柱锥斜交相贯线解析性质分析与特殊点图解方法

Analytical Characteristics of Intersection Curve between a Cylinder and a Cone with Two Obliquely Intersected Axes and Graphical Method of Locating Special Points

轴线相交的圆柱和圆锥两立体相交时,一般情况下会产生两条相贯线。文章分析了在圆柱、圆锥正交和圆柱、圆锥斜交情况下,相贯线随圆柱半径变化而形成的不同形状和特殊点性质。进一步结合解析形式分析,推导了圆柱、圆锥轴线相交并产生左右两条相贯线时,两条相贯线上最里点的分布规律;相贯线形状与圆柱半径取值范围的精确对应关系;并给出了确定相贯线上最里点的辅助球半径公式。最后,文章依据以上结果提出了圆柱、圆锥斜交时相贯线上所有特殊点的图解方法。

In general situations,there will be two spatial intersection curves generated when a cylinder and a cone are intersected with each other.This paper analyzes the various cases of the intersection curves and their special point locations when the axes of the cylinder and the cone are obliquely or perpendicularly intersected.If the intersecting curves are separated as two branches on the left and right,the distribution equation of the innermost points on the two intersection curves is obtained through theoretical analysis.The paper also analyzes the exact geometric conditions for classifying the four forms of the intersection curves and their corresponding cylinder radius distributions.The auxiliary sphere radius corresponding to the inner most point is also obtained.Finally,a graphical method of locating all the special points on the intersection curve is proposed in the paper.