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六次PH 曲线G2 Hermite 插值

  

  • 出版日期:2016-04-28 发布日期:2016-05-20

G2 Hermite Interpolation by Pythagorean Hodograph of Degree Six

  • Online:2016-04-28 Published:2016-05-20

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摘要： 以其在弧长计算与等距线表示上的优势，PH 曲线成为近年来计算机辅助几何设计
研究的焦点问题之一。为此讨论了六次PH 曲线的G2 Hermite 插值问题。在指定自由参数下，对
两类六次PH 曲线分别进行复分析曲线求解，得到满足G2 插值条件的六次PH 曲线和控制顶点。
通过弧长、能量积分、绝对旋转数的衡量，选取较好的插值曲线。进一步，讨论了用六次PH 曲
线G2 Hermite 插值逼近90°和67°圆弧的问题。在同一个自由参数下，选择插值最好的曲线，可
实现六次C1 Hermite 插值逼近圆弧的效果，且逼近90°圆弧时，优于五次G2 Hermite 插值逼近的
PH 曲线，而逼近67°圆弧时，与最好的五次PH 曲线达到的效果几乎相同。

关键词: PH 曲线, G2 Hermite 插值, 圆弧

Abstract: By the advantages of computing arcs and representing offsets, study of phythagorean
hodograph curves is one of the hot topics in recent years. In this paper, G2 Hermite interpolation by
sextic PH curves is studied. Sextic PH curves can be classified into two types and the interpolation
problem can be resolved to get the control points with some free parameter in complex representation.
With the analysis of arc-length, bending energy and absolute rotation number, the better interpolation
curves are selected. Moreover, the sextic PH G2 Hermite interpolation is applied to approximate the 90°
and 67° arcs. The best approximating curves can solve C1 Hermite interpolation by the PH sextics. And
the best curves’ performance is better than the quintic G2 Hermite interpolation curves when
approximating the 90° arc, and is almost same as the latter’s best curve when approximating the 67° arc.

Key words: PH curve, G2 Hermite interpolation, arc