This research uncovers a connection between binary and quaternary subdivision schemes, presenting a generalized formula to derive \((3n+1)\) -point quaternary schemes from \((2n+1)\) -point binary schemes, yielding odd-point quaternary schemes when n is even and even-point quaternary schemes when n is odd. Comprehensive graphical and theoretical analyses demonstrate that these quaternary schemes produce final models similar to their binary counterparts but with fewer iterations, significantly reducing computational costs. Both types exhibit similar shape-preserving properties, with quaternary schemes extending the parametric intervals of continuity and enhancing model smoothness. This method improves computational efficiency and broadens the applicability and flexibility of subdivision schemes, particularly in parametric contexts.
