A General Transformation Framework from 3-Point Relaxed Binary Subdivision Schemes to 4-Point Relaxed Quaternary Subdivision Schemes for Efficient Curve Modeling

Subdivision schemes are widely used in computer-aided geometric design and computer graphics for generating smooth curves from discrete control polygons. This paper presents a general transformation framework for constructing the corresponding 4-point relaxed quaternary subdivision schemes from 3-point relaxed binary subdivision schemes. The proposed framework requires only the subdivision mask of the binary scheme and is applicable to both parametric and non-parametric subdivision schemes. The resulting quaternary schemes preserve the fundamental properties of their binary counterparts, including continuity, polynomial generation, polynomial reproduction, and shape-preserving characteristics. In many cases, they also possess an enlarged continuity parameter range and increased Hölder regularity. Several examples are presented to demonstrate the applicability of the proposed framework and to illustrate the relationship between the binary subdivision schemes and their corresponding quaternary subdivision schemes. The graphical results show that the transformed quaternary schemes preserve the geometric behavior of their binary counterparts while producing smoother limit curves with fewer subdivision iterations. The proposed framework provides a systematic and efficient approach for establishing the connection between binary and quaternary subdivision schemes and for constructing quaternary schemes directly from existing binary schemes.