CLOSED SMOOTH CURVES CONSTRUCTION WITH THE GAUSSIAN INTERPOLATING POLYNOMIAL

Abstract

The paper discusses two algorithms for constructing smooth closed curves using the Gaussian interpolation function.

         In the work of an engineer, there is a constant need to construct geometric objects that have given properties. Components of these objects can be closed curves. In complex technical problems, in addition to closedness, an additional condition of smoothness at all points of the curve may arise. Therefore, the relevance of constructing such smooth closed curves is relevant.

         Gaussian interpolation polynomials, being based on exponential functions, have several advantages over algebraic interpolation polynomials. There is no oscillation that is typical for algebraic polynomials when interpolating larger data sets. Also, using the additional coefficients in the argument of each term, it is possible to control the function when solving any problem that both requires interpolation and implies some additional properties of the interpolant.

Parameterization of the interpolating Gaussian polynomial enables the construction of curves. To make curves closed, we add the condition: the point corresponding to the lowest value of the parameter must coincide with the point corresponding to the highest value. We call this point the gluing point.

         However, until now, the use of Gaussian interpolation polynomial for constructing closed curves was limited due to the fact that the methods of constructing such a polynomial did not provide control over the derivatives at the interpolation points, and therefore over the tangent lines at the points of the parametric curves and the order of smoothness at the gluing point when constructing a closed curve.

         In this paper, two fundamentally different methods of solving the problem of constructing a smooth closed curve with a Gaussian interpolation polynomial are proposed, each, however, with its own advantages and disadvantages. The first method involves setting the values of the derivatives at the points of the interpolation polynomial explicitly, and the second – the smooth closure of the curve at the point of gluing by means of mathematical optimization of the difference of the derivatives in the gluing pint without setting the derivatives explicitly.

         The purpose of the research is to improve the construction of curvilinear contours using the Gaussian interpolation function by introducing the smoothness condition.  Two different methods are proposed, and the results of their work are showcased.

         Keywords: interpolation, Gaussian interpolation curve, curve smoothness, mathematical optimization.