三次Pythagorean Hodograph 曲线在给定总弧长下的$G^2$拼接

Pythagorean-hodograph (PH)曲线因其在弧长和等距线计算方面的优势而被广泛应用于曲线建模中.本文讨论了在总弧长约束下的三次PH曲线$G^2$连续拼接问题.具体地说,给定两个端点和一个拼接点,构造两条三次PH曲线,使其在指定总弧长下插值两个端点,并且在连接点处是$G^2$连续的.这也可以看作是一个曲线延拓问题.根据三次PH曲线的弧长公式和$G^2$连续条件,最终将问题转化为了一个带有约束的极小值问题,同时我们给出了几个具体例子来说明该方法.

$G^{2}$ Blending of Cubic Pythagorean Hodograph Curves with Prescribed Total Arc Length

Pythagorean-hodograph (PH) curve is widely used in curve modeling because of its advantages in arc length and equidistant curve calculation. This paper discusses the $G^{2}$ continuous blending of cubic PH curves under total arc length constraint. Given three points including two end control points and a joint point, construct two cubic PH curves such that they interpolate the end control points and are $G^{2}$ continuous at joint point with prescribed total arc length. It can also be regarded as a curve extension problem. According to the arc length formula of cubic PH curve and the condition of $G^{2}$ blending, the problem is transformed into a constrained minimization problem. Several examples are served to illustrate our method.