Rational B¨¦zier Curve Represents The Circular And Cylindrical Helix

As one of international industrial standards of CAD(Computer Aided Design) systems, the rational Bezier curves have wide applications in CAGD(Computer Aided Geometric Design), CG (Computer Graphics) and GM (Geometric Modeling ).The arc curves play a very important role in CAGD and CG. This paper discusses the representation of the circle and the helix using the rational Bezier curves.Firstly, the paper introduces a brief concise concepts and the development of the Bezier curves and the rational Bezier curves. In the following sections, the paper presents the polynomial Bezier curves and B-spline curves can not represent circles exactly, the rational Bezier curves are introduced, which increase the flexibility of the curves by introducing weights. Because the helix is an irrational curve , the rational Bezier curves can not represent it exactly. The helix is a unique curve, since its curvature and torsion are constant, the helix is the only spatial curve with this property. There are only two plane curves share this property, the circle and the straight line. A practical consequence of this property is that the helix can be moved on to itself, i.e. by applying the appropriate helix motion to any arc of the curve, the moving arc remains on the original helix at any of its positions. This property can be utilized in the approximation of a helix, since, if we can approximate any arc of the helix, the approximating arc can be used repeatedly. In this paper, we propose a new method to approximates the helix, which decreases the error greatly.There are many papers addressing the problem of using rational Be zier curves to represent circular arcs. It is proved that a quadratic rational Bezier curve can not represent an arc with an angle being greater than 180 degree , a rational cubic Bezier can represent a circular arc with an angle being less than 240 degree, while a quartic rational Bezier curve can represents every arc but it can not represents a full circle . This paper discusses the problem that a full circle is represented by a rational quintic Bezier curve, then the necessary conditions for the representation are given. Based on the simplifying of these necessary conditions, the relationships between the six control points and the weights are given. The discussion on the distribution of the parameteris done by extending the parameter space from [0,1] to (-00, +00). Finally,how to choose the weight to make the curve have the ideal parametric distribution is discussed.On the discussion of the approximation of a cylindrical helix by the rational Bezier curves, two methods are addressed, one which guarantees exact slopes at the end points of the approximating curve, and another increases two overlap points. Then we discuss a new method, which decreases the error greatly.