Cubic Hermite Curve Refinement Optimization

Geometric Hermite interpolation is to find a curve which interpolates positions, tangent vectors, curvatures and other conditions at the end. It has widely application in geometric structure and engineering design. This paper presents two new Hermite interpolation methods for the reason that most existed methods may not consider the shape adjustment and the fairing of the interpolation curves. This paper includes three parts.In first chapter, we simply introduce provenience and development of the Computer Aided Geometric Design, summarize the meaning and current research situation in fairing of curves and surfaces and algorithms of curve interpolation.In second chapter, basic knowledge of curves is simply introduced. We mainly present parameter equations of curves, the basic formula of curve theory, the geometric meaning of curvatures and torsions, and the fairing problem of curves. We focus on discussing the energy optimization.In third chapter, we present a scheme for a class of cubic Hermite interpolation curves which is C1 continuous and has two parameters through bringing in a new knot. According to the factor of curvatures and torsions, we present three energy functions, and give the parameters formulas when the energy functions of curve are minimal. And it presents a characteristic of cubic Hermite interpolation curves.In forth chapter, we present another scheme for a class of cubic Hermite interpolation curves which is C1 continuous and has fours parameters through bringing in two new knots. we present two energy functions, then give the parameters formulas when the energy functions of curve are minimum. It also presents a characteristic of cubic Hermite interpolation curves, and analyzes the function of free parameters to modify the interpolation curves.In the paper, we also give some specific examples, these examples are drawn by Matlab according to the algorithms of paper. The methods are effective in generating smooth curves.