A class of splines with constrained length

2016

The actual application for the problem of best approximation of grid function by linear splines was formulated. A mathematical model and a method for its solution were developed. Complexity of the problem was that it was multi extremal and could not be solved analytically. The method was developed in order to solve the problem of dynamic programming scheme, which was extended by us. Given the application of the method to the problem of flow control in the pressure-regulating systems, the pipeline network for transport of substances (pipelines of water, oil, gas, and etc.) that minimizes the amount of substance reservoirs and reduces the discharge of substance from the system. The method and the algorithm developed here may be used in computational mathematics, optimal control and regulation system, and regressive analysis.

Proceedings of the 1986 workshop on Applied computing - SAC '86, 1986

The method applied most frequently in drawing a curve to interpolate a set of given points without spoiling the aim for "shape preserving" is parametrized cubic spline method; however, possibility of producing a bad result on "shape preserving" exists• CATIA 1 system which uses "quintic spline" is not a perfect one. We derive a "quintic spline" from a "cubic spline" and has it modified. Except for the above, we also do the work of error analysis and execute a few examples for comparison.

Numerische Mathematik, 1985

We treat the problem of approximating data that are sampled with error from a function known to be convex and increasing. The approximating function is a polynomial spline with knots at the data points. This paper presents results (analogous to those in [7] and [9]) that describe some approximation properties of polynomial splines and algorithms for determining the existence of a "shape-preserving" approximant for given data.

Computers & Mathematics with Applications, 1997

A review of shape preserving approximation methods and algorithms for approximating univariate functions or discrete data is given. The notion of 'shape' refers to the geometrical behavior of a function's or approximant's graph, and usually includes positivity, monotonicity, and/or convexity. But, in the recent literature, the broader concept of shape also includes symmetry, generalized convexity, unimodality, Lipschitz property, possessing peaks or discontinuities, etc. Special stress is put on shape preserving interpolation methods by polynomials and splines. Of course, this text has no pretensions to be complete.

Computer-Aided Design, 2007

The problem of drawing a smooth obstacle avoiding curve has attracted the attention of many people working in the area of CAD/CAM and its applications. In the present paper we propose a method of constrained curve drawing using certain C 1-quadratic trigonometric splines having shape parameters, which have been recently introduced in [Han X. Quadratic trigonometric polynomial curves with a shape parameter. Computer Aided Geometric Design 2002;19:503-12]. Besides this, we have also presented a simpler approach for studying the approximation properties of the trigonometric spline curves.

Topics in Splines and Applications

A common engineering task consists of interpolating a set of discrete points that arise from measurements and experiments. Another traditional requirement implies creating a curve that mimics a given array of points, namely, a polyline. Any of these problems require building an analytical representation of the given discrete set of points. If the geometrical shape represented by the input polyline is complicated, then we may expect that a global interpolant or polynomial will be of a high degree, to honor all imposed constraints, which makes its use prohibited. Indeed, a global interpolant often experiences inflection points and sudden changes in curvature. To avoid these drawbacks, we often seek solving the interpolation/approximation problem using piecewise polynomial functions called "splines."

In this paper we present an approximation method of curves by a new type of spline functions called fairness cubic splines from a given Lagrangian data set under fairness constraints. An approximating problem of curves is obtained by minimizing a quadratic functional in a parametric space of cubic splines. This method is justified by a convergence result and an analysis of some numerical and graphical examples.

Computer Aided Geometric Design, 2007

We introduce L p (1 p ∞) piecewise polynomial parametric splines of degree 3 or 5 which smoothly interpolate data points. The coefficients of these polynomial splines are calculated by minimizing the L p norm of the second derivative. It is demonstrated that the C 1 -continuous cubic (resp. G 1 -continuous cubic and C 2 -continuous quintic) L p polynomial splines are unique for 1 < p < ∞. The focus is mainly on L 1 polynomial splines which preserve the shape of data even with abrupt changes in direction or spacing. When there are several candidates we introduce a tension parameter which selects from the set of solutions the smallest in norm. The optimisation uses a nonlinear functional; therefore we discretize it so as to obtain a linear problem and we give an upper bound to the error. Computational examples using a primal affine (interior point) algorithm are presented. We will also show how interesting it can be to be able to change the partition associated to the data points in order to obtain convexity properties when possible. Moreover, we will show that the solution is not unvarying by rotation in the L 1 -case and we propose an alternate approach with a local change of coordinates on each segment [P i , P i+1 ]. We show that this method can be applied to any kind of norm for the minimization problem.

Ecosystems and Sustainable Development IX, 2013

The study of natural systems implies considering new modelling methodologies that are able to produce different relationships to those described with mathematical functions, which are derived from the geometry of Euclid. In this paper, the authors propose a geometric model defined by families of cubic Splines, which are the basis for a definition of a numerical methodology for the study and modelling of complex systems. Geometric models are applied in specific cases for types of relationships. One feature of the model is that the polynomials are not represented by their coefficients, because they could be highly dependent of small variations in their values, as it is analyzed in the article. The polynomials will be represented by their values at points considered in their ranges of definition, which will be called nodes. For each variable, a Spline generated from kl cubic polynomials is defined, so the first objective is the analysis of a family of Splines determined by a set of polynomials. Finally, the geometric model is determined on practical examples from experimental data and the advantages of using the new methodology, based on the identification of Splines by their values in a number of points, are discussed, compared with the usual definition from the polynomial coefficients.

Mathematics and Statistics, 2021

Being a continuation of the paper published in Mathematics and Statistics, vol. 7, No. 5, 2019, this article describes the algorithm for the first stage of spline- approximation with an unknown number of elements of the spline and constraints on its parameters. Such problems arise in the computer-aided design of road routes and other linear structures. In this article we consider the problem of a discrete sequence approximation of points on a plane by a spline consisting of line segments conjugated by circular arcs. This problem occurs when designing the longitudinal profile of new and reconstructed railways and highways. At the first stage, using a special dynamic programming algorithm, the number of elements of the spline and the approximate values of its parameters that satisfy all the constraints are determined. At the second stage, this result is used as an initial approximation for optimizing the spline parameters using a special nonlinear programming algorithm. The dynamic programming algorithm is practically the same as in the mentioned article published earlier, with significant simplifications due to the absence of clothoids when connecting straight lines and curves. The need for the second stage is due to the fact that when designing new roads, it is impossible to implement dynamic programming due to the need to take into account the relationship of spline elements in fills and in cuts, if fills will be constructed from soils of cuts. The nonlinear programming algorithm is based on constructing a basis in zero spaces of matrices of active constraints and adjusting this basis when changing the set of active constraints in an iterative process. This allows finding the direction of descent and solving the problem of excluding constraints from the active set without solving systems of linear equations in general or by solving linear systems of low dimension. As an objective function, instead of the traditionally used sum of squares of the deviations of the approximated points from the spline, the article proposes other functions, taking into account the specifics of a specific project task.