GEOMETRIC MODELING OF DISCRETELY DEFINED CURVES: STATE OF THE ART AND SOLUTION METHODS

Abstract

The article examines the current state of research in the field of geometric modeling of discretely defined curves and provides a review of the main approaches to solving the problems of their interpolation, approximation, and reconstruction. The relevance of the study is determined by the widespread use of discretely represented data in modern information technologies, computer graphics, computer-aided design, digital image processing, decision support systems, logistics problems, and other applied fields. In most practical applications, the initial information is presented as a set of discrete points that characterize the shape of an object, the trajectory of a process, or the results of experimental observations. In this regard, the construction of adequate mathematical models that ensure the reconstruction and analysis of geometric relationships is an important direction of modern scientific research.

The paper analyzes the main methods of mathematical representation of curves, including explicit, implicit, and parametric definitions. The geometric characteristics of curves, in particular the tangent, curvature, and arc length, which are widely used in the study of the shape of geometric objects and in the construction of interpolation models, are considered. A classification of curves according to the method of representation and the level of smoothness determined by continuity classes is presented, which is an important characteristic in the development of geometric modeling algorithms.

A comprehensive review of modern methods for modeling discretely defined curves is carried out. Interpolation, approximation, and variational approaches, as well as the features of their practical application, are considered. Methods of global polynomial interpolation, spline technologies, Bézier curves, B-splines, NURBS models, moving least squares methods, and subdivision schemes are analyzed. For each approach, its characteristic features, advantages, and limitations that should be taken into account depending on the specifics of the problem and the properties of the input data are presented.

Particular attention is paid to the problems of continuous and discrete interpolation. Mathematical formulations of the corresponding problems are presented, and the features of constructing interpolation curves using Lagrange polynomials, cubic splines, Hermite splines, and Catmull–Rom schemes are considered. Approaches to ensuring smoothness, geometric consistency, and local shape control of curves are analyzed. Special consideration is given to Chaikin's algorithm and subdivision schemes, which are used for smoothing polygonal lines and constructing smooth limit curves through successive refinement of the initial data.

Based on the conducted review, it has been established that different classes of methods possess their own advantages and application features. Global polynomial interpolation methods provide high accuracy in passing through prescribed points, spline approaches make it possible to obtain smooth curves and perform local shape control, subdivision schemes are effectively used for constructing smooth geometric models, while variational methods and moving least squares methods demonstrate good performance when dealing with irregular and noisy data. At the same time, the choice of a particular approach largely depends on the nature of the input data, as well as on the requirements for accuracy, smoothness, and computational efficiency of the obtained solution.

The main result of the work is the systematization of modern methods of geometric modeling of discretely defined curves, the generalization of their mathematical foundations, the analysis of application features, advantages, and limitations, as well as the identification of relevant directions for further research. The conducted analysis confirms the expediency of developing new approaches to geometric modeling aimed at combining high accuracy, smoothness, local adaptability, and computational efficiency in solving problems of interpolation and reconstruction of discretely represented curves.

Keywords: Mathematical modeling, numerical methods, interpolation, approximation, extrapolation, geometric modeling.