Research On The Problem Of Geometric Hermite Interpolation For Spatial Curves

The interpolation of curves is a basic class of problems in CAGD. For parametric curves, in practical applications, it is required not only to interpolate a sequence of points, but also to interpolate multi-order derivative of these points. In general, the interpolating curve with geometric continuity can also get good results. To solve these problems, a new interpolation method is proposed-geometric Hermite interpolation (GHI). GHI has a wide range of applications in the geometric modeling system. There are already many scholars who have studied the GHI problems, but they just studied planar curves, rarely studied space curves.In this thesis, we first describe several methods of GHI problem of planar curves:(1)Find a solution of a system of linear equations to solve the GHI problem of planar curves.(2)Introduce classical method using Bezier curves by de Boor.(3)Describe the GHI problem of planar curves using B-spline method.Furthermore, we constructed a parametric quartic B-spline curve which interpolates a given special curve, whose position, tangent direction, curvature vector and torsion are prescribed at each endpoint. Moreover, it is shown that under appropriate assumptions the interpolation exists locally with three free variables and the 5th order accuracy.At last, we constructed a parametric cubic B-spline curve with three free variables by different sequence of nodes to interpolate space curve. It is proved that the solution of the interpolation problem exists locally, and gets the same results as the quartic B-spline curve. Besides, the interpolation is also 5th order accurate.