Algorithm for Geometric

We show that the geometric Hermite interpolant can be easily calculated without solving a system of nonlinear equations. In addition we give geometric conditions for the existence and uniqueness of a solution to the interpolation problem. Finally we compare geometric Hermite interpolation with standard cubic Hermite interpolation. x1 Introduction Since parametric representations of curves are not unique, the approximation rates by splines can be signiicantly improved. This surprising fact was rst observed in 1] where a 6{th order cubic interpolation scheme for planar curves had been constructed. By now a number of results of this type have been obtained. Schaback 5] has achieved order 4 with quadratic splines. Degen 2] has attained order 8 with cubic rational splines. So far most of the research has focused on approximations using planar curves. For space curves HH ollig 3] has developed an interpolation scheme using cubic rational splines which is 6{th order accurate. Recently in 4] it has been shown that the performance of standard cubic Hermite interpolation can be improved by interpolating a third point. The resulting method achieves the optimal approximation order 5. We will show that the geometric Hermite interpolation can be performed without solving a system of nonlinear equations. In addition we will give geometric conditions for the existence and uniqueness of a solution. Finally we will compare geometric Hermite interpolation with standard cubic Hermite interpolation.