REVIEW OF CONTINUOUS INTERPOLATION METHODS FOR PLANAR DISCRETELY REPRESENTED CURVES

Abstract

The article presents a comprehensive review of modern continuous interpolation methods used for constructing planar curves based on discrete point sets. Particular attention is paid to polynomial interpolation methods, including classical approaches such as Lagrange and Newton interpolation, as well as their improvements using divided differences. The Hermite method is described, which ensures agreement not only of function values but also of their derivatives at interpolation nodes, allowing for improved curve smoothness. The advantages and limitations of Neville and Aitken methods, which are based on iterative approximation of function values, are analyzed. A separate section is devoted to Bézier curves, which have found wide application in computer graphics due to the convenient control of shape through control points. The benefits of B-splines are considered, as they allow local modifications and continuity of derivatives of a given order without the need to recalculate the entire curve globally. It is substantiated that despite high approximation quality, all continuous methods share common disadvantages: oscillation tendencies, dependence on the number of nodes, difficulty in local adjustment, and relatively high computational complexity. In particular, in the presence of discontinuities or geometric features (self-intersections, cusps, curvature changes), continuous curves become ineffective. In this context, the article highlights the potential of discrete interpolation methods, which ensure stability, absence of parasitic oscillations, ease of implementation, possibility of local correction, and precise modeling of curves of arbitrary complexity.

The article lays the foundation for further research in the direction of formalizing interpolation approaches that meet current demands in engineering design, digital modeling, and graphical applications.

Keywords: interpolation, continuous methods, geometric modeling, local correction.