Optimal Parameterizations of Complex Rational Bézier Curves of Degree One

To apply the complex version of the rational de Casteljau algorithm and the subdivision algorithm in a convenient manner,optimal parameterization of complex rational Bézier curves of degree one is studied.Both algebraic and geometric methods to derive the optimal parameterization are presented.The algebraic method is to deduce the curve's reparameterization under the Möbius transformations by direct algebraic computation,so that the new parameterization is the closest to the arc parameterization under the L2 norm.The geometric method is to directly deduce the optimal parameterization under the Möbius transformations by applying the intrinsic geometric properties of the complex rational Bézier curves of degree one,and then the essence of optimal parameterization is obtained.In addition,a formula for interpolating three given points by a complex rational Bézier curve of degree one is presented as an application of reparameterization.Numerical examples show that the iso-parametric points on the curve are uniform after optimal parameterization.