Based On Quadratic Algebraic Geometry Information Of The Optimal Rational Parametric Curve Endpoint

In the field of Computer Aided Geometric Design(CAGD),there are two common waysto represent curves and surfaces:parametric form and implicit form.Each of the two forms hasits own advantages and disadvantages in practical application. Parametric curves and surfacesare broadly employed in computer graphics for their advantages such as the simple structure,easy display, and independence of the coordinate system.In the process of graphical display, the parametric algorithms will directly affect the drawing of curvesegments.Theoretically speaking, the arc-length parameterization is the ideal optimal parameters. Rationalparameterization is the most common form of parameterization, but the arc-length parameterization ofalgebraic curve can not be expressed as rational parametric equations therefore many researchers turned tothe study of close to the rational parametric equations of arc-length parameterization.A part of algebraic curves or surfaces is intereated merely in geometric modeling. Engineers generallywant to get optimal parametric equation of the curve for engineering demand and attractive appearance.Based on the above issues, this paper will focus on the rational parameterization of quadratic algebraiccurve. In order to solve the problem of optimal rational parameterization,this paper proposed the optimal orclose to optimal rational parameterization formula of any specified segment on the conic curves.Thealgorithm will improve working efficiency in CNC,and will provide a foothold for theoretical study andpractical application of parametric curve.In this paper, we rewrote the equation of algebraic curve segment with the geometricinformation of both ends. The optimal or nearly optimal rational parameterization formula isdetermined according to the principle that parametric speeds at both ends are equal.Comparing with literature[8] and literature[9], the method in this paper has advantage inefficiency and easy to realize. The equation of optimal rational parameterization can beobtained directly by the information of both ends. A large number of numerical experimentaldata shows that our method has more self-adaptability and accuracy than that of literature[10],and if the parametric speed at any end reaches its maximum or minimum value, theparameterization is optimal; otherwise it’s nearly optimal rational parameterization.