有理三次PH曲线的G^1 Hermite插值

G^1 Hermite Interpolation by Rational Pythagorean Hodograph Cubics

探讨了有理PH曲线的G1 Hermite插值问题,运用复数表达将问题转化为包含5个复代数方程的方程组,通过求解这个方程组,得到结论:当插值条件形成凸多边形时,插值问题有2个解,其中之一为多项式解;而当插值条件形成非凸多边形时,只有切方向满足一定条件时,插值问题才有一个解.而对于后一种情况,总可以通过加点的方式细分原逼近曲线,进而得到由两段有理三次PH曲线G1拼接而成的4组样条插值曲线.

The G^1 Hermite interpolation by Rational Pythagorean Hodograph Cubics is discussed. By means of complex representation the problem is transformed to a system of complex equations with 5 complex algebraic equations. By solving this system, the following results are obtained. There are 2 solutions（especially one is a polynomial solution）, when the conditions form a convex polygon. And with the concave polygon conditions, there is no solution unless the tangents satisfy some special conditions. But four solutions constituted with two G^1 PH rational cubic segments always can be got by means of adding points between the former extreme points.