Relationship between the 2n-points binary and (3n-1)-points quaternary approximating subdivision schemes

Geometric objects are primarily represented using curves and surfaces and the subdivision schemes are the basic tools for these representations. This study is based on a new thought that there is a special relation between the binary and some kinds of the quaternary subdivision schemes. Due to the defined relation the quaternary subdivision schemes can also be formulated by the binary subdivision schemes. This study presents the generalized formula in compact form that contains the subdivision rules of (3n-1)-point quaternary approximating subdivision scheme which are based on the predefined 2n-points binary approximating subdivision scheme. Firstly, we derive a relation between the quaternary approximating subdivision scheme and the even-point binary approximating subdivision scheme. By using this relation, we next derive two types of generalized quaternary approximating subdivision scheme that are based on the even and odd values of n. Then we apply these generalized formulas on the known binary schemes for specific values of $n$. This gives us the corresponding quaternary approximating subdivision schemes. We also analyze some of the well-known features of binary and its corresponding quaternary approximating subdivision schemes. These results are equally applicable on parametric and non-parametric subdivision schemes.