关于非退化二阶曲线上的射影变换及对合

On Projective Transformation and Involution of Nondegenerate Second-order Curve

将文献 [1]中的结果推广到非退化二阶曲线的情况 ,得出非退化二阶曲线到自身的双射成为射影变换及对合的充要条件 :非退化二阶曲线Γ到自身的双射 φ为射影变换的充要条件是 φ下的全部交错点共线 ;非退化二阶曲线Γ到自身的一个双射 φ为对合的充要条件是 φ下的全部交错点及全部对应三点形对应边的交点共线。然后 ,再将非退化二阶曲线到自身的双射为对合的充要条件推广到退化二阶曲线两相异点列的情形。

In this paper,a theorem in paper on perspective problems on Pappus Theorem and Pascal Theorem is generalized to the case for nondegenerate second order curve and a necessary and sufficient condition for nondegenerate second order curve from biproject by itself to projective transformation and involution is obtained,namely the biproject φ by itself for nondegenerate second order curve Γ is to be a projective transformation if and only if all alternate points under φ are on one line.At the same time ,the biproject φ by itself for nondegenerate second order curve Γ is to be involution if and only if all alternate points and all the points of intersection of correspondence sides of three point form are on one line.Then,the latter necessary and sufficient condition is generalized to the case for the two different point ranges of degenerate second order curve.