二射影对应面束与二次曲面

The two Projective Pencils of Planes and Quadric

利用正投影法首次系统说明了固有线为底的二射影对应面束,如果平移或旋转其中之一,可以分别形成各种二次柱面、二次锥面、单叶双曲面和双曲抛物面。二射影对应面束,如果选择合适的截平面,可以得到两对对应合同线束。合同线束是否同向共面是二次曲面存在圆截口的充分必要条件。

With the help of the orthographic projection method, the two projective pencils of planes with solid lines as the base are systematically expressed for the first time. When moving or revolving one of them, various types of quadric cylinders, quadric cones, unparted hyperboloids and hyperbolic paraboloids can be produced. When an appropriate cutting plane is selected, two pairs of corresponding contract harnesses can be obtained. Whether or not the harnesses are coplanar is the sufficient and necessary condition for the existence of circular section in quadric surface.