Construction Method of Interpolation Curve with Given Polynomial Reconstruction Precision and Continuity Order

This paper aims at combining the function interpolation in numerical calculation and the parametric curve interpolation in shape design, and provide the general method of constructing the function interpolation with given polynomial reconstruction precision and the parametric curve interpolation with given continuity order. The method takes the cardinal form of Hermite interpolation as a bridge. Firstly, we deduce the expression of the derivative vectors in the cardinal form with the goal of making the function interpolation reaches the given precision. And the coefficients in the derivative vectors are obtained by solving linear equations. Secondly, we substitute the derivative vectors into the cardinal form. Rearranging it in accordance with the interpolation data points, we can obtain the expression of the interpolation basis functions. Lastly, we provide the interpolation curve with the form of the linear combination of the interpolation points and the interpolation basis. The interpolation curve construction method given here does not require one to reverse the control points. Numerical experiment results show that the shape of the resulting curve can be fixed and can do partial adjustment. The reconstruction precision of the degree 2n+1 Hermite interpolation polynomial is generally more than n.