Establishing a Specialized Bridge Between the Even-Point Binary and the Even/Odd-Point Quaternary Subdivision Approaches

Subdivision schemes are powerful tools for generating curves and surfaces in computer graphics. This work explores a novel connection between binary and quaternary schemes, where quaternary schemes can be derived from binary ones. We present a generalized formula for constructing
$(3n-1)$
-point quaternary schemes from existing
$2n$
-point binary schemes. This approach leads to two types of quaternary schemes based on even and odd values of n. We demonstrate the efficiency of these new schemes by applying them to known binary schemes and analyzing their properties. Our results show that the derived quaternary schemes achieve similar final models as their binary counterparts, but with fewer iterations, leading to significant computational cost reduction. This effectiveness is validated through graphical and theoretical analyses, confirming the applicability of our method to both parametric and non-parametric settings.