Research On Parameterization Methods For Data Points On Parametric Curve

In the fields of Computer Graphics and Computer-Aided Geometric Design, there are a lot of methods to define and express a curve. One of which, Parameter Spline Curve is widely used. Usually the parameters of the request structure interpolation curve must be smooth. In order to meet this requirement, not only it needs good interpolation methods, but also needs good methods to determine parameters of the data nodes. The same set of data points, even using the same interpolation method, if using different parameters methods, will receive different interpolation curve. So the parameters of the data points should reflect the nature of the interpolation curve which designers want to be.Generally speaking, the curve parameterization will make it closer to the arc length parameterization, This is because arc length parameterization are ideal parameterized method, which enables parameters uniformly distributed points on the curve corresponds to the parameters of uniform distribution, for the practical application of great significance, and now the parameters of methods, are making an effort to move closer to the arc length parameterization, but the true sense of the arc-length parameterization is also a wide gap between the ideal, but also measuring the distance has a standard, therefore, arc-length parameterization is very difficult.So far, although there are several data points of the parametric methods used in practice, how to obtain a satisfactory outcome of the parameters of this problem has still not been solved satisfactorily.At present, there are such parameters of the methods, commonly used now, uniform parameterization method, accumulated chord length parameterization method, the centripetal parameterization, adapting chord length parameterization method and the ZCM method.Uniform parameterization method is the simplest method, which considers all the parameters are uniformly increased. But in fact the distributions of data points under normal circumstances are uneven, so the use of this method of curve constructed less effective. accumulative chord length method are now widely used and accepted method parameters, in this method, the chord length was seen as the approximate length, through the iterative, accumulative chord length parameterization method has become the method of arc-length parameterization, However, only parametric curve is a straight line, the parametric curve interpolation in accordance with only the most suitable arc length.In recent years, technique based on optimization to determine the node parameters has been widely developed, experiments show that these parameters of the method of curve constructed are better, but more difficult to achieve.Centripetal parameterization, by Lee (Lee, 1989) from Boeing Company proposed, which uses accumulative chord length to calculate the square root of the node parameters. According to his introduction, at all non-uniform distribution of experimental data points, the method gives a better parameterization than the results of the first two methods.Foley developed an adapting chord length parameterization method, which considers distance apart from the adjacent chord length between two points, but also the length of two adjacent chord length of the angle between two adjacent angles, to obtain good parameters results.Professor Caiming Zhang proposed ZCM parameterization method, the method is a holistic approach, the data points the parameters of the results of the accuracy of the merits of the second experiment showed that under normal circumstances the use of its structural parameters interpolation curve better than the effect of parameters to the heart or amendment chord length parameterization.This article presents a parameterization method based on minimal energy with the quadratic accuracy; the method uses local optimization of minimum energy curves bound to determine the node parameters, so the structure of the spline curve is smoothing. In some cases of distribution of data points, this method can get the best results.At last I select some representative samples of data, and then use the classical parameterization methods and energy model method to do experiments separately. According to experiment results, I proposed the direction of improvement for the next step.