Research Article The Representation of Circular Arc by Using Rational Cubic Timmer Curve Muhammad Abbas, 1,2 Norhidayah Ramli, 2 Ahmad Abd. Majid, 2 and Jamaludin Md. Ali 2 1 Department of Mathematics, University of Sargodha, 40100 Sargodha, Pakistan 2 School of Mathematical Sciences, Universiti Sains Malaysia, 11800 Penang, Malaysia Correspondence should be addressed to Muhammad Abbas; m.abbas@uos.edu.pk Received 24 June 2013; Accepted 12 December 2013; Published 16 January 2014 Academic Editor: Metin O. Kaya Copyright © 2014 Muhammad Abbas et al. his is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In CAD/CAM systems, rational polynomials, in particular the B´ ezier or NURBS forms, are useful to approximate the circular arcs. In this paper, a new representation method by means of rational cubic Timmer (RCT) curves is proposed to efectively represent a circular arc. he turning angle of a rational cubic B´ ezier and rational cubic Ball circular arcs without negative weight is still not more than 4/3 and , respectively. he turning angle of proposed approach is more than B´ ezier and Ball circular arcs with easier calculation and determination of control points. he proposed method also provides the easier modiication in the shape of circular arc showing in several numerical examples. 1. Introduction he study of curves plays a signiicant role in Computer Aided Geometric Design (CAGD) and Computer Graphics (CG) in particular parametric forms because it is easy to model curves interactively [1]. CAGD copes with the representation of free form curves. In parametric form, it is important which basis functions are used to represent the circular arcs. Circular arcs are extensively used in the ields of CAGD and CAD/CAM systems, since circular arcs can be represented by parametric (rational) polynomials instead of polynomials in explicit form. Faux and Pratt [2] represented only an elliptic segment whose turning angle is less than  by a rational quadratic B´ ezier curve. Generally, a rational cubic B´ ezier is used to extend the turning angle for conics. he maximum turning angle of a rational cubic circular arc is not more than 4/3 [3]. Only the negative weight conditions can extend its expressing range to 2 (not equal to 2) and such conditions are not cooperative in CAD systems because they lose the convex hull property [4]. A rational quartic B´ ezier curve can express any circular arc whose central angle is less than 2, and it requires at least rational B´ ezier curve of degree ive to represent a full circle without resorting to negative weights [5]. Fang [6] presented a special representation for conic sections by a rational quartic B´ ezier curve which has the same weight for all control points but the middle one. G.-J. Wang and G.-Z. Wang [7] presented the necessary and suicient conditions for the rational cubic B´ ezier representation of conics by applying coordinate transformation and parameter transformation. Hu and Wang [8] derived the necessary and suicient conditions on control points and weights for the rational quartic B´ ezier representation of conics by using two specials kinds, degree reducible and improperly parameterization. Usually rational cubic and rational quartic B´ ezier curves are used for representation of conic sections or circular arcs representation which can also be found in several papers [9–12]. Hu and Wang [13] constructed necessary and suicient conditions for conic representation in rational low degree B´ ezier form and the transformation formula from Bernstein basis to Said-Ball basis. In this work, we present a rational cubic representation of a circular arc using Timmer basis functions. It is able to approximate a circular arc up to 2 (but not including 2) without resorting to negative weights in contrast of rational cubic B´ ezier and rational cubic Ball circular arcs Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2014, Article ID 408492, 6 pages http://dx.doi.org/10.1155/2014/408492