material of a topic, and at the end we can produce a unique way of teaching but in a different way. In our thesis we study different approaches to teaching in a several numerical analysis books in the topic of cubic spline interpolation. What is cubic spline interpolation? Cubic spline interpolation is a type of interpolation of data points. Interpolation is a method of constructing a curve between some data points. We chose cubic spline interpolation because it is better than other kinds of interpolation. Cubic spline interpolation has a smaller curvature compared with other types of interpolation. Therefore, cubic spline interpolation produces a smooth curve. In this research we study different approaches of teaching cubic spline interpolation to find a good way for presenting the cubic spline interpolation topic, because this topic may be complicated for students to understand. To reach a good process of presentation of cubic spline interpolation we compare each part of different approaches in the books we have studied for teaching cubic spline interpolation by asking questions and then answering those questions. In this way we will show how we can evaluate each answer. Evaluating each answer we will obtain a good result which will prepare us for writing our own chapter in order to present cubic spline interpolation in our way