Parameterization Based On Maximom Curvature Minimization Model

With the rapid development of Computer Aided Design (CAD), the modern industrial design has been gradually inseparable with the application and theoretical support of Computer Aided Geometric Design Technology (CAGD). As an important aspect of Computer Aided Geometric Design, the parameter interpolation curve has also been widely used. In practical applications, interpolation curve is often required to achieve continuous and smooth. Before constructing the parametric interpolation curve, parameters of nodes need to be determined in advance, that is data point parameterization. Therefore, in order to achieve the purpose, not only need an appropriate interpolation method, but also a good method for determining parametric knots, which can reflect the nature of the data points suggested.From the theoretical side, the best effect is the arc length parameterization. However, it’s very difficult to achieve. After decades of development, in the practical application there are various data point parameterization methods, but they still has their drawbacks and limitations. For example, the uniform parameterization method is simple to understand, but is not suited to the data points distribution which distances between consecutive data points vary greatly. Accumulated chord length parameterization method is considered to be closest to arc length parameterization, however, it’s most appropriate only in the case that the curve is a straight line. Parameterization method based on optimization technology has been rapidly developed in recent years, their effect is good but they are not easy to achieve.This paper analyzes the classic data point parameterization algorithms currently published, on this basis, presents a new research ideas-parameterization method based on the maximum curvature minimization model. The method designs a new fairing constraint model, interpolating every three data points to construct a quadratic Lagrange interpolation curve, then using the quadratic curve curvature model to find the maximum curvature of the curve. Through minimizing the maximum we can get local parameter values. Finally transform the local parameters into the global parameters through a unified field parameter transformation model. The method reduces the curvature variation by controlling the maximum curvature of each local curve to stay minimum. In comparison of existing fairing constraint model, using the maximum curvature minimization model to construct the objective function, instead of using the approximate model, can obtain exact solution in use of the formula method. So this method can obtain smooth interpolation curve.When the knots determined by the new method are used to construct interpolation curve, the constructed curve have good precision. We also give some comparisons of the new method with existing methods, and our method can perform better in interpolation error, and the interpolated curve is more smooth.