Anal. Theory Appl. Vol. 27, No. 1 (2011), 10–20 DOI10.1007/s10496-011-0010-1 A UNIFIED THREE POINT APPROXIMATING SUBDIVISION SCHEME Ghulam Mustafa, Faheem Khan, Muhammad Sadia Hashmi, Muhammad Zeshan Afzal (The Islamia University of Bahawalpur, Pakistan) Received Nov. 4, 2008 c  Editorial Board of Analysis in Theory & Applications and Springer-Verlag Berlin Heidelberg 2011 Abstract. In this paper, we propose a three point approximating subdivision scheme, with three shape parameters, that unifies three different existing three point approximating schemes. Some sufficient conditions for subdivision curve C 0 to C 3 continuity and conver- gence of the scheme for generating tensor product surfaces for certain ranges of parameters by using Laurent polynomial method are discussed. The systems of curve and surface de- sign based on our scheme have been developed successfully in garment CAD especially for clothes modelling. Key words: approximating subdivision scheme, shape parameters, Laurent polynomial AMS (2010) subject classification: 65D17, 65D07, 65D05 1 Introduction In recent years, the subdivision scheme became one of the most popular methods of creating geometric object in computer aided geometric design and in animation industry. Their popular- ity is due to the facts that subdivision algorithms are easy to implement and suitable for computer applications. If the limit curve / surface approximates the initial control polygon and that after subdivision, the newly generated control points are not in the limit curve / surface, the scheme is said to be approximating. It is called interpolating if after subdivision, the control points of the original control polygon and the new generated control points are interpolated on the limit curve / surface. The important schemes for applications should allow to control the shape of the limit curve and be capable of reproducing families of curves widely used in computer graphics. A wide variety of schemes that has been proposed in the literature which posses shape parameters [2,4,5,8,9,15] are interpolating [3,11,16] , presented approximating subdivision schemes with tension ∗ Supported by the Indigenous PhD Scholarship Scheme of Higher Education Commission (HEC) Pakistan.