Reparameterization Of Rational Bézier Curves Using Piecewise M(?)bius Transformation

In this paper,we propose a reparameterization method for rational Bézier curves of arbitrary degree.By using the M(?)bius transformation,this method can keep the degree and parametric domain of the parameterized curves unchanged,and possess the flexibility of parameterized curves at the same time.In this paper,we regard the deviation of the new parametric speed from unit-speed in the L2 norm as the objective function to describe how good the new parameter approximates the arc-length parameter.And the parameter in the M(?)bius transformation can be obtained by minimizing the objective function.In this paper,we firstly present a reparameterization method based on the M(?)bius transformation.Firstly,the M(?)bius transformation is applied to the rational Bézier curves,and then the parameter in the M(?)bius transformation is obtained by minimizing the objective function.Thus we obtain a rational Bézier curve representation approximating the arc-length parameter.Considering that not all the M(?)bius transformations can meet users' needs in practical applications,the new parameter is ideally close to the arc-length parameter.We also propose a reparameterization method based on piecewise M(?)bius transformation.This method chooses the maximum points of the curvature of the original curve as the points of segmentation to segment the original curve,and then the different M(?)bius transformations are applied to each curve.In order to make the parameterized curve have better smoothness and better approximation of arc-length parameter,we require the parameterized curve to satisfy the C1 continuity at the segment points to ensure the continuity of the parameter rate.At the same time,it is also required that the parameter values of the parameterized curve at the corresponding nodes are equal to the arc-length parameter.According to these two constraints,the rational Bézier curve representation with approximating arc-length parameter is obtained by minimizing the objective function to obtain all the unknown parameters in the piecewise M(?)bius transformation.Finally numerical examples results show that the parameter of the rational Bézier curve reparameterized by the piecewise M(?)bius transformation has a good approximation effect of arc-length parameter.