To the theory of optimal splines

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Abstract

On June 18, 2008 at the Plenary Meeting of the International Conference “Differential Equations and Topology” dedicated to the 100th anniversary of Pontryagin, the report [1] was submitted by Isaev and Leitmann. This report in a summary form included a section dedicated to the research of scientists of TsAGI in the field of automation of full life-cycle (i.e. engineering-design-manufacturing, or CAE/CAD/CAM, or CALS-technologies) of wind tunnel models [2]. Within this framework, methods of geometric modeling [3], [4] were intensively developed, new classes of optimal splines have been built, including the Pontryagin splines and the Chebyshev splines [5], [6], [7], [8]. This paper reviews some results on the Pontryagin splines. We also give some results on the Lurie splines, that arise in the problem of interpolation of a cylindrical type surface given by the family of table coplanar planes.

Introduction

If we use the Pontryagin maximum principle [9], [10] and tools of differential geometry [11], [12], [13] for constructing a mathematical model of the controlled interpolation process, we will get [5], [6], [7] a formulation and methods of solution of a new problem in the computational geometry – the problem of the controlled Hermitian interpolation. The possibility of optimal controlling the local behavior of functions (curves and surfaces) distinguishes this statement from the classical problems of approximation, interpolation and smoothing [14], [15], [16], [17], [18]. As a result, we have a new class of functions: P-class of Pontryagin splines, or P-splines. P-splines possess natural parametrization and internal extremal properties. Some subclasses of Pontryagin splines will be presented: Pκ0,Pκ0,Pκ2,Pκ2, arising up at consideration of different controls and series of functionals.

Section snippets

General statement of the optimization problem and choice of the natural method

We have the 2D Cartesian system of coordinates Oxy and a finite ordered set of points:{xj,yj};j=0,1,,n;orΔx={x0,x1,,xn};Δy={y0,y1,,yn}.It is required to construct a flat curve passing through all these points in the increasing order of their subscripts (1).

A plane curve is an one-dimensional manifold, it can be defined as a function of scalar argument by one of the following methods:

  • (1)

    The coordinate method (one coordinate is defined as a function of the other one):y=y(x)orx=x(y);It is assumed

Pontryagin splines in R3 (3D P-splines)

Let a set of points be given in the three-dimensional Cartesian space:{xj,yj,zj};j=0,1,,N,orΔx=(x0,x1,,xN);Δy=(y0,y1,,yN);Δz=(z0,z1,,zN).It is required to draw a curve r = r(s) passing through these points.

We consider curves with the continuous piecewise-smooth curvature.

Classification of P-splines

Properties of P-splines (and primarily, the structure of the optimal control) are determined by the following factors:

  • the dimension n of the Euclidean (physical) space En;

  • the type of the control u;

  • the form of the functional (criterion of optimization);

  • constraints on the control (u  U) and phase coordinates x  X.

Since between the type of control u and the functional there may be some connections, a sketch of the classification of P-splines was outlined in [7]. In this classification, the decisive

General statement of the optimization problem and basic restrictions

Let the initial information be given in the form of an ordered set of lines Γ0, Γ1,  , ΓN, and possibly some other boundary conditions. These lines represent the arcs of simple plane curves lying in parallel planes. Beginnings and ends of these curves are connected by space curves l1 and l2; (see Fig. 2).

It is required to construct a surface S passing through all the lines l1, l2, Γi, i = 0, 1,  , N, and, besides, the lines l1, l2, Γ0(x = x0) and ΓN(x = xN) should be edges. We assume that the normal to the

Conclusion

The article describes the methods for solving problems of optimal Hermitian interpolation of curves and surfaces. The proposed approach is characterized by the consideration of processes of interpolation from the standpoint of controlled dynamic systems, using the theory of optimal processes and internal (invariant) properties of geometric objects.

For the mathematical description of planar and spatial curves, one of the most convenient is a combination of methods of differential geometry and

Acknowledgements

The author expresses his gratitude to his colleagues A.I. Malakhov, S.A. Plotnikov and V.V. Sonin for their important participation in the development of methods of optimal splines.

The author is also warmly thankful to V.F. Demyanov (the St.-Petersburg State University) for his remarks on the English translation of this paper.

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