Construction of Local Adjustable C2 Parametic Quartic Interpolation Curve

To construct interpolation curve and surface with quartic polynomials as basic functions is an effective approach. It makes the curve and surface constructed have the advantages such as having simple construction and being easy to compute. In applications, C2 curves and surfaces satisfy the requirements of the most applications. In shape design, the freedom degrees can be used to increase the flexibility of design and construction to control the shapes of the curve and surface, therefore making the shape of the curve and surface more desirable. The authors discuss the problem of constructing local adjustment C2 quartic spline interpolation curve. The method for determining the freedom degrees with local method is presented. The tangent vector at every data point is identified through the localized quadratic spline function at first, and the tangent vectors and data points determine approximately the shape of the quartic spline curve. Then the freedom degrees are determined by minimizing the change rate of the spline curve. For the imperfect part of the spline curve, the corresponding tangent vectors are modified by the following way. A desirable moving vector is defined which makes the curve have better shape if it varies along the moving vector. An objective function is defined by the integral of the squared vector product of the moving vector and the tangent vector of the curve. The imperfect part of the curve is modified by minimizing the objective function. The comparisons of the new method with other methods and the examples of locally adjusting the curve by minimizing the vector product are also included.