Research On B-spline Curve And Surface Based On Modified Chord Length Parameterization Method

With the gradual improvement of the combination of industrial design and digitalization,free curve and surface are more and more widely used i n the field of reverse engineering.Interpolation approximation of data by B-spline curves and surfaces is a key technology in reverse engineering,among which the key content of B-spline curves and surfaces is to construct a reasonable data point parameterization method.The distribution of joint parameters on the parameter axis is related to the shape of the generated interpolation curve.Therefore,the smoothness of data points is of great significance to the study of data point parameterization methods.Based on the in-depth study of the existing classical data point parameterization methods,this paper proposes an improved chord length parameterization method.The main contents of this paper are as follows:(1)Research the theory of B-spline curve and surface.The relevant important properties of the B-spline basis function are summarized,and then based on the² continuous cubic B-spline curve,the interpolation conditions are given from the closed-loop B-spline curve and the open-loop B-spline curve.So as to obtain the control points of the interpolation curve by solving the parameter equation to determine the interpolation curve;from the curve to the surface,the basis function and construction method of the B-spline surface are explained,and finally several widely used classical data point parameterization methods are introduced.(2)An improved parameterization method of chord length is established.This method constructs a direction vector based on two consec utive data points at the beginning,and then uses the angle relationship between the chord length to construct the first interpolated arc,and generates the remaining interpolated arcs through several adaptive iterative processes,using the interpolated ar c The length replaces the chord length to calculate the global node parameters,and at the same time,the generated interpolated arc is controlled by the chord length and angle two variables,so that the obtained modified chord length is relatively close to the actual interpolated curve arc length,achieving a more ideal parameterization effect.(3)A smoothness criterion of B-spline curve is proposed.Aiming at the smoothness of the curve interpolated at the same set of data points,a function of the radius of curvature changing with the cumulative curve length is established,which intuitively reflects the change of the curvature of the interpolation curve with the cumulative curve length.In addition,in order to evaluate the fluctuation of the curvature of the interpolation curve,the curve smoothness indexis proposed by the calculation method of separating the interpolation curve and so on.In addition,the deviation error of evaluation curve swing is introduced.(4)Several typical data point distributions are selected for curve interpolation test,and the cubic B-spline interpolation curve constructed by the improved chord length parameterization method,chord length parameterization method,uniform parameterization method,and centripetal parame terization method respectively analyze and compare.Then combine two point clouds examples of head and feet,and use Gaussian curvature to analyze different parameterization methods to construct B-spline interpolation surface smoothness.Based on the chord length parameterization method,this paper proposes an improved chord length parameterization method.The chord length is corrected by constructing an interpolation arc,and the curve smoothness index is selected as the criterion for the overall smoothnes s of the curve.The proposed method.It is also suitable for point cloud surface interpolation,effectively improving the smoothness of the interpolation curve and surface in reverse engineering design,which is beneficial to its application and promotion.