轴线相交的柱锥曲面交于平面曲线的理论探索

Theoretical Research on the Intersection Curve of Cylinder and Cone with Intersecting Axes

用解析几何的方法证明了工程图学中当柱锥曲面轴线相交并且满足一定的几何条件时其相贯线是平面曲线的结论。其过程是:分别建立柱、锥曲面方程,给出内切于球的两曲面立体的相贯线投影方程,进而将该投影方程转变为双曲线方程和渐近线方程的形式。分别讨论了3种柱锥曲面相贯线的投影方程,证明了相贯线的平面投影可以用两相交直线表示的结论。该结论验证了解析法与投影法的一致性,消除了学习者的疑虑。

The fact is proved using analytic geometry that in engineering graphics the intersection curve of a cylinder and a cone, which have two axes intersected, can be a plane curve under certain geometric conditions. The process is： forming two surface equations and then getting intersection curve＇s projection equation of a cylinder and a cone with a same inscribed sphere. Next, the equation is transformed into hyperbola equation and asymptote equation. Through studying three kinds of intersection curve＇s projection equations separately, a conclusion is obtained that intersection curve＇s projection can be two intersection lines. This conclusion is consistent with the descriptive geometry and thus solves the doubt of the learner.