Mathematical Modeling-Driven Shape Digitization: A Perspective of Mongolian Motifs and Patterns

Human civilization embodies a rich cultural heritage shaped over long historical periods by numerous ethnic groups, each employing distinctive motifs and patterns in religious spaces, architecture, clothing, utensils, and other artifacts. Such motifs commonly originate from elementary geometric primitives that are organized through symmetric or asymmetric compositions to convey symbolic and esthetic meaning. This study focuses on Mongolian patterns derived from the nomadic heritage of Mongolia and still prevalent in contemporary design. These patterns draw inspiration from nature, geometry, animals, plants, and symbolic forms. This article proposes a mathematical modeling-driven digitization framework for the systematic analysis and digitization of Mongolian patterns, with the objective of generating accurate digital representations in the form of computer-aided design (CAD) models. A concise review of related work is first presented, followed by a structured digitization framework and a taxonomy of representative Mongolian motifs. A case study demonstrates that, when combined through distance-preserving and shape-preserving geometric operations such as translation, rotation, and reflection, four fundamental geometric entities, namely the circle, circular arc, spiral, and astroid, are sufficient to retain the intrinsic symmetry and compositional coherence of complex patterns observed in selected artifacts. Furthermore, the proposed analytical modeling approach enables the generation of vector-based line drawings that support precise CAD model construction. Accordingly, this study establishes a computational design workflow that integrates cultural heritage patterns into CAD-based modeling environments, thereby supporting digital preservation and fabrication with high geometric fidelity.