On The Arc-angle Reparameterization Algorithm For Parametric Curves

Rational parameterization of curves and surfaces is an important problem in computer-aided geometric design.Parameterization with nice properties plays an important role in various specific applications.Different from the arc-length parameterization,the arc-angle parameterization can reflect how much the curve bends.Given a rational parametric curve,one can obtain the rational approximation of its arc-angle reparameterization with the help of a piecewise rational transformation with low degree.However,this transformation is only applicable to parametric curves whose angular speed does not have zeros on the unit interval.This paper studies the problem how to compute the arc-angle reparameterization for a rational parametric curve when the angular speed function has zeros in the unit interval.In this paper,an arc-angle reparameterization algorithm with radical transformation is proposed.Via such a transformation,a rational parameterization whose angular speed function has zeros can be converted into an approximate arc-angle parameterization with C~1continuity.In the first phase,the zeros on the unit interval are removed by the first reparameterization.The main idea is to introduce the constraint of continuity and use the piecewise radical transformation to obtain the C~1continuous parameterization whose angular speed has no zeros.Then one computes the C~1 reparameterization for the obtained representation with the C~1 optimal arc-angle reparameterization algorithm,resulting in an approximation of the arc-angle reparameterization of the given curve.Experiments show that for the rational parametric curves with angular speed having zeros,the algorithm cannot only improve the angular speed uniformity effectively,but also can guarantee the continuity of the angular speed function.