Bézier Curves With Modification Parameters Approximation To Real Plane Algebraic Curves

parametric curves have important applications in CAGD. Therefore , the conversion between the two curves has been paid more attention. We can always convert a parameter curve into an algebraic curve, but most of the algebraic curve can not be accurately parameterized. Therefore , the problem of approximation parametric representation of algebraic curves is an important issue in CAGD.The approximation parameteric representation of algebraic curves means that parametrization may be achieved with a small number of curves, and we try to find high accuracy of approximation.To solve the above problems, we can select two endpoints of curves, and combine with the algorithm makes the parametric curves to be an optimal approximation of algebraic curves. The paper discusses the methods of determining modification of Bézier curves, such that we can globally or locally adjust the curve shape.we introduce the conception Bézier curves with two modification parameters, and give an extension of Bézier curves with modification parameters. We convert a quardratic Bézier curve with two modification parameters into a cubic Bézier curve. Then we use the method of parallelogram to determine the parameters, such that the cubic Bézier curve given an optimal approximation to algebraic curves. It is very important that the approximation error of parametric curves to algebraic curves, and the approximation error rely on the selection of parameters. The method of parallelogram is proposed to determine the parameters. We get the approximation parametric representaion of real quadratic algebraic curves. Futhermore, a error bound and numrical examples are provided. In addition, we also extend our method to approximate other algebraic curves.