THREE-DIMENSIONAL FRACTAL GEOMETRY MODELING AND DIGITAL HOLOGRAPHY BASED ON R-FUNCTIONS

This paper is devoted to modern research in the field of digital modeling of complex-shaped geometric objects and the determination of their optical properties, which currently represent one of the most relevant challenges in contemporary science. In particular, the problem of realistic representation of three-dimensional objects with fractal geometry and their holographic reconstruction in a full 3D format is of significant scientific and practical importance for such fields as industry, medicine, engineering, architecture, materials science, virtual reality (VR), and digital art. Fractal structures possess a number of unique properties, including self-similarity, unlimited detail, and high spatial complexity, which makes them an effective mathematical basis for modeling natural objects such as plants, vascular systems, bone tissues, crystalline structures, and surface reliefs. At the same time, the geometric representation of fractal forms using classical methods is challenging, and their mathematical modeling requires the application of high-precision and formally rigorous techniques. At present, the mathematical description of fractal objects is often based on statistical, stochastic, or iterative algorithms. However, such approaches are generally characterized by insufficient analytical rigor and smoothness, blurred boundaries, and the lack of a holistic spatial description. In this regard, there arises a need to develop methods for modeling complex fractal forms based on strict analytical expressions, in particular using the R-functions apparatus. An additional challenging task is the reconstruction of holographic images of the modeled fractal objects, which requires high-precision optical modeling. The application of holographic technologies based on the principles of interference and diffraction makes it possible to adequately reproduce the spatial and structural features of fractal objects in a digital environment.