REVIEW OF SURFACE GEOMETRIC MODELING METHODS

Abstract

The article presents a comprehensive analysis of surface geometric modeling methods, which form the foundation of modern digital design, engineering analysis, and computer graphics. Particular attention is given to the comparison of continuous and discrete approaches to the representation of geometric forms, their theoretical foundations, advantages, limitations, and areas of effective application. Classical continuous methods are considered, including B-splines, NURBS, cubic and bicubic interpolation functions, as well as radial basis functions. These approaches provide high smoothness (up to C2 continuity), analytical accuracy, and efficiency in modeling both standard and complex surfaces. The study analyzes the mathematical properties of continuous models, the specifics of constructing functional dependencies between control points, weight coefficients, and object geometry. At the same time, key limitations are outlined: difficulty of local editing, inefficiency in handling discontinuities and topological features, and significant computational load when processing dense meshes or large data sets.

A special focus is placed on the review of discrete modeling methods based on polygonal meshes, compositional approaches, adaptive discretization, and isotropic geometry. Examples of practical applications of discrete models in engineering design, visualization, digital prototyping, surface reconstruction from point clouds, and lighting design are provided. The contribution of Ukrainian researchers (Naidish, Pugachov, Botvinovska, Ausheva) is highlighted, particularly in the development of hybrid models that combine the strengths of continuous and discrete methods. The findings demonstrate that discrete methods, due to their flexibility, locality, computational efficiency, and implementation simplicity, are a promising tool for solving modern surface modeling problems—especially in high-dynamic environments requiring variable levels of detail and processing of large volumes of structured and unstructured data.

Keywords: geometric modeling, continuous methods, discrete methods, computer graphics.