  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.
