Constructing Planar $C^1$ Cubic Hermite Interpolation Curves Via Approximate Energy Minimization

Abstract:

      The methods for constructing planar $C^1$ cubic Hermite interpolation curves via approximate energy minimization are studied. The main purpose of the proposed methods are to obtain the optimal tangent vectors of the $C^1$ cubic Hermite interpolation curves. By minimizing the appropriate approximate functions of the strain energy, the curvature variation energy and the combined energy, the linear equation systems for solving the optimal tangent vectors are obtained. It is found that there is no unique solution for the minimization of approximate curvature variation energy minimization, while there is unique solution for the minimization of approximate strain energy and the minimization of approximate combination energy because the coefficient matrix of the equation system is strictly diagonally dominant. Some examples are provided to illustrate the effectiveness of the proposed method in constructing planar $C^1$ cubic Hermite interpolation curves.