Topological Properties of Geometric Figures in Computer Graphics and Virtual Modeling

Geometric modeling is a cornerstone of computer graphics and virtual modeling, offering precise methods to represent, analyze, and visualize objects. By employing mathematical principles such as parametric equations, spline functions, and intersection techniques, geometric modeling constructs complex curves, surfaces, and transitions. These methods facilitate the design of intricate shapes and enhance computational efficiency in engineering, cultural heritage preservation, and virtual environments. The modeling process involves selecting geometric parameters, refining mathematical relationships, and implementing computational tools for visualization. Practical applications include designing durable components in engineering, reconstructing artifacts with high accuracy, and creating immersive 3D environments. Smooth transitions between intersecting surfaces, such as spheres and cylinders, demonstrate the utility of advanced modeling techniques in preserving both functionality and aesthetics. This paper explores the foundational principles of geometric modeling and its applications, highlighting advancements that have reduced computational loads, improved design precision, and expanded its utility across diverse fields, including healthcare, robotics, and gaming. The study emphasizes the transformative impact of geometric modeling in modern technological and cultural domains.