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The Representation of Circular Arc by Using Rational Cubic Timmer Curve

DOI: 10.1155/2014/408492

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Abstract:

In CAD/CAM systems, rational polynomials, in particular the Bézier 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 effectively represent a circular arc. The turning angle of a rational cubic Bézier and rational cubic Ball circular arcs without negative weight is still not more than and , respectively. The turning angle of proposed approach is more than Bézier and Ball circular arcs with easier calculation and determination of control points. The proposed method also provides the easier modification in the shape of circular arc showing in several numerical examples. 1. Introduction The study of curves plays a significant 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 fields 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 Bézier curve. Generally, a rational cubic Bézier is used to extend the turning angle for conics. The maximum turning angle of a rational cubic circular arc is not more than [3]. Only the negative weight conditions can extend its expressing range to (not equal to ) and such conditions are not cooperative in CAD systems because they lose the convex hull property [4]. A rational quartic Bézier curve can express any circular arc whose central angle is less than , and it requires at least rational Bézier curve of degree five to represent a full circle without resorting to negative weights [5]. Fang [6] presented a special representation for conic sections by a rational quartic Bézier 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 sufficient conditions for the rational cubic Bézier representation of conics by applying coordinate transformation and parameter transformation. Hu and Wang [8] derived the necessary and sufficient conditions on control points and weights for the rational quartic Bézier representation of conics by using two specials kinds, degree reducible and improperly parameterization. Usually rational cubic and rational

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