Research On The Problem Of The Hermite Interpolation By Cubic B(?)zier Curves

A special kind of plane curve is called PH (Pythagorean Hodograph) curve which is introduced by Farouki and Sakkalis in 1990. Based on this Pottmamn proposed rational PH curve in 1995. The property of PH curves provides a number of advantages in applications than traditional polynomial parametric curves,such as, arc length and offset curves of PH curve can be expressed exact, offset curves in the field of industrial design e.g.3D.NC machining,design of automobile contour,the railway track,the robot motion control and so on have a wide range of applications. Because of these obvious advantages, about PH curve interpolation problem has been extensive and in-depth study.Hermite interpolation, the generation of smooth curve segments satisfying given end conditions, is a popular modeling method in computer-aided geometric design (CAGD). Rational B(?)zier curve is a common model in the system.Related to interpolating G2 Hermite data by a rational B(?)zier curve, as early as the 1980s, de Boor obtained an interpolation of high accuracy for second-order geometric contact by a planar cubic B(?)zier curve.The methods depend on the solution of a system of two simultaneous quadratic equations. Degen improved this method by introducing a rational cubic B(?)zier curve in 1993. D.J.Walton presented a G2 Hermite interpolation method by expand the four degrees of freedom base on G1 Hermite interpolation methods,but the solvability condition is complex.In this paper, based on the G1 Hermite interpolation by a cubic PH curve,we present a simple method for G1 Hermite interpolation by a rational cubic B(?)zier curve. The method has a well-understood geometrical meaning and the weights of the rational cubic B(?)zier curve are expressed by the curvature of the endpoints. The numerical examples show the effectiveness of the algorithm.