Modelling of curve of arbitrary topology is an important area of research because curve constitutes a basic tool for the geometric design in Computer Aided Geo- metric Design and Computer Graphics etc. The main purpose of this dissertation is to introduce new subdivision schemes because the existing one is not computa- tionally fast and the proposed one allow geometric designer to create different and high order of derivative continuous curves in CAGD which is novel. This disser- tation represents various flavours of curve subdivision schemes w.r.t. points, basis functions, continuity and smoothness, which have main concerned in geometric designing. For more attractive feature of these subdivision schemes, several tools have been discussed such as Laurent polynomial method has been used to investi- gate the smoothness of the subdivision schemes, H ̈lder exponent and support size. o Geometrical performances of the proposed subdivision schemes have been followed by explanatory examples.